HaskellForMaths 0.4.3 → 0.4.4

raw patch · 24 files changed

+1302/−95 lines, 24 filesPVP: major bump suggested

API removals or changes: PVP suggests a major version bump

API changes (from Hackage documentation)

Files

• HaskellForMaths.cabal +7/−3
• Math/Algebra/Group/PermutationGroup.hs +2/−1
• Math/Algebras/GroupAlgebra.hs +35/−26
• Math/Algebras/Structures.hs +1/−1
• Math/Algebras/VectorSpace.hs +10/−5
• Math/Combinatorics/CombinatorialHopfAlgebra.hs +660/−0
• Math/Combinatorics/Design.hs +1/−1
• Math/Combinatorics/GraphAuts.hs +6/−1
• Math/Combinatorics/IncidenceAlgebra.hs +15/−12
• Math/Combinatorics/Poset.hs +46/−11
• Math/Core/Utils.hs +75/−4
• Math/NumberTheory/Prime.hs +2/−1
• Math/NumberTheory/QuadraticField.hs +116/−0
• Math/Test/TAlgebras/TGroupAlgebra.hs +13/−13
• Math/Test/TAlgebras/TOctonions.hs +2/−1
• Math/Test/TAlgebras/TStructures.hs +12/−2
• Math/Test/TAlgebras/TTensorProduct.hs +3/−2
• Math/Test/TAlgebras/TVectorSpace.hs +9/−8
• Math/Test/TCombinatorics/TCombinatorialHopfAlgebra.hs +178/−0
• Math/Test/TCombinatorics/TDigraph.hs +1/−0
• Math/Test/TCore/TUtils.hs +50/−0
• Math/Test/TNumberTheory/TQuadraticField.hs +47/−0
• Math/Test/TPermutationGroup.hs +2/−2
• Math/Test/TestAll.hs +9/−1

@@ -1,5 +1,5 @@    Name:                HaskellForMaths
-   Version:             0.4.3
+   Version:             0.4.4
    Category:            Math
    Description:         A library of maths code in the areas of combinatorics, group theory, commutative algebra, and non-commutative algebra. The library is mainly intended as an educational resource, but does have efficient implementations of several fundamental algorithms.
    Synopsis:            Combinatorics, group theory, commutative algebra, non-commutative algebra
@@ -29,16 +29,19 @@         Math/Test/TAlgebras/TOctonions.hs
         Math/Test/TAlgebras/TMatrix.hs
         Math/Test/TAlgebras/TGroupAlgebra.hs
-        Math/Test/TCombinatorics/TPoset.hs
+        Math/Test/TCombinatorics/TCombinatorialHopfAlgebra.hs
         Math/Test/TCombinatorics/TDigraph.hs
         Math/Test/TCombinatorics/TFiniteGeometry.hs
         Math/Test/TCombinatorics/TGraphAuts.hs
         Math/Test/TCombinatorics/TIncidenceAlgebra.hs
         Math/Test/TCombinatorics/TMatroid.hs
+        Math/Test/TCombinatorics/TPoset.hs
         Math/Test/TCommutativeAlgebra/TPolynomial.hs
         Math/Test/TCommutativeAlgebra/TGroebnerBasis.hs
         Math/Test/TCore/TField.hs
+        Math/Test/TCore/TUtils.hs
         Math/Test/TNumberTheory/TPrimeFactor.hs
+        Math/Test/TNumberTheory/TQuadraticField.hs
         Math/Test/TProjects/TMiniquaternionGeometry.hs
 
    Library
@@ -55,6 +58,7 @@         Math.Algebras.Octonions, Math.Algebras.Quaternions,
         Math.Algebras.Structures, Math.Algebras.TensorAlgebra,
         Math.Algebras.TensorProduct, Math.Algebras.VectorSpace,
+        Math.Combinatorics.CombinatorialHopfAlgebra,
         Math.Combinatorics.Graph, Math.Combinatorics.GraphAuts, Math.Combinatorics.StronglyRegularGraph,
         Math.Combinatorics.Design, Math.Combinatorics.FiniteGeometry, Math.Combinatorics.Hypergraph,
         Math.Combinatorics.LatinSquares, Math.Combinatorics.Poset, Math.Combinatorics.IncidenceAlgebra,
@@ -62,7 +66,7 @@         Math.Common.IntegerAsType, Math.Common.ListSet,
         Math.CommutativeAlgebra.Polynomial, Math.CommutativeAlgebra.GroebnerBasis,
         Math.Core.Utils, Math.Core.Field,
-        Math.NumberTheory.Prime, Math.NumberTheory.Factor,
+        Math.NumberTheory.Prime, Math.NumberTheory.Factor, Math.NumberTheory.QuadraticField,
         Math.Projects.RootSystem,
         Math.Projects.Rubik, Math.Projects.MiniquaternionGeometry,
         Math.Projects.ChevalleyGroup.Classical, Math.Projects.ChevalleyGroup.Exceptional,

@@ -81,7 +81,8 @@     cycleOf' y ys = let y' = y .^ g in if y' == x then reverse (y:ys) else cycleOf' y' (y:ys)
 
 instance (Ord a, Show a) => Show (Permutation a) where
-    show g = show (toCycles g)
+    show g | g == 1 = "1"
+           | otherwise = show (toCycles g)
 
 parity g = let cs = toCycles g in (length (concat cs) - length cs) `mod` 2
 -- parity' g = length (filter (even . length) $ toCycles g) `mod` 2

@@ -10,61 +10,71 @@ -- would be entered as @p [[1,2,3],[4,5]]@, and displayed as [[1,2,3],[4,5]]. -- -- Given a field K and group G, the group algebra KG is the free K-vector space over the elements of G.--- Elements of the group algebra consists of arbitrary K-linear combinations of elements of G.+-- Elements of the group algebra consist of arbitrary K-linear combinations of elements of G. -- For example, @p [[1,2,3]] + 2 * p [[1,2],[3,4]]@-module Math.Algebras.GroupAlgebra where+module Math.Algebras.GroupAlgebra (GroupAlgebra, p) where  import Math.Core.Field-import Math.Core.Utils+import Math.Core.Utils hiding (elts)  import Math.Algebras.VectorSpace import Math.Algebras.TensorProduct import Math.Algebras.Structures  import Math.Algebra.Group.PermutationGroup hiding (p, action)-import qualified Math.Algebra.Group.PermutationGroup as P+-- import qualified Math.Algebra.Group.PermutationGroup as P -import Math.Algebra.LinearAlgebra hiding (inverse, (*>) )+import Math.Algebra.LinearAlgebra (solveLinearSystem) -- hiding (inverse, (*>) )  import Math.CommutativeAlgebra.Polynomial import Math.CommutativeAlgebra.GroebnerBasis  +type GroupAlgebra k = Vect k (Permutation Int)++instance (Eq k, Num k) => Algebra k (Permutation Int) where+    unit x = x *> return 1+    mult = nf . fmap (\(g,h) -> g*h)++{- instance Mon (Permutation Int) where     munit = 1     mmult = (*) -type GroupAlgebra k = Vect k (Permutation Int)- -- Monoid Algebra instance instance (Eq k, Num k) => Algebra k (Permutation Int) where-    unit 0 = zero -- V []-    unit x = V [(munit,x)]-    mult = nf . fmap (\(a,b) -> a `mmult` b)+    unit x = x *> return munit+    mult = nf . fmap (\(g,h) -> g `mmult` h)+-}  -- Set Coalgebra instance -- instance SetCoalgebra (Permutation Int) where {}  instance (Eq k, Num k) => Coalgebra k (Permutation Int) where-    counit (V ts) = sum [x | (m,x) <- ts] -- trace-    comult = fmap (\m -> (m,m)) -- diagonal+    -- counit (V ts) = sum [x | (g,x) <- ts] -- trace+    counit = unwrap . linear counit' where counit' g = 1 -- trace+    comult = fmap (\g -> (g,g))                          -- diagonal  instance (Eq k, Num k) => Bialgebra k (Permutation Int) where {} -- should check that the algebra and coalgebra structures are compatible  instance (Eq k, Num k) => HopfAlgebra k (Permutation Int) where-    antipode (V ts) = nf $ V [(g^-1,x) | (g,x) <- ts]+    antipode = nf . fmap inverse+    -- antipode (V ts) = nf $ V [(g^-1,x) | (g,x) <- ts]  -- |Construct a permutation, as an element of the group algebra, from a list of cycles. -- For example, @p [[1,2],[3,4,5]]@ constructs the permutation (1 2)(3 4 5), which is displayed -- as [[1,2],[3,4,5]]. p :: [[Int]] -> GroupAlgebra Q-p cs = return $ P.p cs+p = return . fromCycles   instance (Eq k, Num k) => Module k (Permutation Int) Int where     action = nf . fmap (\(g,x) -> x .^ g) +instance (Eq k, Num k) => Module k (Permutation Int) [Int] where+    action = nf . fmap (\(g,xs) -> xs -^ g)+ -- use *. instead -- r *> m = action (r `te` m) @@ -73,7 +83,7 @@ -- Find the inverse of a group algebra element using Groebner basis techniques -- This is overkill, but it was what I had to hand at first inv x@(V ts) =-    let gs = P.elts $ map fst $ terms x -- all elements in the group generated by the terms+    let gs = elts $ map fst $ terms x -- all elements in the group generated by the terms         cs = map (glexvar . X) gs         x' = V $ map (\(g,c) -> (g, unit c)) ts         one = x' * (V $ zip gs cs)@@ -86,27 +96,26 @@        -- sum [-c *> p g | V [ (Glex (M 1 [(X g, 1)]), 1), (Glex (M 0 []), c) ] <- solution]        -- should extract the solution into a group algebra element, but having trouble getting types right --- The following code can be made to work over an arbitrary field by uncommenting the commented code+-- The following code can be made to work over an arbitrary field by using ScopedTypeVariables and var instead of glexvar. -- However, we should then probably also change the signature of p to p :: Fractional k => [[Int]] -> GroupAlgebra k--- instance Fractional k => HasInverses (GroupAlgebra k) where  -- |Note that the inverse of a group algebra element can only be efficiently calculated -- if the group generated by the non-zero terms is very small (eg \<100 elements). instance HasInverses (GroupAlgebra Q) where     inverse x@(V ts) =-        let gs = P.elts $ map fst $ terms x -- all elements in the group generated by the terms-            -- cs = map (var . X) gs :: [Vect k (Glex (X (Permutation Int)))]-            cs = map (glexvar . X) gs-            x' = V $ map (\(g,c) -> (g, unit c)) ts-            one = x' * (V $ zip gs cs)-            m = [ [coeff (mvar (X j)) c | j <- gs] | i <- gs, let c = coeff i one]-            b = 1 : replicate (length gs - 1) 0-        in case solveLinearSystem m b of+        let gs = elts $ map fst ts -- all elements in the group generated by the terms+            n = length gs+            y = V $ zip gs $ map (glexvar . X) [1..n] -- x1*1+x2*g2+...+xn*gn+            x' = V $ map (\(g,c) -> (g, unit c)) ts -- lift the coefficients in x into the polynomial algebra+            one = x' * y+            m = [ [coeff (mvar (X j)) c | j <- [1..n]] | i <- gs, let c = coeff i one] -- matrix of the linear system+            b = 1 : replicate (n-1) 0+        in case solveLinearSystem m b of -- find v such that m v == b - ie find the values of x1, x2, ... xn             Just v -> nf $ V $ zip gs v             Nothing -> error "GroupAlgebra.inverse: not invertible"  maybeInverse x@(V ts) =-    let gs = P.elts $ map fst $ terms x -- all elements in the group generated by the terms+    let gs = elts $ map fst $ terms x -- all elements in the group generated by the terms         cs = map (glexvar . X) gs         x' = V $ map (\(g,c) -> (g, unit c)) ts         one = x' * (V $ zip gs cs)

