Flag statistics from the Ehrhart h * -polynomial of multi-hypersimplices

It is known that the normalized volume of standard hypersimplices (deﬁned as some slices of the unit hypercube) are the Eulerian numbers. More generally, a recent conjecture of Stanley relates the Ehrhart series of hypersimplices with descents and excedences in permutations. This conjecture was proved by Nan Li, who also gave a generalization to colored permutations. In this article, we give another generalization to colored permutations, using the ﬂag statistics introduced by Foata and Han. We obtain in particular a new proof of Stanley’s conjecture, and some combinatorial identities relating pairs of Eulerian statistics on colored permutations.


Introduction
A modern combinatorial definition of the Eulerian numbers A n,k is given by counting descents in permutations: A n,k := #{σ ∈ S n : des(σ) = k − 1}. ( Foata suggested in [6] the problem that we describe below.It is known that the Eulerian numbers A n,k satisfy this is essentially a calculation due to Laplace (see [6] for details).But the combinatorial definition can be easily translated in the following way: The problem is to find a measure-preserving bijection between the two sets, to explain why they have the same volume.A simple solution was given by Stanley [12].The set {v ∈ [0, 1] n : k ≤ v i ≤ k + 1} is in fact a convex integral polytope known as the hypersimplex, and in this context we can consider the h * -polynomial, which is a generalization of the volume.This led to a recent conjecture by Stanley about the h * -polynomial of the hypersimplex (more precisely, a partially open version of the hypersimplex), which was proved by Nan Li [10] in two different ways.It is remarkable that two Eulerian statistics are needed to state the conjecture, which says that the h * -polynomial of a hypersimplex is the descent generating function for permutations with a given number of excedences.Nan Li also extended the result to colored permutations by considering the hypercube [0, r] n for some integer r > 0, and the polytopes {v ∈ [0, r] n : k ≤ v i ≤ k + 1} are the multi-hypersimplices referred to in the title of this article.We would like to mention that besides Stanley's conjecture, some recent works deals with the geometry and combinatorics of hypersimplices, see [8,11].
The goal of this article is to give another generalization of Stanley's conjecture to colored permutations.Our result is stated in terms of the flag descents and flag excedences in colored permutations, and relies on some related work by Foata and Han [5].Our method gives in particular a new proof of Stanley's conjecture in the uncolored case.Our method can roughly be described as follows.We first consider the case of the half-open hypercube [0, r) n , where an analog of Stanley's conjecture in terms of descents and inverse descents can be proved in a rather elementary way.We can relate the half-open hypercube [0, r) n with the usual hypercube [0, r] n via an inclusion-exclusion argument.Then, it remains only to prove an identity relating two generating functions for colored permutations.
This article is organized as follows.Section 2 contains some preliminaries.Our main results are Theorems 17 and 18, whose particular case r = 1 gives Stanley's conjecture.The core of the proof is in Sections 3 and 4, but it also relies on some combinatorial results on colored permutations which are in Sections 5, 6 and 7.
polynomial E(X , t) is defined as the unique polynomial in t such that, for any integer t > 0, where tX := {tx : x ∈ X }.The h * -polynomial of X is defined as From a general result of Stanley [13], the series E * (X , z) is in fact a polynomial with positive integral coefficients.As it often happens, it is an interesting problem to find their combinatorial meaning for special polytopes.Perhaps the most basic example is the unit hypercube [0, 1] n which has the nth Eulerian polynomial as h * -polynomial, as will be detailed below.Although we do not use this language here, a general method to find the h * -polynomial of a polytope is to use unimodular shellable triangulations (as was done for example in [10]) and it is essentially the idea behind what follows.