@@ -45,7 +45,7 @@  instance (Eq k, Num k, Eq b, Ord b, Show b, Algebra k b) => Num (Vect k b) where     x+y = x <+> y-    negate x = neg x+    negate x = negatev x     -- negate (V ts) = V $ map (\(b,x) -> (b, negate x)) ts     x*y = mult (x `te` y)     fromInteger n = unit (fromInteger n)

@@ -8,8 +8,6 @@ import qualified Data.List as L import qualified Data.Set as S -- only needed for toSet --- toSet = S.toList . S.fromList- infixr 7 *> infixl 7 <* infixl 6 <+>, <->@@ -42,8 +40,12 @@  terms (V ts) = ts +-- |Return the coefficient of the specified basis element in a vector coeff b v = sum [k | (b',k) <- terms v, b' == b] +-- |Remove the term for a specified basis element from a vector+removeTerm b v = v <-> coeff b v *> return b+ -- Deprecated zero = V [] @@ -71,13 +73,16 @@ sumv :: (Ord b, Eq k, Num k) => [Vect k b] -> Vect k b sumv = foldl (<+>) zerov --- |Negation of vector-neg :: (Eq k, Num k) => Vect k b -> Vect k b+-- Deprecated neg (V ts) = V $ map (\(b,x) -> (b,-x)) ts +-- |Negation of a vector+negatev :: (Eq k, Num k) => Vect k b -> Vect k b+negatev (V ts) = V $ map (\(b,x) -> (b,-x)) ts+ -- |Subtraction of vectors (<->) :: (Ord b, Eq k, Num k) => Vect k b -> Vect k b -> Vect k b-(<->) u v = u <+> neg v+(<->) u v = u <+> negatev v  -- |Scalar multiplication (on the left) smultL :: (Eq k, Num k) => k -> Vect k b -> Vect k b

@@ -0,0 +1,660 @@+-- Copyright (c) 2012, David Amos. All rights reserved.++{-# LANGUAGE MultiParamTypeClasses, FlexibleInstances, NoMonomorphismRestriction, ScopedTypeVariables, DeriveFunctor #-}++-- |A module defining the following Combinatorial Hopf Algebras, together with coalgebra or Hopf algebra morphisms between them:+--+-- * SSym, the Malvenuto-Reutnenauer Hopf algebra of permutations+--+-- * YSym, the (dual of the) Loday-Ronco Hopf algebra of binary trees+--+-- * QSym, the Hopf algebra of quasi-symmetric functions (having a basis indexed by compositions)+--+-- * Sh, the Shuffle Hopf algebra+module Math.Combinatorics.CombinatorialHopfAlgebra where++-- Sources:++-- Structure of the Malvenuto-Reutenauer Hopf algebra of permutations+-- Marcelo Aguiar and Frank Sottile+-- http://www.math.tamu.edu/~sottile/research/pdf/SSym.pdf++-- Structure of the Loday-Ronco Hopf algebra of trees+-- Marcelo Aguiar and Frank Sottile+-- http://www.math.tamu.edu/~sottile/research/pdf/Loday.pdf++-- Hopf Structures on the Multiplihedra+-- Stefan Forcey, Aaron Lauve and Frank Sottile+-- http://www.math.tamu.edu/~sottile/research/pdf/MSym.pdf+++import Data.List as L+import Data.Maybe (fromJust)+import qualified Data.Set as S++import Math.Core.Field+import Math.Core.Utils++import Math.Algebras.VectorSpace hiding (E)+import Math.Algebras.TensorProduct+import Math.Algebras.Structures++import Math.Combinatorics.Poset++-- import Math.Algebra.Group.PermutationGroup+import Math.CommutativeAlgebra.Polynomial+++-- SHUFFLE ALGEBRA+-- This is just the tensor algebra, but with shuffle product (and deconcatenation coproduct)++-- |A basis for the shuffle algebra. As a vector space, the shuffle algebra is identical to the tensor algebra.+-- However, we consider a different algebra structure, based on the shuffle product. Together with the+-- deconcatenation coproduct, this leads to a Hopf algebra structure.+newtype Shuffle a = Sh [a] deriving (Eq,Ord,Show)++-- |Construct a basis element of the shuffle algebra+sh :: [a] -> Vect Q (Shuffle a)+sh = return . Sh++shuffles (x:xs) (y:ys) = map (x:) (shuffles xs (y:ys)) ++ map (y:) (shuffles (x:xs) ys)+shuffles xs [] = [xs]+shuffles [] ys = [ys]++instance (Eq k, Num k, Ord a) => Algebra k (Shuffle a) where+    unit x = x *> return (Sh [])+    mult = linear mult'+        where mult' (Sh xs, Sh ys) = sumv [return (Sh zs) | zs <- shuffles xs ys]++deconcatenations xs = zip (inits xs) (tails xs)++instance (Eq k, Num k, Ord a) => Coalgebra k (Shuffle a) where+    counit = unwrap . linear counit' where counit' (Sh xs) = if null xs then 1 else 0+    comult = linear comult'+        where comult' (Sh xs) = sumv [return (Sh us, Sh vs) | (us, vs) <- deconcatenations xs]++instance (Eq k, Num k, Ord a) => Bialgebra k (Shuffle a) where {}++instance (Eq k, Num k, Ord a) => HopfAlgebra k (Shuffle a) where+    antipode = linear (\(Sh xs) -> (-1)^length xs *> return (Sh (reverse xs)))+++-- SSYM: PERMUTATIONS+-- (This is permutations considered as combinatorial objects rather than as algebraic objects)++-- Permutations with shifted shuffle product+-- This is the Malvenuto-Reutenauer Hopf algebra of permutations, SSym.+-- It is neither commutative nor co-commutative++-- ssymF xs is the fundamental basis F_xs (Aguiar and Sottile)++-- |The fundamental basis for the Malvenuto-Reutenauer Hopf algebra of permutations, SSym.+newtype SSymF = SSymF [Int] deriving (Eq)++instance Ord SSymF where+    compare (SSymF xs) (SSymF ys) = compare (length xs, xs) (length ys, ys)++instance Show SSymF where+    show (SSymF xs) = "F " ++ show xs++-- |Construct a fundamental basis element in SSym.+-- The list of ints must be a permutation of [1..n], eg [1,2], [3,4,2,1].+ssymF :: [Int] -> Vect Q SSymF+ssymF xs | L.sort xs == [1..n] = return (SSymF xs)+         | otherwise = error "Not a permutation of [1..n]"+         where n = length xs++-- so this is a candidate mult. It is associative and SSymF [] is obviously a left and right identity+-- (need quickcheck properties to prove that)+shiftedConcat (SSymF xs) (SSymF ys) = let k = length xs in SSymF (xs ++ map (+k) ys)++prop_Associative f (x,y,z) = f x (f y z) == f (f x y) z++-- > quickCheck (prop_Associative shiftedConcat)+-- +++ OK, passed 100 tests.+++instance (Eq k, Num k) => Algebra k SSymF where+    unit x = x *> return (SSymF [])+    mult = linear mult'+        where mult' (SSymF xs, SSymF ys) =+                  let k = length xs+                  in sumv [return (SSymF zs) | zs <- shuffles xs (map (+k) ys)]+++-- standard permutation, also called flattening, eg [6,2,5] -> [3,1,2]+flatten xs = let mapping = zip (L.sort xs) [1..]+        in [y | x <- xs, let Just y = lookup x mapping] ++instance (Eq k, Num k) => Coalgebra k SSymF where+    counit = unwrap . linear counit' where counit' (SSymF xs) = if null xs then 1 else 0+    comult = linear comult'+        where comult' (SSymF xs) = sumv [return (SSymF (st us), SSymF (st vs)) | (us, vs) <- deconcatenations xs]+              st = flatten++instance (Eq k, Num k) => Bialgebra k SSymF where {}++instance (Eq k, Num k) => HopfAlgebra k SSymF where+    antipode = linear antipode'+        where antipode' (SSymF []) = return (SSymF [])+              antipode' x@(SSymF xs) = (negatev . mult . (id `tf` antipode) . removeTerm (SSymF [],x) . comult . return) x+        -- This expression for antipode is derived from mult . (id `tf` antipode) . comult == unit . counit+        -- It's possible because this is a graded, connected Hopf algebra. (connected means the counit is projection onto the grade 0 part)+-- It would be nicer to have an explicit expression for antipode.+{-+instance (Eq k, Num k) => HopfAlgebra k SSymF where+    antipode = linear antipode'+        where antipode' (SSymF v) = sumv [lambda v w *> return (SSymF w) | w <- L.permutations v]+              lambda v w = length [s | s <- powerset [1..n-1],  odd (length s), descentSet (w^-1 * v_s) `isSubset` s]+                         - length [s | s <- powerset [1..n-1],  even (length s), descentSet (w^-1 * v_s) `isSubset` s]+-}++-- |An alternative \"monomial\" basis for the Malvenuto-Reutenauer Hopf algebra of permutations, SSym.+-- This basis is related to the fundamental basis by Mobius inversion in the poset of permutations with the weak order.+newtype SSymM = SSymM [Int] deriving (Eq)++instance Ord SSymM where+    compare (SSymM xs) (SSymM ys) = compare (length xs, xs) (length ys, ys)++instance Show SSymM where+    show (SSymM xs) = "M " ++ show xs++-- |Construct a monomial basis element in SSym.+-- The list of ints must be a permutation of [1..n], eg [1,2], [3,4,2,1].+ssymM :: [Int] -> Vect Q SSymM+ssymM xs | L.sort xs == [1..n] = return (SSymM xs)+         | otherwise = error "Not a permutation of [1..n]"+         where n = length xs++inversions xs = let ixs = zip [1..] xs+                in [(i,j) | ((i,xi),(j,xj)) <- pairs ixs, xi > xj]++weakOrder xs ys = inversions xs `isSubsetAsc` inversions ys++mu (set,po) x y = mu' x y where+    mu' x y | x == y    = 1+            | po x y    = negate $ sum [mu' x z | z <- set, po x z, po z y, z /= y]+            | otherwise = 0++-- |Convert an element of SSym represented in the monomial basis to the fundamental basis+toSSymF :: (Eq k, Num k) => Vect k SSymM -> Vect k SSymF+toSSymF = linear toSSymF'+    where toSSymF' (SSymM u) = sumv [mu (set,po) u v *> return (SSymF v) | v <- set, po u v]+              where set = L.permutations u+                    po = weakOrder++-- |Convert an element of SSym represented in the fundamental basis to the monomial basis+toSSymM :: (Eq k, Num k) => Vect k SSymF -> Vect k SSymM+toSSymM = linear toSSymM'+    where toSSymM' (SSymF u) = sumv [return (SSymM v) | v <- set, po u v]+              where set = L.permutations u+                    po = weakOrder++-- (p,q)-shuffles: permutations of [1..p+q] having at most one descent, at position p+-- denoted S^{(p,q)} in Aguiar&Sottile+-- (Grassmannian permutations?)+-- pqShuffles p q = [u++v | u <- combinationsOf p [1..n], let v = [1..n] `diffAsc` u] where n = p+q++-- The inverse of a (p,q)-shuffle.+-- The special form of (p,q)-shuffles makes an O(n) algorithm possible+-- pqInverse :: Int -> Int -> [Int] -> [Int]+{-+-- incorrect+pqInverse p q xs = pqInverse' [1..p] [p+1..p+q] xs+    where pqInverse' (l:ls) (r:rs) (x:xs) =+              if x <= p then l : pqInverse' ls (r:rs) xs else r : pqInverse' (l:ls) rs xs+          pqInverse' ls rs _ = ls ++ rs -- one of them is null+-}+-- pqInverseShuffles p q = shuffles [1..p] [p+1..p+q]+++instance (Eq k, Num k) => Algebra k SSymM where+    unit x = x *> return (SSymM [])+    mult = toSSymM . mult . (toSSymF `tf` toSSymF)++{-+mult2 = linear mult'+    where mult' (SSymM u, SSymM v) = sumv [alpha u v w *> return (SSymM w) | w <- L.permutations [1..p+q] ]+                                     where p = length u; q = length v++alpha u v w = length [z | z <- pqInverseShuffles p q, let uv = shiftedConcat u v,+                          uv * z `weakOrder` w, u and v are maximal, ie no transposition of adjacents in either also works]+    where p = length u+          q = length v+-- so we need to define (*) for permutations in row form+-}++instance (Eq k, Num k) => Coalgebra k SSymM where+    counit = unwrap . linear counit' where counit' (SSymM xs) = if null xs then 1 else 0+    -- comult = (toSSymM `tf` toSSymM) . comult . toSSymF+    comult = linear comult'+        where comult' (SSymM xs) = sumv [return (SSymM (flatten ys), SSymM (flatten zs))+                                        | (ys,zs) <- deconcatenations xs,+                                          minimum (infinity:ys) > maximum (0:zs)] -- ie deconcatenations at a global descent+              infinity = maxBound :: Int++instance (Eq k, Num k) => Bialgebra k SSymM where {}++instance (Eq k, Num k) => HopfAlgebra k SSymM where+    antipode = toSSymM . antipode . toSSymF+++-- YSYM: PLANAR BINARY TREES+-- These are really rooted planar binary trees.+-- It's because they're planar that we can distinguish left and right child branches.+-- (Non-planar would be if we considered trees where left and right children are swapped relative to one another as the same tree)+-- It is neither commutative nor co-commutative++-- |A type for (rooted) planar binary trees. The basis elements of the Loday-Ronco Hopf algebra are indexed by these.+--+-- Although the trees are labelled, we're really only interested in the shapes of the trees, and hence in the type PBT ().+-- The Algebra, Coalgebra and HopfAlgebra instances all ignore the labels.+-- However, it is convenient to allow labels, as they can be useful for seeing what is going on, and they also make it possible+-- to define various ways to create trees from lists of labels.+data PBT a = T (PBT a) a (PBT a) | E deriving (Eq, Show, Functor)++instance Ord a => Ord (PBT a) where+    compare u v = compare (shapeSignature u, prefix u) (shapeSignature v, prefix v)++-- |The fundamental basis for (the dual of) the Loday-Ronco Hopf algebra of binary trees, YSym.+newtype YSymF a = YSymF (PBT a) deriving (Eq, Ord, Functor)++instance Show a => Show (YSymF a) where+    show (YSymF t) = "F(" ++ show t ++ ")"++-- |Construct the element of YSym in the fundamental basis indexed by the given tree+ysymF :: PBT a -> Vect Q (YSymF a)+ysymF t = return (YSymF t)++{-+depth (T l x r) = 1 + max (depth l) (depth r)+depth E = 0+-}+nodecount (T l x r) = 1 + nodecount l + nodecount r+nodecount E = 0++-- in fact leafcount t = 1 + nodecount t (easiest to see with a picture)+leafcount (T l x r) = leafcount l + leafcount r+leafcount E = 1++prefix E = []+prefix (T l x r) = x : prefix l ++ prefix r++-- The shape signature uniquely identifies the shape of a tree.+-- Trees with distinct shapes have distinct signatures.+-- In addition, if sorting on shapeSignature, smaller trees sort before larger trees,+-- and leftward leaning trees sort before rightward leaning trees+shapeSignature t = shapeSignature' (nodeCountTree t)+    where shapeSignature' E = [0] -- not [], otherwise we can't distinguish T (T E () E) () E from T E () (T E () E)+          shapeSignature' (T l x r) = x : shapeSignature' r ++ shapeSignature' l++nodeCountTree E = E+nodeCountTree (T l _ r) = T l' n r'+    where l' = nodeCountTree l+          r' = nodeCountTree r+          n = 1 + (case l' of E -> 0; T _ lc _ -> lc) + (case r' of E -> 0; T _ rc _ -> rc)++leafCountTree E = E+leafCountTree (T l _ r) = T l' n r'+    where l' = leafCountTree l+          r' = leafCountTree r+          n = (case l' of E -> 1; T _ lc _ -> lc) + (case r' of E -> 1; T _ rc _ -> rc)++-- A tree that counts nodes in left and right subtrees+lrCountTree E = E+lrCountTree (T l _ r) = T l' (lc,rc) r'+    where l' = lrCountTree l+          r' = lrCountTree r+          lc = case l' of E -> 0; T _ (llc,lrc) _ -> 1 + llc + lrc+          rc = case r' of E -> 0; T _ (rlc,rrc) _ -> 1 + rlc + rrc++shape :: PBT a -> PBT ()+shape t = fmap (\_ -> ()) t++-- label the nodes of a tree in infix order while preserving its shape+numbered t = numbered' 1 t+    where numbered' _ E = E+          numbered' i (T l x r) = let k = nodecount l in T (numbered' i l) (i+k) (numbered' (i+k+1) r)+-- could also pair the numbers with the input labels+++splits E = [(E,E)]+splits (T l x r) = [(u, T v x r) | (u,v) <- splits l] ++ [(T l x u, v) | (u,v) <- splits r]++instance (Eq k, Num k, Ord a) => Coalgebra k (YSymF a) where+    counit = unwrap . linear counit' where counit' (YSymF E) = 1; counit' (YSymF (T _ _ _)) = 0+    comult = linear comult'+        where comult' (YSymF t) = sumv [return (YSymF u, YSymF v) | (u,v) <- splits t]+              -- using sumv rather than sum to avoid requiring Show a+    -- so again this is a kind of deconcatenation coproduct++multisplits 1 t = [ [t] ]+multisplits 2 t = [ [u,v] | (u,v) <- splits t ]+multisplits n t = [ u:ws | (u,v) <- splits t, ws <- multisplits (n-1) v ]++graft [t] E = t+graft ts (T l x r) = let (ls,rs) = splitAt (leafcount l) ts+                     in T (graft ls l) x (graft rs r)++instance (Eq k, Num k, Ord a) => Algebra k (YSymF a) where+    unit x = x *> return (YSymF E)+    mult = linear mult'+        where mult' (YSymF t, YSymF u) = sumv [return (YSymF (graft ts u)) | ts <- multisplits (leafcount u) t]+              -- using sumv rather than sum to avoid requiring Show a++instance (Eq k, Num k, Ord a) => Bialgebra k (YSymF a) where {}++instance (Eq k, Num k, Ord a) => HopfAlgebra k (YSymF a) where+    antipode = linear antipode'+        where antipode' (YSymF E) = return (YSymF E)+              antipode' x = (negatev . mult . (id `tf` antipode) . removeTerm (YSymF E,x) . comult . return) x+++-- |An alternative "monomial" basis for (the dual of) the Loday-Ronco Hopf algebra of binary trees, YSym.