We define the standardization std(v) of v to be the unique permutation σ ∈ S n such that for all i < j we have v i ≤ v j iff σ i < σ j .For each permutation σ ∈ S n , let For example, one can check that if x < y < z, std(x, y, x, z, y, x, x) = 1527634.Note that the unit hypercube [0, 1] n is the disjoint union of the subsets S σ for σ ∈ S n .Geometrically, each S σ is a unit simplex where some facets are removed.So it is not a polytope in the usual sense; we should call it a "partially open" polytope.But note that Equations (2) and (3) make sense even when X is not a polytope, so in particular E * (S σ , z) is well defined, and we have: The number E(S σ , t) counts such sequences with the additional condition that all elements are integers between 0 and t.By defining we have a bijection with integer sequences satisfying 0 The expansion permits to finish the proof.Note that their coefficients are the numbers A n,k defined in Equation (1).
For example, A 0 (z) = A 1 (z) = 1, A 2 (z) = 1 + z, A 3 (z) = 1 + 4z + z 2 , etc. Since the h * -polynomial is additive with respect to disjoint union, we get from the previous lemma that Note that, since the Ehrhart polynomial of [0, 1] n is clearly (t + 1) n , we have proved the classical identity: Let us turn to the case of the half-open hypercube [0, 1) n .Note that we are now dealing with a half-open polytope, i.e., a polytope where some of the (n − 1)-dimensional faces are removed.In this case, it is not a priori clear that the h * -polynomial is a polynomial with nonnegative integral coefficients.We can decompose [0, 1) n as the disjoint union of the polytopes: These are also simplices where some facets are removed, and we get: Proof.It is similar to the one of Lemma 2.
Thus, the half-open hypercube has the h * -polynomial zA n (z).Besides, its Ehrhart polynomial is clearly E([0, 1) n , t) = t n .Once again we get Identity (4), with an additional factor z. Let us present another example of a h * -polynomial that will be used in the sequel.It is presented in [14,Section 7.19] in the context of quasi-symmetric functions.Let λ be a Young diagram (we use the French notation).Let Z λ (respectively, R λ ) denote the set of fillings of λ with integers (respectively, real numbers).And let Y λ denote the set of semi-standard fillings of λ by real numbers in (0, 1] where semi-standard mean weakly increasing in rows and strictly increasing in columns.Clearly, Y λ is a (partially open) convex polytope in R λ .A semi-standard tableau with largest entry less than t is just an element of Z λ ∩ tY λ , so that the Schur function s λ where t variables are set to 1 and the others to 0. Let SY T (λ) denote the set of standard tableaux of shape λ, and recall that a descent of a standard tableau is an entry i such that the entry i + 1 is in an upper row.Let des(T ) denote the number of descents of a standard tableau, then we have: z des(T )+1 . ( Let us sketch the proof.The reading word w(T ) of a semi-standard tableau T ∈ Y λ is defined by ordering its entries row by row, from left to right and from top to bottom.Then, the standardization std(T ) is defined to be the unique standard tableau U of the same shape such that std(w(T )) = w(U ).The set Y λ is partitioned into the subsets where U ∈ SY T (λ).Now, the previous proposition is a consequence of the following: Proof.This is essentially the same as Proposition 4.
It is in order to precise the link with quasi-symmetric functions mentionned above.Semi-standard tableaux of shape λ can be seen as (P, ω)-partitions in the sense of [14].In general, (P, ω)-partitions are the integral points of a polytope called the order polytope, and quasi-symmetric functions are useful in this context to compute the h * -polynomial.