+newtype YSymM = YSymM (PBT ()) deriving (Eq, Ord)++instance Show YSymM where+    show (YSymM t) = "M(" ++ show t ++ ")"++-- |Construct the element of YSym in the monomial basis indexed by the given tree+ysymM :: PBT () -> Vect Q YSymM+ysymM t = return (YSymM t)+++trees 0 = [E]+trees n = [T l () r | i <- [0..n-1], l <- trees (n-1-i), r <- trees i]++-- |The covering relation for the Tamari partial order on binary trees+covers E = []+covers (T t@(T u x v) y w) = [T t' y w | t' <- covers t]+                          ++ [T t y w' | w' <- covers w]+                          ++ [T u y (T v x w)]+                          -- Note that this preserves the descending property, and hence the bijection with permutations+                          -- If we were to swap x and y, we would preserve the binary search tree property instead (if our trees had it)+covers (T E x u) = [T E x u' | u' <- covers u]  ++-- |The up-set of a binary tree in the Tamari partial order+tamariUpSet t = upSet' [] [t]+    where upSet' interior boundary =+              if null boundary+              then interior+              else let interior' = setUnionAsc interior boundary+                       boundary' = toSet $ concatMap covers boundary+                   in upSet' interior' boundary'++-- tamariOrder1 u v = v `elem` upSet u++tamariOrder u v = weakOrder (minPerm u) (minPerm v)+-- It should be possible to unpack this to be a statement purely about trees, but probably not worth++-- |Convert an element of YSym represented in the monomial basis to the fundamental basis+toYSymF :: (Eq k, Num k) => Vect k YSymM -> Vect k (YSymF ())+toYSymF = linear toYSymF'+    where toYSymF' (YSymM t) = sumv [mu (set,po) t s *> return (YSymF s) | s <- set]+              where po = tamariOrder+                    set = tamariUpSet t -- [s | s <- trees (nodecount t), t `tamariOrder` s]++-- |Convert an element of YSym represented in the fundamental basis to the monomial basis+toYSymM :: (Eq k, Num k) => Vect k (YSymF ()) -> Vect k YSymM+toYSymM = linear toYSymM'+    where toYSymM' (YSymF t) = sumv [return (YSymM s) | s <- tamariUpSet t]+                            -- sumv [return (YSymM s) | s <- trees (nodecount t), t `tamariOrder` s]+++instance (Eq k, Num k) => Algebra k YSymM where+    unit x = x *> return (YSymM E)+    mult = toYSymM . mult . (toYSymF `tf` toYSymF)++instance (Eq k, Num k) => Coalgebra k YSymM where+    counit = unwrap . linear counit' where counit' (YSymM E) = 1; counit' (YSymM (T _ _ _)) = 0+    -- comult = (toYSymM `tf` toYSymM) . comult . toYSymF+    comult = linear comult'+        where comult' (YSymM t) = sumv [return (YSymM r, YSymM s) | (rs,ss) <- deconcatenations (underDecomposition t),+                                                                    let r = foldl under E rs, let s = foldl under E ss]+++instance (Eq k, Num k) => Bialgebra k YSymM where {}++instance (Eq k, Num k) => HopfAlgebra k YSymM where+    antipode = toYSymM . antipode . toYSymF +++-- QSYM: QUASI-SYMMETRIC FUNCTIONS+-- The following is the Hopf algebra QSym of quasi-symmetric functions+-- using the monomial basis (indexed by compositions)++-- compositions in ascending order+-- might be better to use bfs to get length order+-- |List the compositions of an integer n. For example, the compositions of 4 are [[1,1,1,1],[1,1,2],[1,2,1],[1,3],[2,1,1],[2,2],[3,1],[4]]+compositions :: Int -> [[Int]]+compositions 0 = [[]]+compositions n = [i:is | i <- [1..n], is <- compositions (n-i)]++-- can retrieve subsets of [1..n-1] from compositions n as follows+-- > map (tail . scanl (+) 0) (map init $ compositions 4)+-- [[],[3],[2],[2,3],[1],[1,3],[1,2],[1,2,3]]++-- quasi shuffles of two compositions+quasiShuffles :: [Int] -> [Int] -> [[Int]]+quasiShuffles (x:xs) (y:ys) = map (x:) (quasiShuffles xs (y:ys)) +++                              map (y:) (quasiShuffles (x:xs) ys) +++                              map ((x+y):) (quasiShuffles xs ys)+quasiShuffles xs [] = [xs]+quasiShuffles [] ys = [ys]+++-- |A type for the monomial basis for the quasi-symmetric functions, indexed by compositions.+newtype QSymM = QSymM [Int] deriving (Eq)++instance Ord QSymM where+    compare (QSymM xs) (QSymM ys) = compare (length xs, xs) (length ys, ys)++instance Show QSymM where+    show (QSymM xs) = "M " ++ show xs++-- |Construct the element of QSym in the monomial basis indexed by the given composition+qsymM :: [Int] -> Vect Q QSymM+qsymM = return . QSymM++instance (Eq k, Num k) => Algebra k QSymM where+    unit x = x *> return (QSymM [])+    mult = linear mult'+        where mult' (QSymM alpha, QSymM beta) = sum [return (QSymM gamma) | gamma <- quasiShuffles alpha beta]++instance (Eq k, Num k) => Coalgebra k QSymM where+    counit = unwrap . linear counit' where counit' (QSymM alpha) = if null alpha then 1 else 0+    comult = linear comult' where+        comult' (QSymM gamma) = sum [return (QSymM alpha, QSymM beta) | (alpha,beta) <- deconcatenations gamma]++instance (Eq k, Num k) => Bialgebra k QSymM where {}++instance (Eq k, Num k) => HopfAlgebra k QSymM where+    antipode = linear antipode' where+        antipode' (QSymM alpha) = (-1)^length alpha * sum [return (QSymM (reverse beta)) | beta <- coarsenings alpha]++coarsenings (x1:x2:xs) = coarsenings ((x1+x2):xs) ++ map (x1:) (coarsenings (x2:xs))+coarsenings xs = [xs] -- for xs a singleton or null++refinements (x:xs) = [y++ys | y <- compositions x, ys <- refinements xs]+refinements [] = [[]]+++newtype QSymF = QSymF [Int] deriving (Eq)++instance Ord QSymF where+    compare (QSymF xs) (QSymF ys) = compare (length xs, xs) (length ys, ys)++instance Show QSymF where+    show (QSymF xs) = "F " ++ show xs++-- |Construct the element of QSym in the fundamental basis indexed by the given composition+qsymF :: [Int] -> Vect Q QSymF+qsymF = return . QSymF++-- |Convert an element of QSym represented in the monomial basis to the fundamental basis+toQSymF :: (Eq k, Num k) => Vect k QSymM -> Vect k QSymF+toQSymF = linear toQSymF'+    where toQSymF' (QSymM alpha) = sumv [(-1) ^ (length beta - length alpha) *> return (QSymF beta) | beta <- refinements alpha]++-- |Convert an element of QSym represented in the fundamental basis to the monomial basis+toQSymM :: (Eq k, Num k) => Vect k QSymF -> Vect k QSymM+toQSymM = linear toQSymM'+    where toQSymM' (QSymF alpha) = sumv [return (QSymM beta) | beta <- refinements alpha] -- ie beta <- up-set of alpha++instance (Eq k, Num k) => Algebra k QSymF where+    unit x = x *> return (QSymF [])+    mult = toQSymF . mult . (toQSymM `tf` toQSymM)++instance (Eq k, Num k) => Coalgebra k QSymF where+    counit = unwrap . linear counit' where counit' (QSymF xs) = if null xs then 1 else 0+    comult = (toQSymF `tf` toQSymF) . comult . toQSymM++instance (Eq k, Num k) => Bialgebra k QSymF where {}++instance (Eq k, Num k) => HopfAlgebra k QSymF where+    antipode = toQSymF . antipode . toQSymM+++-- QUASI-SYMMETRIC POLYNOMIALS++-- the above induces Hopf algebra structure on quasi-symmetric functions via+-- m_alpha -> sum [product (zipWith (^) (map x_ is) alpha | is <- combinationsOf k [] ] where k = length alpha++xvars n = [glexvar ("x" ++ show i) | i <- [1..n] ]++-- compare with Reynolds operator+-- so a basis for quasi-symmetric functions over xvars n consists of [quasiSymM xs is | m <- [0..], is <- compositions m]+quasiSymM xs is = sum [product (zipWith (^) xs' is) | xs' <- combinationsOf r xs]+    where r = length is+++-- MAPS BETWEEN (POSETS AND) HOPF ALGEBRAS++-- A descending tree is one in which a child is always less than a parent.+descendingTree [] = E+descendingTree [x] = T E x E+descendingTree xs = T l x r+    where x = maximum xs+          (ls,_:rs) = L.break (== x) xs+          l = descendingTree ls+          r = descendingTree rs+-- This is a bijection from permutations to "ordered trees".+-- It is order-preserving on trees with the same nodecount.+-- We can recover the permutation by reading the node labels in infix order.+-- This is the map called lambda in Loday.pdf+++-- |A Hopf algebra morphism from SSymF to YSymF+descendingTreeMap :: (Eq k, Num k) => Vect k SSymF -> Vect k (YSymF ())+descendingTreeMap = nf . fmap (YSymF . shape .  descendingTree')+    where descendingTree' (SSymF xs) = descendingTree xs+-- This is the map called Lambda in Loday.pdf, or tau in MSym.pdf+-- It is an algebra morphism.++-- One of the ideas in the MSym paper is to look at the intermediate result (fmap descendingTree' x),+-- which is an "ordered tree", and consider the map as factored through this++-- The map is surjective but not injective. The fibers tau^-1(t) are intervals in the weak order on permutations++-- "inverse" for descendingTree+-- These are the maps called gamma in Loday.pdf+minPerm t = minPerm' (lrCountTree t)+    where minPerm' E = []+          minPerm' (T l (lc,rc) r) = minPerm' l ++ [lc+rc+1] ++ map (+lc) (minPerm' r)++maxPerm t = maxPerm' (lrCountTree t)+    where maxPerm' E = []+          maxPerm' (T l (lc,rc) r) = map (+rc) (maxPerm' l) ++ [lc+rc+1] ++ maxPerm' r+++-- The composition of [1..n] obtained by treating each left-facing leaf as a cut+-- Specifically, we visit the nodes in infix order, cutting after a node if it does not have an E as its right child+-- This is the map called L in Loday.pdf+leftLeafComposition E = []+leftLeafComposition t = cuts $ tail $ leftLeafs t+    where leftLeafs (T l x E) = leftLeafs l ++ [False]+          leftLeafs (T l x r) = leftLeafs l ++ leftLeafs r+          leftLeafs E = [True]+          cuts bs = case break id bs of+                    (ls,r:rs) -> (length ls + 1) : cuts rs+                    (ls,[]) -> [length ls]++leftLeafComposition' (YSymF t) = QSymF (leftLeafComposition t)++-- |A Hopf algebra morphism from YSymF to QSymF+leftLeafCompositionMap :: (Eq k, Num k) => Vect k (YSymF a) -> Vect k QSymF+leftLeafCompositionMap = nf . fmap leftLeafComposition'+++-- The descent set of a permutation is [i | x_i > x_i+1], where we start the indexing from 1+descents [] = []+descents xs = map (+1) $ L.elemIndices True $ zipWith (>) xs (tail xs)++-- The composition of [1..n] obtained by treating each descent as a cut+descentComposition [] = []+descentComposition xs = dc $ zipWith (>) xs (tail xs) ++ [False]+    where dc bs = case break id bs of+                  (ls,r:rs) -> (length ls + 1) : dc rs+                  (ls,[]) -> [length ls]++-- |A Hopf algebra morphism from SSymF to QSymF+descentMap :: (Eq k, Num k) => Vect k SSymF -> Vect k QSymF+descentMap = nf . fmap (\(SSymF xs) -> QSymF (descentComposition xs))+-- descentMap == leftLeafCompositionMap . descendingTreeMap++underComposition (QSymF ps) = foldr under (SSymF []) [SSymF [1..p] | p <- ps]+    where under (SSymF xs) (SSymF ys) = let q = length ys+                                            zs = map (+q) xs ++ ys -- so it has a global descent at the split+                                        in SSymF zs+-- This is a poset morphism (indeed, it forms a Galois connection with descentComposition)+-- but it does not extend to a Hopf algebra morphism.+-- (It does extend to a coalgebra morphism.)+-- (It is picking the maximum permutation having a given descent composition,+-- so there's an element of arbitrariness to it.)+-- This is the map called Z (Zeta?) in Loday.pdf++{-+-- This is O(n^2), whereas an O(n) implementation should be possible+-- Also, we would really like the associated composition (obtained by treating each global descent as a cut)?+globalDescents xs = globalDescents' 0 [] xs+    where globalDescents' i ls (r:rs) = (if minimum (infinity:ls) > maximum (0:r:rs) then [i] else [])+                                     ++ globalDescents' (i+1) (r:ls) rs+          globalDescents' n _ [] = [n]+          infinity = maxBound :: Int+-- The idea is that this leads to a map from SSymM to QSymM++globalDescentComposition [] = []+globalDescentComposition (x:xs) = globalDescents' 1 x xs+    where globalDescents' i minl (r:rs) = if minl > maximum (r:rs)+                                          then i : globalDescents' 1 r rs+                                          else globalDescents' (i+1) r rs+          globalDescents' i _ [] = [i]++globalDescentMap :: (Eq k, Num k) => Vect k SSymM -> Vect k QSymM+globalDescentMap = nf . fmap (\(SSymM xs) -> QSymM (globalDescentComposition xs))+-}++-- A multiplication operation on trees+-- (Connected with their being cofree)+-- (intended to be used as infix)+under E t = t+under (T l x r) t = T l x (under r t)++isUnderIrreducible (T l x E) = True+isUnderIrreducible _ = False++underDecomposition (T l x r) = T l x E : underDecomposition r+underDecomposition E = []+++-- GHC7.4.1 doesn't like the following type signature - a bug.+-- ysymmToSh :: (Eq k, Num k) => Vect k (YSymM) => Vect k (Shuffle (PBT ()))+ysymmToSh = fmap ysymmToSh'+    where ysymmToSh' (YSymM t) = Sh (underDecomposition t)+-- This is a coalgebra morphism (but not an algebra morphism)+-- It shows that YSym is co-free+{-+-- This one not working yet - perhaps it needs an nf, or to go via S/YSymF, or ...+ssymmToSh = nf . fmap ssymmToSh'+    where ssymmToSh' (SSymM xs) = (Sh . underDecomposition . shape . descendingTree) xs+-}