The generalization of Stanley's bijection
In this section we adapt Stanley's bijection from [12] to the case of the half-open hypercube [0, r) n (for some integer r > 0), and r-colored permutations.Note that a similar generalization was given by Steingrímsson in [ We will not use here the group structure of colored permutations.Still, note that flag descents are indeed related with it [2].
To define our generalization of Stanley's bijection, let (a i ) 1≤i≤n ∈ [0, r) n , and let a 0 = 0. Then the map φ((a i ) 1≤i≤n ) = (b i ) 1≤i≤n is defined as follows: From this definition we get: where the modulo means that we take the unique representative in [0, r).In fact, it is elementary to check that Equations ( 6) and ( 7) define two inverse bijections from [0, r) n to itself. and For each colored permutation (σ, c) we define the translated simplex: n where c i = v i and σ = std(v 1 mod 1, . . ., v n mod 1).
Proof.From ( 6) and keeping the notation we get Proof.Let v ∈ T (σ,c) , then the condition v i ≤ v i+1 or v i > v i+1 only depends on (σ, c).So from its definition in (6), we see that the restriction of φ to T (σ,c) is equal to an affine map that sends Z n to itself.It follows that φ(T (σ,c) ) has the same h * -polynomial as T (σ,c) .Besides, since T (σ,c) is a translation of T σ by an integer vector, they have the same h *polynomial, which is therefore z des(σ −1 )+1 by Lemma 4.
Proof.Using Lemma 9, we get: From Lemma 10 and the fact that φ is a bijection, we have: From Lemma 11 and the fact that the h * -polynomial is additive with respect to disjoint union, we get the result.
4 From the half-open hypercube to the closed hypercube Definition 13.The multi-hypersimplices are the polytopes defined by: These polytopes form a particular class of the ones introduced by Lam and Postnikov [9] under the same name.The polytopes B (1) n,k are simply called the hypersimplices, and they can be described geometrically as truncated simplices (this fact is essentially due to Coxeter [4,Section 8.7]).
Note that Proof.For each ∆ ⊂ {1, . . ., n}, let By removing the coordinates equal to r, we see that H ∆ is in bijection with A (r) n−j,k−rj where j = #∆.The bijection preserves integral points, and this holds with the convention that both polytopes are empty if k − rj < 0. Hence: The previous proposition is conveniently rewritten in terms of the generating functions.Let then these two series are related as stated below.
Theorem 15.The following identity holds: Proof.From Proposition 14, we get: and the result follows.
Together with Proposition 12, the relation in the previous theorem shows that a generalization of Stanley's conjecture can be obtained via an identity on generating functions.This identity will be presented in the next sections.We first need some definitions to state the result.
In particular, let us mention that the statistics fexc and fdes * are equidistributed on S (r) n , see [5,Theorem 1.4].We will also give a proof that fdes and fdes * are equidistributed in the next section.
Proof.Let C n,k (z) denote the right-hand side of the equation, and we use the same convention as with n,k (z)y k , and x n n! .
From Proposition 12 and Theorem 24 in the sequel, we have Comparing with Theorem 15 shows that we have B (r) (x, y, z) = C (r) (x, y, z), which proves the theorem.
We have in fact another result, which is not trivially equivalent to the previous one: Proof.This is a consequence of the previous theorem, together with the bijection in Section 7.
Proof.Let w k = k j=1 c j + des(σ 1 , . . ., σ k ) for k = 1, . . .n so that c i = w i mod r.Clearly, w 1 , . . ., w n is a nondecreasing sequence and w n = cdes(σ, c).We can write w n = qr + c n for a unique q.This integer q counts the number of positive multiples of r that are smaller than w n .Using the fact that w k − w k−1 ≤ r and w 1 < r, we have: To count the cardinality of this set, we distinguish two cases.If w i−1 − w i = r, it means that c i = r − 1 and σ i−1 > σ i .From the definition of the bijection, this is equivalent to c i = c i+1 and σ i−1 > σ i .Otherwise, w i−1 − w i < r.We can see that this case is equivalent to c i−1 > c i .Hence, we obtain q = des(σ, c ), and w n = fdes(σ, c ).
It is also possible to define a bijection α * by α * (σ, c) = (σ, c ) where As in the case of the previous proposition, we can prove fdes * (σ, c ) = cdes(σ, c).In particular, it follows that fdes and fdes * are equidistributed.
The bijection α only changes the colors c i , and not the permutation σ, so we have: .
But the right-hand side clearly can be factorized, so that with the notation we have: A formula for the case r = 1 is given in the proposition below.This is in fact a particular case of a result of Garsia and Gessel [7,Theorem 2.3], but we also include a short proof based on the Robinson-Schensted correspondence.
Proposition 21.For r = 1, we have: Proof.Let Par(n) denote the set of integer partitions of n.By the Robinson-Schensted correspondence, we have: y des(P )+1 z des(Q)+1 .
So, using Equation ( 5), we get: By the Cauchy identity on Schur functions, we have This ends the proof.
From Equation ( 9) and the previous proposition, we deduce: Note that another consequence of Equation ( 9), together with Equation (4), is the following (which is not a new result, see for example [2]).