@@ -15,7 +15,7 @@ import Math.Algebra.Group.PermutationGroup hiding (elts, order, isMember)
 import Math.Algebra.Group.SchreierSims as SS
 import Math.Combinatorics.Graph as G hiding (to1n, incidenceMatrix)
-import Math.Combinatorics.GraphAuts (refine', isSingleton, graphAuts, incidenceAuts) -- , removeGens)
+import Math.Combinatorics.GraphAuts (graphAuts, incidenceAuts) -- , removeGens)
 import Math.Combinatorics.FiniteGeometry
 
 -- Cameron & van Lint, Designs, Graphs, Codes and their Links

@@ -1,6 +1,11 @@ -- Copyright (c) David Amos, 2009. All rights reserved.
 
-module Math.Combinatorics.GraphAuts where
+module Math.Combinatorics.GraphAuts (isVertexTransitive, isEdgeTransitive,
+                                     isArcTransitive, is2ArcTransitive, is3ArcTransitive, isnArcTransitive,
+                                     isDistanceTransitive,
+                                     graphAuts, incidenceAuts,
+                                     graphIsos, incidenceIsos,
+                                     isGraphIso, isIncidenceIso) where
 
 import Data.Either (lefts)
 import qualified Data.List as L

@@ -5,6 +5,8 @@  module Math.Combinatorics.IncidenceAlgebra where +import Math.Core.Utils+ import Math.Combinatorics.Digraph import Math.Combinatorics.Poset @@ -111,14 +113,14 @@   -- |The unit of the incidence algebra of a poset-unitIA :: (Eq k, Num k, Ord t) => Poset t -> Vect k (Interval t)+unitIA :: (Eq k, Num k, Ord a) => Poset a -> Vect k (Interval a) unitIA poset@(Poset (set,_)) = sumv [return (Iv poset (x,x)) | x <- set] -basisIA :: Num k => Poset t -> [Vect k (Interval t)]+basisIA :: Num k => Poset a -> [Vect k (Interval a)] basisIA poset = [return (Iv poset xy) | xy <- intervals poset]  -- |The zeta function of a poset-zetaIA :: (Eq k, Num k, Ord t) => Poset t -> Vect k (Interval t)+zetaIA :: (Eq k, Num k, Ord a) => Poset a -> Vect k (Interval a) zetaIA poset = sumv $ basisIA poset  -- Then for example, zeta^2 counts the number of points in each interval@@ -132,7 +134,7 @@  -- calculate the mobius function of a poset, with memoization -- |The Mobius function of a poset-muIA :: (Eq k, Num k, Ord t) => Poset t -> Vect k (Interval t)+muIA :: (Eq k, Num k, Ord a) => Poset a -> Vect k (Interval a) muIA poset@(Poset (set,po)) = sumv [mus M.! (x,y) *> return (Iv poset (x,y)) | x <- set, y <- set]     where mu (x,y) | x == y    = 1                    | po x y    = negate $ sum [mus M.! (x,z) | z <- set, po x z, po z y, z /= y]@@ -152,7 +154,7 @@ -- Stanley, Enumerative Combinatorics I, p144 -- |The inverse of an element in the incidence algebra of a poset. -- This is only defined for elements which are non-zero on all intervals (x,x)-invIA :: (Eq k, Fractional k, Ord t) => Vect k (Interval t) -> Maybe (Vect k (Interval t))+invIA :: (Eq k, Fractional k, Ord a) => Vect k (Interval a) -> Maybe (Vect k (Interval a)) invIA f | f == zerov = Nothing -- error "invIA 0"         | any (==0) [f' (x,x) | x <- set] = Nothing -- error "invIA: not invertible"         | otherwise = Just g@@ -163,23 +165,24 @@                    | otherwise = (-1 / f' (x,x)) * sum [f' (x,z) * (g's M.! (z,y)) | z <- interval poset (x,y), x /= z]           g's = M.fromList [(xy, g' xy) | xy <- intervals poset] -invIA' f = case invIA f of-           Just g -> g-           Nothing -> error "invIA': not invertible"+instance (Eq k, Fractional k, Ord a, Show a) => HasInverses (Vect k (Interval a)) where+    inverse f = case invIA f of+                Just g -> g+                Nothing -> error "IncidenceAlgebra.inverse: not invertible"  -- Then for example we can count multichains or chains using the incidence algebra - see Stanley  -- |A function (ie element of the incidence algebra) that counts the total number of chains in each interval-numChainsIA :: (Ord a) => Poset a -> Vect Q (Interval a)-numChainsIA poset = invIA' (2 *> unitIA poset <-> zetaIA poset)+numChainsIA :: (Ord a, Show a) => Poset a -> Vect Q (Interval a)+numChainsIA poset = (2 *> unitIA poset <-> zetaIA poset)^-1  -- The eta function on intervals (x,y) is 1 if x -< y (y covers x), 0 otherwise etaIA poset = let DG vs es = hasseDigraph poset               in sumv [return (Iv poset (x,y)) | (x,y) <- es]  -- |A function (ie element of the incidence algebra) that counts the number of maximal chains in each interval-numMaximalChainsIA :: (Ord a) => Poset a -> Vect Q (Interval a)-numMaximalChainsIA poset = invIA' (unitIA poset <-> etaIA poset)+numMaximalChainsIA :: (Ord a, Show a) => Poset a -> Vect Q (Interval a)+numMaximalChainsIA poset = (unitIA poset <-> etaIA poset)^-1   -- In order to quickCheck this, we would need

@@ -1,11 +1,12 @@ -- Copyright (c) 2011, David Amos. All rights reserved. -{-# LANGUAGE NoMonomorphismRestriction, TupleSections #-}+{-# LANGUAGE NoMonomorphismRestriction #-}   module Math.Combinatorics.Poset where  import Math.Common.ListSet as LS -- set operations on strictly ascending lists+import Math.Core.Utils -- for set/multiset operations on ordered lists import Math.Algebra.Field.Base import Math.Combinatorics.FiniteGeometry import Math.Algebra.LinearAlgebra@@ -102,7 +103,7 @@ posetB n = Poset ( powerset [1..n], LS.isSubset )  --- LATTICE OF PARTITIONS OF [1..N] ORDERED BY REFINEMENT+-- LATTICE OF SET PARTITIONS OF [1..N] ORDERED BY REFINEMENT  partitions [] = [[]] partitions [x] = [[[x]]]@@ -113,7 +114,7 @@ isRefinement a b = and [or [acell `isSubset` bcell | bcell <- b] | acell <- a] -- if we know that a and b are appropriately sorted, then this can probably be done more efficiently --- |posetP n is the lattice of partitions of [1..n] ordered by refinement+-- |posetP n is the lattice of set partitions of [1..n], ordered by refinement posetP :: Int -> Poset [[Int]] posetP n = Poset ( partitions [1..n], isRefinement ) @@ -130,17 +131,57 @@  intervalPartitions2 [] = [[]] intervalPartitions2 [x] = [[[x]]]-intervalPartitions2 (x:xs) = let ips = intervalPartitions xs in+intervalPartitions2 (x:xs) = let ips = intervalPartitions2 xs in     map ([x]:) ips ++ [ (x:head):tail | (head:tail) <- ips] -- we're guaranteed that x+1 is at the head of the head  +-- LATTICE OF INTEGER PARTITIONS OF N ORDERED BY REFINEMENT++integerPartitions1 n = ips (reverse [1..n]) n+    where ips [] 0 = [[]]+          ips [] _ = []+          ips (x:xs) n | x > n     = ips xs n+                       | otherwise = map (x:) (ips (x:xs) (n-x)) ++ ips xs n++-- For example, integerPartitions 5 -> [ [5], [4,1], [3,2], [3,1,1], [2,2,1], [2,1,1,1], [1,1,1,1,1] ]+integerPartitions n = dfs ([],n,n)+    where dfs (xs, 0, _) = [reverse xs]+          dfs (xs, r, i) = concatMap dfs [ (i':xs, r-i', i') | i' <- reverse [1..min r i] ]++isIPRefinement ys xs = dfs xs ys+    where dfs (x:xs) (y:ys) | x < y = False+                            | x == y = dfs xs ys+                            | otherwise = or [dfs xs' ys' | y' <- y:ys, let ys' = L.delete y' (y:ys),+                                                                        let xs' = insertDesc (x-y') xs]+          dfs [] [] = True+          insertDesc = L.insertBy (flip compare)++{-+-- In theory it feels like this ought to be faster for large n, but in practice it's unclear+isIPRefinement2 ys xs = isIPRefinement (ys \\ xs) (xs \\ ys)+    where (\\) = diffDesc+-}++-- |posetIP n is the poset of integer partitions of n, ordered by refinement+posetIP :: Int -> Poset [Int]+posetIP n = Poset (integerPartitions n, isIPRefinement)+++-- Could also implement the Young lattice (or the part up to n)+-- Of integer partitions <= n, ordered by inclusion+-- Kassel, Turaev, Braid Groups, p202+++-- INTEGER COMPOSITIONS++ -- LATTICE OF SUBSPACES OF Fq^n  subspaces fq n = [] : concatMap (flatsPG (n-1) fq) [0..n-1] -- note that flatsPG returns the subspaces as a matrix of row vectors in reduced row echelon form --- inSpanRE m v returns whether the vector v is in the span of the matrix m, where m is required to be in row echelon form+-- inSpanRE m v returns whether the vector v is in the span of the rows of the matrix m, where m is required to be in row echelon form isSubspace s1 s2 = all (inSpanRE s2) s1  -- This is the projective geometry PG(n,q)@@ -253,12 +294,6 @@ -- This returns all automorphisms -- What we really want is to return generators of the permutation group -----pairs (x:xs) = map (x,) xs ++ pairs xs -- TupleSections-pairs [] = []  -- |A linear extension of a poset is a linear ordering of the elements which extends the partial order. -- Equivalently, it is an ordering [x1..xn] of the underlying set, such that if xi <= xj then i <= j.

@@ -11,6 +11,7 @@  toSet = S.toList . S.fromList +{- -- Merge two ordered listsets. Elements appearing in both inputs appear only once in the output mergeSet (x:xs) (y:ys) =     case compare x y of@@ -18,7 +19,73 @@     EQ -> x : mergeSet xs ys     GT -> y : mergeSet (x:xs) ys mergeSet xs ys = xs ++ ys+-} +-- |The set union of two ascending lists. If both inputs are strictly increasing, then the output is their union+-- and is strictly increasing. The code does not check that the lists are strictly increasing.+setUnionAsc :: Ord a => [a] -> [a] -> [a]+setUnionAsc (x:xs) (y:ys) =+    case compare x y of+    LT -> x : setUnionAsc xs (y:ys)+    EQ -> x : setUnionAsc xs ys+    GT -> y : setUnionAsc (x:xs) ys+setUnionAsc xs ys = xs ++ ys++-- |The multiset sum of two ascending lists. If xs and ys are ascending, then multisetSumAsc xs ys == sort (xs++ys).+-- The code does not check that the lists are ascending.+multisetSumAsc :: Ord a => [a] -> [a] -> [a]+multisetSumAsc (x:xs) (y:ys) =+    case compare x y of+    LT -> x : multisetSumAsc xs (y:ys)+    EQ -> x : y : multisetSumAsc xs ys+    GT -> y : multisetSumAsc (x:xs) ys+multisetSumAsc xs ys = xs ++ ys++-- |The multiset sum of two descending lists. If xs and ys are descending, then multisetSumDesc xs ys == sort (xs++ys).+-- The code does not check that the lists are descending.+multisetSumDesc :: Ord a => [a] -> [a] -> [a]+multisetSumDesc (x:xs) (y:ys) =+    case compare x y of+    GT -> x : multisetSumDesc xs (y:ys)+    EQ -> x : y : multisetSumDesc xs ys+    LT -> y : multisetSumDesc (x:xs) ys+multisetSumDesc xs ys = xs ++ ys+++-- |The multiset or set difference between two ascending lists. If xs and ys are ascending, then diffAsc xs ys == xs \\ ys,+-- and diffAsc is more efficient. If xs and ys are sets (that is, have no repetitions), then diffAsc xs ys is the set difference.+-- The code does not check that the lists are ascending.+diffAsc :: Ord a => [a] -> [a] -> [a]+diffAsc (x:xs) (y:ys) = case compare x y of+                        LT -> x : diffAsc xs (y:ys)+                        EQ -> diffAsc xs ys+                        GT -> diffAsc (x:xs) ys+diffAsc xs [] = xs+diffAsc [] _ = []++-- |The multiset or set difference between two descending lists. If xs and ys are descending, then diffDesc xs ys == xs \\ ys,+-- and diffDesc is more efficient. If xs and ys are sets (that is, have no repetitions), then diffDesc xs ys is the set difference.+-- The code does not check that the lists are descending.+diffDesc :: Ord a => [a] -> [a] -> [a]+diffDesc (x:xs) (y:ys) = case compare x y of+                        GT -> x : diffDesc xs (y:ys)+                        EQ -> diffDesc xs ys+                        LT -> diffDesc (x:xs) ys+diffDesc xs [] = xs+diffDesc [] _ = []+++isSubsetAsc = isSubMultisetAsc++isSubMultisetAsc (x:xs) (y:ys) =+    case compare x y of+    LT -> False+    EQ -> isSubMultisetAsc xs ys+    GT -> isSubMultisetAsc (x:xs) ys+isSubMultisetAsc [] ys = True+isSubMultisetAsc xs [] = False++ pairs (x:xs) = map (x,) xs ++ pairs xs pairs [] = [] @@ -29,11 +96,15 @@ -- fold a comparison operator through a list foldcmpl p (x1:x2:xs) = p x1 x2 && foldcmpl p (x2:xs) foldcmpl _ _ = True--- This can be expressed as a pure fold:--- foldcmpl cmp (x:xs) = snd $ foldl (\(bool,x') x -> (bool && cmp x' x, x)) (True,x)+ -- foldcmpl _ [] = True--- However, that is less efficient, as we can't abort as soon as we fail--- (What about using the Maybe monad?)+-- foldcmpl p xs = and $ zipWith p xs (tail xs)++isWeaklyIncreasing :: Ord t => [t] -> Bool+isWeaklyIncreasing = foldcmpl (<=)++isStrictlyIncreasing :: Ord t => [t] -> Bool+isStrictlyIncreasing = foldcmpl (<)  -- for use with L.sortBy cmpfst x y = compare (fst x) (fst y)