Identities on bi-Eulerian generating functions
We keep the definition of A (r) n (y, z) and A (r) (x, y, z) as before, but in this section we only need the formula in Equation (10).We recall that C n (y, z), and C(x, y, z) were defined in the proof of Theorem 17.The goal of this section is to prove the following relation between the two generating functions for colored permutations: Theorem 24.C (r) (x, y, z) = e (1−z)y r x A (r) (x, y, z).
The particular case q = 1 of [5,Theorem 5.11], after an easy simplification, gives the following formula: where .
Next, we define a linear operator β on power series in z by β(z k ) = z k/r .So: From Equation (12), we get n≥0 β(W n (y, z)) x and from the definition of F k we get .
From Equation ( 13) and the previous equation, and after the substitution (x, y) ← (xy r (1− z), y −1 ), we reach: Besides, from Equation (10), we have: After the substition x ← x(1 − y r )(1 − z), we obtain: Proof of Theorem 24.Since Theorem 24 is a relation on exponential generating functions, it is convenient to use the Laplace transform.It sends a function f (x) to

L
x k k! , x, s = 1 s k+1 .We have: By Theorem 26, with s = s − (1 − z)y r , the latter expression is equal to we get: Besides, from Theorem 25, we also get: So we have proved L C (r) (x, y, z), x, s = L e (1−z)y r x A (r) (x, y, z), x, s , which completes the proof of Theorem 24.

Another combinatorial model
We give in this section a bijective proof of by defining an involution I on colored permutations such that y fexc(σ,c) z fdes(σ,c)/r = y fexc(I(σ,c)) z fdes * (I(σ,c))/r .
Let (σ, c) ∈ S n .We consider (σ, c) as a word whose successive letters are (σ 1 , c 1 ), (σ 2 , c 2 ), . . ., (σ n , c n ).Note that the pair (σ i , c i ) is considered as a letter with color c i .Then, we consider the unique factorization where each block B i contains letters of the same color, and m is minimal.The involution I is defined by permuting the blocks, following these two conditions: • each zero colored block stays at the same location, • each maximal sequence of nonzero colored blocks B j . . .B k is replaced with B k . . .B j (maximal means that B j−1 is zero colored or j = 1, and B k+1 is zero colored or k = n).
Proof.This is immediate, since the letters with color 0 are unchanged by I (and the sum of the colors is also preserved).
Proof.We compute fdes(σ, c)/r on one side and fdes * (I(σ, c))/r and the other side, by examining the different contributions to each quantity.First, each pair of letters (σ i , c i )(σ i+1 , c i+1 ) where σ i > σ i+1 inside a given block B j contribute by 1 to each side (since c i = c i+1 by definition of the blocks).It remains to consider the term r × #{i : c i > c i+1 } + c n in the definition of fdes, and the term r × #{i : c i < c i+1 } + c 1 in the definition of fdes * .
Let us write B i > B i+1 or B i < B i+1 to mean that the color of the block B i is greater or smaller than that of B i+1 (by definition they cannot be equal).Let j < k be such that B j and B k are zero colored blocks, but B j+1 , . . ., B k−1 are not.In the factor B j . . .B k of (σ, c), there is a contribution

#
i : j ≤ i < k and B i > B i+1to fdes(σ, c)/r .But in the factor B j B k−1 . . .B j+1 B k of I(σ, c), there is the same contribution to fdes * (I(σ, c))/r .Now, let B j be the first zero colored block of (σ, c).If j > 1, the prefix B 1 . . .B j of (σ, c) contributes by # i : 1 ≤ i < j and B i > B i+1to fdes(σ, c)/r , and the prefix B j−1 . . .B 1 B j of I(σ, c) contributes by1 + # i : j − 1 > i ≥ 1 and B i+1 < B ito fdes * (I(σ, c))/r (the 1 come from the term c 1 in the definition of fdes * since B j−1 is a nonzero colored block).The two numbers are easily seen to be equal.Similarly, let B k be the last zero colored block of (σ, c).If k < m, the suffix B k . . .B m of (σ, c) contributes by1 + # i : k ≤ i < m and B i > B i+1to fdes(σ, c)/r (the 1 come from the term c n in the definition of fdes since B m is a nonzero colored block), and the suffix B k B m . . .B k+1 of I(σ, c) contributes by 1 + # i : m > i ≥ k + 1 and B i+1 < B i to fdes * (I(σ, c))/r (the 1 come from the fact that B k < B m ).The two numbers are easily seen to be equal.Checking the respective definitions of fdes and fdes * , we can see that what we have counted proves the proposition.