@@ -3,7 +3,8 @@ {-# LANGUAGE NoMonomorphismRestriction #-}  -- |A module providing functions to test for primality, and find next and previous primes.-module Math.NumberTheory.Prime where+module Math.NumberTheory.Prime (primes, isTrialDivisionPrime, isMillerRabinPrime,+                                isPrime, notPrime, prevPrime, nextPrime) where  import System.Random import System.IO.Unsafe

@@ -0,0 +1,116 @@+-- Copyright (c) 2011, David Amos. All rights reserved.++{-# LANGUAGE MultiParamTypeClasses, TypeSynonymInstances, FlexibleInstances, OverlappingInstances #-}++-- |A module for arithmetic in quadratic number fields. A quadratic number field is a field of the form Q(sqrt d),+-- where d is a square-free integer. For example, we can perform the following calculation in Q(sqrt 2):+--+-- > (1 + sqrt 2) / (2 + sqrt 2)+--+-- It is also possible to mix different square roots in the same calculation. For example:+--+-- > (1 + sqrt 2) * (1 + sqrt 3)+--+-- Square roots of negative numbers are also permitted. For example:+--+-- > i * sqrt(-3)+module Math.NumberTheory.QuadraticField where++import Prelude hiding (sqrt)++import Data.List as L+import Math.Core.Field+import Math.Core.Utils (powersetdfs)+import Math.Algebras.VectorSpace+import Math.Algebras.TensorProduct+import Math.Algebras.Structures+import Math.NumberTheory.Factor++import Math.Algebra.LinearAlgebra hiding (inverse, (*>) )+import Math.CommutativeAlgebra.Polynomial+++-- Q(sqrt n)++-- |A basis for quadratic number fields Q(sqrt d), where d is a square-free integer.+data QNFBasis = One | Sqrt Integer deriving (Eq,Ord)++instance Show QNFBasis where+    show One = "1"+    show (Sqrt d) | d == -1 = "i"+                  | otherwise = "sqrt(" ++ show d ++ ")"++-- |The type for elements of quadratic number fields+type QNF = Vect Q QNFBasis++-- |Although this has the same name as the Prelude.sqrt function, it should be thought of as more like a constructor+-- for creating elements of quadratic fields.+--+-- Note that for d positive, sqrt d means the positive square root, and sqrt (-d) should be interpreted as the square root+-- with positive imaginary part, that is i * sqrt d. This has the consequence that for example, sqrt (-2) * sqrt (-3) = - sqrt 6.+sqrt :: Integer -> QNF+sqrt d | fr == 1   = fromInteger sq+       | otherwise = fromInteger sq * return (Sqrt fr)+    where (sq,fr) = squaredFree 1 1 (pfactors d)+          squaredFree squared free (d1:d2:ds) =+              if d1 == d2 then squaredFree (d1*squared) free ds else squaredFree squared (d1*free) (d2:ds)+          squaredFree squared free ds = (squared, free * product ds)++sqrt2 = sqrt 2+sqrt3 = sqrt 3+sqrt5 = sqrt 5+sqrt6 = sqrt 6+sqrt7 = sqrt 7++i :: QNF+i = sqrt (-1)++instance (Eq k, Num k) => Algebra k QNFBasis where+    unit x = x *> return One+    mult = linear mult'+         where mult' (One,x) = return x+               mult' (x,One) = return x+               mult' (Sqrt m, Sqrt n) | m == n = unit (fromInteger m)+                                      | otherwise = let (i,d) = interdiff (pfactors m) (pfactors n) 1 1+                                                    in fromInteger i *> return (Sqrt d)+               -- if squarefree a == product ps, b == product qs+               -- then sqrt a * sqrt b = product (intersect ps qs) * sqrt (product (symdiff ps qs))+               -- the following calculates these two products+               -- in particular, it correctly handles the case that either or both contain -1+               interdiff (p:ps) (q:qs) i d =+                   case compare p q of+                   LT -> interdiff ps (q:qs) i (d*p)+                   EQ -> interdiff ps qs (i*p) d+                   GT -> interdiff (p:ps) qs i (d*q)+               interdiff ps qs i d = (i, d * product (ps ++ qs))++{-+instance HasConjugation Q QNFBasis where+    conj = (>>= conj') where+        conj' One = return One+        conj' sqrt_d = -1 *> return sqrt_d+    -- ie conj = linear conj', but avoiding unnecessary nf call+    sqnorm x = coeff One (x * conj x)+-}++newtype XVar = X Int deriving (Eq, Ord, Show)++instance Fractional QNF where+    recip x@(V ts) =+        let ds = [d | (Sqrt d, _) <- terms x]+            fs = (if any (<0) ds then [-1] else []) ++ pfactors (foldl lcm 1 ds) -- lcm is always positive+            rs = map (\d -> case d of 1 -> One; d' -> Sqrt d') $+                 map product $ powersetdfs $ fs +            -- for example, for x == sqrt2 + sqrt3, we would have rs == [One, Sqrt 2, Sqrt 3, Sqrt 6]+            n = length rs+            y = V $ zip rs $ map (glexvar . X) [1..n] -- x1*1+x2*r2+...+xn*rn+            x' = V $ map (\(s,c) -> (s, unit c)) ts -- lift the coefficients in x into the polynomial algebra+            one = x' * y+            m = [ [coeff (mvar (X j)) c | j <- [1..n]] | i <- rs, let c = coeff i one] -- matrix of the linear system+            b = 1 : replicate (n-1) 0+        in case solveLinearSystem m b of -- find v such that m v == b - ie find the values of x1, x2, ... xn+            Just v -> nf $ V $ zip rs v+            Nothing -> error "QNF.recip 0"+    fromRational q = fromRational q *> 1++

@@ -14,41 +14,41 @@ import Math.Algebras.TensorProduct import Math.Algebras.Structures import Math.Algebras.GroupAlgebra+import Math.Core.Field import Math.Core.Utils -import Math.Test.TAlgebras.TStructures--instance Arbitrary (GroupAlgebra Integer) where-    arbitrary = do ts <- arbitrary :: Gen [(Permutation Int, Integer)]-                   return $ nf $ V $ take 10 ts+import Math.Test.TAlgebras.TVectorSpace -- for instance Arbitrary Q and (Vect k b)+import Math.Test.TAlgebras.TStructures -- for quickcheck properties  prop_Algebra_GroupAlgebra (k,x,y,z) = prop_Algebra (k,x,y,z)-    where types = (k,x,y,z) :: (Integer, GroupAlgebra Integer, GroupAlgebra Integer, GroupAlgebra Integer)+    where types = (k,x,y,z) :: (Q, GroupAlgebra Q, GroupAlgebra Q, GroupAlgebra Q)  -- have to split the 8-tuple into two 4-tuples to avoid having to write Arbitrary instance prop_Algebra_Linear_GroupAlgebra ((k,l,m,n),(x,y,z,w)) = prop_Algebra_Linear (k,l,m,n,x,y,z,w)-    where types = (k,l,m,n,x,y,z,w) :: (Integer, Integer, Integer, Integer,-                   GroupAlgebra Integer, GroupAlgebra Integer, GroupAlgebra Integer, GroupAlgebra Integer)+    where types = (k,l,m,n,x,y,z,w) :: (Q, Q, Q, Q,+                   GroupAlgebra Q, GroupAlgebra Q, GroupAlgebra Q, GroupAlgebra Q)  prop_Coalgebra_GroupAlgebra x = prop_Coalgebra x-    where types = x :: GroupAlgebra Integer+    where types = x :: GroupAlgebra Q  prop_Coalgebra_Linear_GroupAlgebra (k,l,x,y) = prop_Coalgebra_Linear (k,l,x,y)-    where types = (k,l,x,y) :: (Integer, Integer, GroupAlgebra Integer, GroupAlgebra Integer)+    where types = (k,l,x,y) :: (Q, Q, GroupAlgebra Q, GroupAlgebra Q)  prop_Bialgebra_GroupAlgebra (k,x,y) = prop_Bialgebra (k,x,y)-    where types = (k,x,y) :: (Integer, GroupAlgebra Integer, GroupAlgebra Integer)+    where types = (k,x,y) :: (Q, GroupAlgebra Q, GroupAlgebra Q)  prop_HopfAlgebra_GroupAlgebra x = prop_HopfAlgebra x-    where types = x :: GroupAlgebra Integer+    where types = x :: GroupAlgebra Q   quickCheckGroupAlgebra = do-    putStrLn "Checking that group algebra is an algebra, coalgebra, bialgebra, and Hopf algebra..."+    putStrLn "Testing that group algebra is an algebra, coalgebra, bialgebra, and Hopf algebra..."     quickCheck prop_Algebra_GroupAlgebra     quickCheck prop_Coalgebra_GroupAlgebra     quickCheck prop_Bialgebra_GroupAlgebra     quickCheck prop_HopfAlgebra_GroupAlgebra+    quickCheck (prop_AlgebraAntiMorphism (antipode :: GroupAlgebra Q -> GroupAlgebra Q))+    quickCheck (prop_CoalgebraAntiMorphism (antipode :: GroupAlgebra Q -> GroupAlgebra Q))   testlistGroupAlgebra = TestList [

@@ -4,7 +4,8 @@  import Test.QuickCheck -import Math.Algebra.Field.Base+import Math.Core.Field+-- import Math.Algebra.Field.Base import Math.Algebras.VectorSpace import Math.Algebras.TensorProduct import Math.Algebras.Quaternions

@@ -32,7 +32,7 @@ -- if we had a way to convert a bilinear function to a tensor function  prop_Algebra_Linear ::-    (Num k, Ord b, Algebra k b) =>+    (Eq k, Num k, Ord b, Algebra k b) =>     (k, k, k, k, Vect k b, Vect k b, Vect k b, Vect k b) -> Bool prop_Algebra_Linear (k,l,m,n,x,y,z,w) = --    (unit (k * m + l * n) :: Vect k b) == (add (smultL k (unit m)) (smultL l (unit n)) :: Vect k b) &&@@ -43,7 +43,6 @@ prop_Coalgebra_Linear (k,l,x,y) =     prop_Linear counit' (k,l,x,y) &&     prop_Linear comult (k,l,x,y)--- now need instances for GroupAlgebra etc   -- ALGEBRAS@@ -95,6 +94,17 @@ prop_CoalgebraMorphism f x =     (counit . f) x == counit x &&     ( (f `tf` f) . comult) x == (comult . f) x++-- An antihomomorphism is like a homomorphism except that it reverses the order of multiplication+prop_AlgebraAntiMorphism f (k,x,y) =+    (f . unit) k == unit k &&+    (f . mult) (x `te` y) == (mult . (f `tf` f) . twist) (x `te` y) ++prop_CoalgebraAntiMorphism f x =+    (counit . f) x == counit x &&+    (twist . (f `tf` f) . comult) x == (comult . f) x++prop_HopfAlgebraMorphism f x = (f . antipode) x == (antipode . f) x   -- BIALGEBRAS

@@ -8,7 +8,8 @@ import Test.QuickCheck import Math.Algebras.VectorSpace import Math.Algebras.TensorProduct-import Math.Algebra.Field.Base+import Math.Core.Field+-- import Math.Algebra.Field.Base import Math.Test.TAlgebras.TVectorSpace  import Prelude as P@@ -88,7 +89,7 @@     id = Linear P.id     (Linear f) . (Linear g) = Linear (f P.. g) -instance Num k => Arrow (Linear k) where+instance (Eq k, Num k) => Arrow (Linear k) where     arr f = Linear (fmap f) -- requires nf call afterwards     first (Linear f) = Linear f *** Linear P.id     second (Linear f) = Linear P.id *** Linear f

@@ -8,7 +8,8 @@ import Test.QuickCheck import Math.Algebras.VectorSpace import Math.Algebras.TensorProduct-import Math.Algebra.Field.Base+import Math.Core.Field+-- import Math.Algebra.Field.Base  -- import Control.Monad -- MonadPlus @@ -40,10 +41,10 @@                    d <- arbitrary :: Gen Integer                    return (if d == 0 then fromInteger n else fromInteger n / fromInteger d) -instance (Num k, Ord b, Arbitrary k, Arbitrary b) => Arbitrary (Vect k b) where+instance (Eq k, Num k, Ord b, Arbitrary k, Arbitrary b) => Arbitrary (Vect k b) where     arbitrary = do ts <- arbitrary :: Gen [(b, k)] -- ScopedTypeVariables-                   return $ nf $ V $ take 10 ts-                   -- we impose complexity bound of 10 terms, to avoid unbounded running time.+                   return $ nf $ V $ take 3 ts+-- we impose a complexity bound of 3 terms to limit to 27 terms when testing associativity and ensure reasonable running time  prop_VecSpQn (a,b,x,y,z) = prop_VecSp (a,b,x,y,z)     where types = (a,b,x,y,z) :: (Q, Q, Vect Q EBasis, Vect Q EBasis, Vect Q EBasis)@@ -91,7 +92,7 @@ type LinFun k a b = [(a, Vect k b)] -- a way of representing a linear function as data -linfun :: (Eq a, Ord b, Num k) => LinFun k a b -> Vect k a -> Vect k b+linfun :: (Eq k, Num k, Eq a, Ord b) => LinFun k a b -> Vect k a -> Vect k b linfun avbs = linear f where     f a = case lookup a avbs of           Just vb -> vb@@ -165,11 +166,11 @@     where types = (a,u1,u2,v1,v2) :: (Q, Vect Q EBasis, Vect Q EBasis, Vect Q EBasis, Vect Q EBasis) -} -tensor :: (Num k, Ord a, Ord b) => Vect k (Either a b) -> Vect k (a, b)+tensor :: (Eq k, Num k, Ord a, Ord b) => Vect k (Either a b) -> Vect k (a, b) tensor uv = nf $ V [( (a,b), x*y) | (a,x) <- u, (b,y) <- v]     where V u = p1 uv; V v = p2 uv -bilinear :: (Num k, Ord a, Ord b, Ord c) =>+bilinear :: (Eq k, Num k, Ord a, Ord b, Ord c) =>     ((a, b) -> Vect k c) -> Vect k (Either a b) -> Vect k c bilinear f = linear f . tensor @@ -177,7 +178,7 @@  polymult = bilinear (\(E i, E j) -> return (E (i+j))) -prop_Bilinear :: (Num k, Ord a, Ord b, Ord t) =>+prop_Bilinear :: (Eq k, Num k, Ord a, Ord b, Ord t) =>      (Vect k (Either a b) -> Vect k t) -> (k, Vect k a, Vect k a, Vect k b, Vect k b) -> Bool prop_Bilinear f (a,u1,u2,v1,v2) =     prop_Linear (\v -> f (u1 `dsume` v)) (a,v1,v2) &&

@@ -0,0 +1,178 @@+-- Copyright (c) 2012, David Amos. All rights reserved.++{-# LANGUAGE FlexibleInstances #-}++module Math.Test.TCombinatorics.TCombinatorialHopfAlgebra where++import Data.List as L++import Math.Core.Field++import Math.Algebras.VectorSpace hiding (E)+import Math.Algebras.Structures++import Math.Combinatorics.CombinatorialHopfAlgebra++import Math.Test.TAlgebras.TVectorSpace hiding (T, f)+import Math.Test.TAlgebras.TTensorProduct+import Math.Test.TAlgebras.TStructures++import Test.QuickCheck+import Test.HUnit++quickCheckCombinatorialHopfAlgebra = do+    quickCheckShuffleAlgebra+    quickCheckSSymF+    quickCheckSSymM+    quickCheckYSymF+    quickCheckYSymM+    quickCheckQSymM+    quickCheckQSymF+    quickCheckCHAIsomorphism+    quickCheckCHAMorphism+++instance Arbitrary a => Arbitrary (Shuffle a) where+    arbitrary = fmap (Sh . take 3) arbitrary++quickCheckShuffleAlgebra = do+    putStrLn "Checking shuffle algebra"+    -- quickCheck (prop_Algebra :: (Q, Vect Q (Shuffle Int), Vect Q (Shuffle Int), Vect Q (Shuffle Int)) -> Bool) -- too slow+    quickCheck (prop_Coalgebra :: Vect Q (Shuffle Int) -> Bool)+    quickCheck (prop_Bialgebra :: (Q, Vect Q (Shuffle Int), Vect Q (Shuffle Int)) -> Bool) -- slow+    quickCheck (prop_HopfAlgebra :: Vect Q (Shuffle Int) -> Bool)+++instance Arbitrary SSymF where+    arbitrary = do xs <- elements permsTo3+                   return (SSymF xs)+        where permsTo3 = concatMap (\n -> L.permutations [1..n]) [0..3]++instance Arbitrary SSymM where+    arbitrary = do xs <- elements permsTo3+                   return (SSymM xs)+        where permsTo3 = concatMap (\n -> L.permutations [1..n]) [0..3]++quickCheckSSymF = do+    putStrLn "Checking SSymF"+    -- quickCheck (prop_Algebra :: (Q, Vect Q SSymF, Vect Q SSymF, Vect Q SSymF) -> Bool) -- too slow+    quickCheck (prop_Coalgebra :: Vect Q SSymF -> Bool)+    quickCheck (prop_Bialgebra :: (Q, Vect Q SSymF, Vect Q SSymF) -> Bool)+    quickCheck (prop_HopfAlgebra :: Vect Q SSymF -> Bool)++quickCheckSSymM = do+    putStrLn "Checking SSymM"+    -- quickCheck (prop_Algebra :: (Q, Vect Q SSymM, Vect Q SSymM, Vect Q SSymM) -> Bool) -- too slow+    quickCheck (prop_Coalgebra :: Vect Q SSymM -> Bool)+    -- quickCheck (prop_Bialgebra :: (Q, Vect Q SSymM, Vect Q SSymM) -> Bool) -- too slow+    quickCheck (prop_HopfAlgebra :: Vect Q SSymM -> Bool)+++instance Arbitrary (YSymF ()) where+    arbitrary = fmap (YSymF . shape . descendingTree . take 3) (arbitrary :: Gen [Int])+-- We use descendingTree because it can make trees of interesting shapes from a given list+-- but we could equally have used other tree construction methods such as binary search tree++instance Arbitrary (YSymF Int) where+    arbitrary = fmap (YSymF . descendingTree . take 3) (arbitrary :: Gen [Int])+-- It seems to all work even if we leave the labels on. Perhaps we should really put random labels on though,+-- rather than leaving the descendingTree labels++instance Arbitrary (YSymM) where+    arbitrary = fmap (YSymM . shape . descendingTree . take 3) (arbitrary :: Gen [Int])++quickCheckYSymF = do+    putStrLn "Checking YSymF"+    -- quickCheck (prop_Algebra :: (Q, Vect Q (YSymF ()), Vect Q (YSymF ()), Vect Q (YSymF ())) -> Bool) -- too slow+    quickCheck (prop_Coalgebra :: Vect Q (YSymF ()) -> Bool)+    quickCheck (prop_Bialgebra :: (Q, Vect Q (YSymF ()), Vect Q (YSymF ())) -> Bool)+    quickCheck (prop_HopfAlgebra :: Vect Q (YSymF ()) -> Bool)++quickCheckYSymM = do+    putStrLn "Checking YSymM"+    -- quickCheck (prop_Algebra :: (Q, Vect Q YSymM, Vect Q YSymM, Vect Q YSymM) -> Bool) -- too slow+    quickCheck (prop_Coalgebra :: Vect Q YSymM -> Bool)+    -- quickCheck (prop_Bialgebra :: (Q, Vect Q YSymM, Vect Q YSymM) -> Bool)+    quickCheck (prop_HopfAlgebra :: Vect Q YSymM -> Bool)+++instance Arbitrary QSymM where+    arbitrary = do xs <- elements compositionsTo3+                   return (QSymM xs)+        where compositionsTo3 = concatMap compositions [0..3]++instance Arbitrary QSymF where+    arbitrary = do xs <- elements compositionsTo3+                   return (QSymF xs)+        where compositionsTo3 = concatMap compositions [0..3]++quickCheckQSymM = do+    putStrLn "Checking QSymM"+    quickCheck (prop_Algebra :: (Q, Vect Q QSymM, Vect Q QSymM, Vect Q QSymM) -> Bool) -- too slow+    quickCheck (prop_Coalgebra :: Vect Q QSymM -> Bool)+    quickCheck (prop_Bialgebra :: (Q, Vect Q QSymM, Vect Q QSymM) -> Bool)+    quickCheck (prop_HopfAlgebra :: (Vect Q QSymM) -> Bool)++quickCheckQSymF = do+    putStrLn "Checking QSymF"+    quickCheck (prop_Algebra :: (Q, Vect Q QSymF, Vect Q QSymF, Vect Q QSymF) -> Bool) -- too slow+    quickCheck (prop_Coalgebra :: Vect Q QSymF -> Bool)+    quickCheck (prop_Bialgebra :: (Q, Vect Q QSymF, Vect Q QSymF) -> Bool)+    quickCheck (prop_HopfAlgebra :: (Vect Q QSymF) -> Bool)++quickCheckCHAIsomorphism = do+    putStrLn "Checking CHA isomorphism (change of basis)"+    putStrLn "Checking bijections"+    quickCheck (prop_Id (toSSymF . toSSymM) :: Vect Q SSymF -> Bool)+    quickCheck (prop_Id (toSSymM . toSSymF) :: Vect Q SSymM -> Bool)+    quickCheck (prop_Id (toYSymF . toYSymM) :: Vect Q (YSymF ()) -> Bool)+    quickCheck (prop_Id (toYSymM . toYSymF) :: Vect Q YSymM -> Bool)+    quickCheck (prop_Id (toQSymF . toQSymM) :: Vect Q QSymF -> Bool)+    quickCheck (prop_Id (toQSymM . toQSymF) :: Vect Q QSymM -> Bool)+    putStrLn "Checking morphisms"+    putStrLn "SSym"+    -- quickCheck (prop_AlgebraMorphism toSSymF :: (Q, Vect Q SSymM, Vect Q SSymM) -> Bool) -- too slow+    -- quickCheck (prop_AlgebraMorphism toSSymM :: (Q, Vect Q SSymF, Vect Q SSymF) -> Bool) -- too slow+    quickCheck (prop_CoalgebraMorphism toSSymF :: Vect Q SSymM -> Bool)+    quickCheck (prop_CoalgebraMorphism toSSymM :: Vect Q SSymF -> Bool)+    quickCheck (prop_HopfAlgebraMorphism toSSymM :: Vect Q SSymF -> Bool)+    quickCheck (prop_HopfAlgebraMorphism toSSymF :: Vect Q SSymM -> Bool)+    putStrLn "YSym"+    -- quickCheck (prop_AlgebraMorphism toYSymF :: (Q, Vect Q YSymM, Vect Q YSymM) -> Bool) -- too slow+    quickCheck (prop_AlgebraMorphism toYSymM :: (Q, Vect Q (YSymF ()), Vect Q (YSymF ())) -> Bool)+    quickCheck (prop_CoalgebraMorphism toYSymF :: Vect Q YSymM -> Bool)+    quickCheck (prop_CoalgebraMorphism toYSymM :: Vect Q (YSymF ()) -> Bool)+    quickCheck (prop_HopfAlgebraMorphism toYSymF :: Vect Q YSymM -> Bool)+    quickCheck (prop_HopfAlgebraMorphism toYSymM :: Vect Q (YSymF ()) -> Bool)+    putStrLn "QSym"+    quickCheck (prop_AlgebraMorphism toQSymF :: (Q, Vect Q QSymM, Vect Q QSymM) -> Bool)+    quickCheck (prop_AlgebraMorphism toQSymM :: (Q, Vect Q QSymF, Vect Q QSymF) -> Bool)+    quickCheck (prop_CoalgebraMorphism toQSymF :: Vect Q QSymM -> Bool)+    quickCheck (prop_CoalgebraMorphism toQSymM :: Vect Q QSymF -> Bool)+    quickCheck (prop_HopfAlgebraMorphism toQSymM :: Vect Q QSymF -> Bool)+    quickCheck (prop_HopfAlgebraMorphism toQSymF :: Vect Q QSymM -> Bool)+    where prop_Id f x = f x == x ++quickCheckCHAMorphism = do+    putStrLn "Checking morphisms between CHAs"+    quickCheck (prop_AlgebraMorphism descendingTreeMap :: (Q, Vect Q SSymF, Vect Q SSymF) -> Bool)+    quickCheck (prop_CoalgebraMorphism descendingTreeMap :: Vect Q SSymF -> Bool)+    quickCheck (prop_HopfAlgebraMorphism descendingTreeMap :: Vect Q SSymF -> Bool)+    quickCheck (prop_AlgebraMorphism descentMap :: (Q, Vect Q SSymF, Vect Q SSymF) -> Bool)+    quickCheck (prop_CoalgebraMorphism descentMap :: Vect Q SSymF -> Bool)+    quickCheck (prop_HopfAlgebraMorphism descentMap :: Vect Q SSymF -> Bool)+    quickCheck (prop_AlgebraMorphism leftLeafCompositionMap :: (Q, Vect Q (YSymF ()), Vect Q (YSymF ())) -> Bool)+    quickCheck (prop_CoalgebraMorphism leftLeafCompositionMap :: Vect Q (YSymF ()) -> Bool)+    quickCheck (prop_HopfAlgebraMorphism leftLeafCompositionMap :: Vect Q (YSymF ()) -> Bool)+    quickCheck (\x -> descentMap x == (leftLeafCompositionMap . descendingTreeMap) (x :: Vect Q SSymF))+    -- Coalgebra morphisms showing that various Hopf algebras are cofree+    quickCheck (prop_CoalgebraMorphism ysymmToSh :: Vect Q YSymM -> Bool)+++testlistCHA = TestList [+    TestCase $ assertEqual "toYSymF" (toYSymF $ ysymM $ T (T E () E) () (T (T E () E) () E))+               ( ysymF (T (T E () E) () (T (T E () E) () E)) - ysymF (T (T E () E) () (T E () (T E () E)))+               - ysymF (T E () (T E () (T (T E () E) () E))) + ysymF (T E () (T E () (T E () (T E () E)))) ), -- Loday.pdf, p10+    TestCase $ assertEqual "leftLeafComposition" [2,3,2,1]+               (leftLeafComposition $ T (T (T E 1 E) 2 (T (T E 3 E) 4 E)) 5 (T (T E 6 E) 7 (T E 8 E))) -- Loday.pdf, p6+    ]

@@ -5,6 +5,7 @@ import Test.HUnit import Control.Monad (when, unless) +import Math.Core.Utils (pairs) import Math.Combinatorics.Digraph import Math.Combinatorics.Poset 

@@ -0,0 +1,50 @@+-- Copyright (c) 2012, David Amos. All rights reserved.++module Math.Test.TCore.TUtils where++import Data.List as L++import Test.QuickCheck++import Math.Core.Utils++quickCheckUtils = do+    putStrLn "Testing Math.Core.Utils"+    quickCheck prop_setUnionAsc+    quickCheck prop_multisetSumAsc+    quickCheck prop_multisetSumDesc+    quickCheck prop_diffAsc+    quickCheck prop_diffDesc++prop_setUnionAsc xs ys = setUnionAsc xs' ys' == zs'+    where xs' = toSet xs :: [Int]+          ys' = toSet ys+          zs' = toSet (xs++ys)++prop_multisetSumAsc xs ys = multisetSumAsc xs' ys' == zs'+    where xs' = L.sort xs :: [Int]+          ys' = L.sort ys+          zs' = L.sort (xs ++ ys)++prop_multisetSumDesc xs ys = multisetSumDesc xs' ys' == zs'+    where xs' = reverse (L.sort xs) :: [Int]+          ys' = reverse (L.sort ys)+          zs' = reverse (L.sort (xs ++ ys))++prop_diffAsc xs ys = diffAsc xs' ys' == xs' \\ ys'+    where xs' = L.sort xs :: [Int]+          ys' = L.sort ys++prop_diffDesc xs ys = diffDesc xs' ys' == xs' \\ ys'+    where xs' = reverse (L.sort xs) :: [Int]+          ys' = reverse (L.sort ys)++-- !! Feels like we need a better negative test+-- xs is never submultiset of symmetric difference xs ys, unless null ys+prop_isSubMultisetAsc xs ys = isSubMultisetAsc xs' zs'+                           && (isSubMultisetAsc zs' xs' `implies` null ys)+                           && (isSubMultisetAsc xs' ys' `implies` (length xs <= length ys)) +    where xs' = L.sort xs :: [Int]+          ys' = L.sort ys+          zs' = multisetSumAsc xs' ys'+          implies p q = not p || q

@@ -0,0 +1,47 @@+-- Copyright (c) 2012, David Amos. All rights reserved.++-- {-# LANGUAGE #-}+++module Math.Test.TNumberTheory.TQuadraticField where++import Prelude hiding (sqrt)++import Math.Core.Field+import Math.NumberTheory.QuadraticField++import Test.HUnit+++testlistQuadraticField = TestList [+    testlistMult,+    testlistRecip+    ]+++testcaseMult x y z = TestCase $ assertEqual (show x ++ "*" ++ show y ++ "==" ++ show z) z (x*y)++testlistMult = TestList [+    testcaseMult (sqrt 2) (sqrt 2) 2,+    testcaseMult (sqrt 2) (sqrt 3) (sqrt 6),+    testcaseMult (sqrt 2) (sqrt 6) (2 * sqrt 3),+    testcaseMult i i (-1),+    testcaseMult i (i * sqrt 3) (-1 * sqrt 3),+    testcaseMult (i * sqrt 2) (i * sqrt 3) (-1 * sqrt 6)+    ]+-- We don't bother to test multiplication of sums, because it's obvious by definition that it will work+++testcaseRecip x = TestCase $ assertBool ("recip " ++ show x) (x * recip x == 1)++testlistRecip = TestList [+    testcaseRecip (sqrt 2),+    testcaseRecip i,+    testcaseRecip (sqrt 2 + sqrt 3),+    testcaseRecip (sqrt 2 + 2 * sqrt 3),+    testcaseRecip (sqrt 2 + sqrt 6),+    testcaseRecip (sqrt 2 + sqrt 3 + sqrt 5),+    testcaseRecip (i + 3*sqrt 2 + 2*sqrt 3 - sqrt 5 + 5*sqrt 11)+    ]+-- These tests could be replaced with QuickCheck equivalents, provided we limited the Arbitrary instance+-- to avoid having to solve too large a linear system

@@ -116,8 +116,8 @@ ccTest = and ccTests
 
 ccTests =
-    [conjClassReps (graphAuts2 (c 5)) == [(p [],1),(p [[1,2],[3,5]],5),(p [[1,2,3,4,5]],2),(p [[1,3,5,2,4]],2)]
-    ,conjClassReps (graphAuts2 (q 3)) ==
+    [conjClassReps (graphAuts (c 5)) == [(p [],1),(p [[1,2],[3,5]],5),(p [[1,2,3,4,5]],2),(p [[1,3,5,2,4]],2)]
+    ,conjClassReps (graphAuts (q 3)) ==
         [(p [],1)
         ,(p [[0,1],[2,3],[4,5],[6,7]],3)
         ,(p [[0,1],[2,5],[3,4],[6,7]],6)

@@ -10,11 +10,13 @@ import Math.Test.TRootSystem
 
 import Math.Test.TCore.TField
+import Math.Test.TCore.TUtils
 
 import Math.Test.TAlgebras.TGroupAlgebra
 import Math.Test.TAlgebras.TOctonions
 import Math.Test.TAlgebras.TTensorAlgebra
 import Math.Test.TAlgebras.TTensorProduct
+import Math.Test.TCombinatorics.TCombinatorialHopfAlgebra
 import Math.Test.TCombinatorics.TDigraph
 import Math.Test.TCombinatorics.TFiniteGeometry
 import Math.Test.TCombinatorics.TGraphAuts
@@ -24,12 +26,14 @@ import Math.Test.TCommutativeAlgebra.TPolynomial
 import Math.Test.TCommutativeAlgebra.TGroebnerBasis
 import Math.Test.TNumberTheory.TPrimeFactor
+import Math.Test.TNumberTheory.TQuadraticField
 import Math.Test.TProjects.TMiniquaternionGeometry
 
 
 import Test.QuickCheck
 import Test.HUnit
 
+-- legacy tests - should really be converted to HUnit
 testall = and
     [Math.Test.TGraph.test
     ,Math.Test.TDesign.test
@@ -43,6 +47,7 @@ quickCheckAll =
     do
     -- quickCheck prop_NonCommRingNPoly
+    quickCheckUtils
     quickCheck prop_GroupPerm
     quickCheckField
     quickCheckTensorProduct
@@ -54,9 +59,11 @@     putStrLn "Testing miniquaternion geometries..."
     quickCheck prop_NearFieldF9
     quickCheck prop_NearFieldJ9
+    quickCheckCombinatorialHopfAlgebra
 
 hunitAll = runTestTT $ TestList [
     testlistGroupAlgebra,
+    testlistCHA,
     testlistDigraph,
     testlistFiniteGeometry,
     testlistGraphAuts,
@@ -65,5 +72,6 @@     testlistPoset,
     testlistPolynomial,
     testlistGroebnerBasis,
-    testlistPrimeFactor
+    testlistPrimeFactor,
+    testlistQuadraticField
     ]