On q-integrals over order polytopes ( extended abstract )

A q-integral over an order polytope coming from a poset is interpreted as a generating function of linear extensions of the poset. As an application, the q-beta integral and a q-analog of Dirichlet’s integral are computed. A combinatorial interpretation of a q-Selberg integral is also obtained. Résumé. Une q-intégrale sur un polytope provenant d’un poset est interprété comme une série génératrice d’extensions linéaires de la poset. En application, l’intégrale q-bêta et un q-analogue de l’intégrale de Dirichlet sont calculés. Une interprétation combinatoire de une intégrale q-Selberg est également obtenue.


Introduction
In this extended abstract we give a combinatorial interpretation of q-integrals over order polytopes.The motivation of this extended abstract is to generalize Stanley's combinatorial interpretation of the Selberg integral.
The Selberg integral is the following integral first evaluated by Selberg [8] in 1944: = n j=1 Γ(α + (j − 1)γ)Γ(β + (j − 1)γ)Γ(1 + jγ) where n is a positive integer and α, β, γ are complex numbers such that Re(α) > 0, Re(β) > 0, and Re(γ) > − min{1/n, Re(α)/(n − 1), Re(β)/(n − 1)}.Stanley [9, Exercise 1.10 (b)] found a combinatorial interpretation of the above integral when α − 1, β − 1 and 2γ are nonnegative integers in terms of permutations.His idea is to interprete the integral as the probability that a random permutation satisfies certain properties.This idea uses the fact that a real number x ∈ (0, 1) can be understood as the probability that a random number selected from (0, 1) lies on an interval of length x is equal to x. Generalizing this fact to q-integrals is not obvious.We instead consider a different approach by interpreting q-integrals as generating functions in q.
Throughout this extended abstract we assume 0 < q < 1.We will use the following notation for q-series: We also use the notation [n] := {1, 2, . . ., n}.We denote by S n the set of permutations on [n].
In order to state our main results, we need several definitions.First, recall that the q-integral of a function Note that the limit as q → 1, the q-integral becomes the usual integral.It is well known that b a For a permutation π = π 1 . . .π n ∈ S n we will denote the region The q-integral over this region is defined as follows.
Definition 1.1.For a permutation π = π 1 . . .π n ∈ S n , we define where For example, Note that since s i and t i are constants when x i+1 , . . ., x n and q are fixed, the above definition makes sense as a q-integral.
Let P be a poset on a set {x 1 , x 2 , . . ., x n }.By abuse of notation, we will consider x i as an element of P and also as a variable.For i ∈ [n], we denote by P i the subposet of P consisting of x i , x i+1 , . . ., x n .Let O b a (P ) be the polytope We will also use O(P ) in place of O 1 0 (P ).We define the q-integral over this polytope as follows.
Definition 1.2.The q-integral of f (x 1 , . . ., x n ) over the polytope O b a (P ) is defined by where By definition we have Note that if P is the chain The remainder of this extended abstract is organised as follows.In Section 2 we will find a formula for the q-integral over a simplex.In Section 3 we show that the q-volume of the order polytope O(P ) can be written as a generating function for linear extensions of the poset P .In Section 4 we find a relation between q-integrals over order polytopes O(P ) and O(Q) for two posets P and Q such that Q is obtained from P by adding a chain.In Section 5 we show that the q-beta integral and a q-analog of Dirichlet's integral can be computed using our methods.We also discuss a connection with the general q-beta integral of Andrews and Askey [2].In Section 6 we express a q-Selberg integral in terms of the q-volume of certain order polytope and find a combinatorial interpretation for it.

q-integrals over simplices
In this section we compute q-integrals of a multivariate function over the simplex Let Des(π) be the set of descents of π.We define des(π) and maj(π) to be the number of descents of π and the sum of descents of π, respectively.Lemma 2.1.For π ∈ S n and r, s ∈ {0, 1, 2, . . .} ∪ {∞}, we have The following theorem gives a formula for the volume of the simplex.
Theorem 2.2.For π ∈ S n and real numbers a < b, we have When a = 0 and b = 1 in Theorem 2.2 we obtain the following corollary.
3 q-integrals over order polytopes In this section we consider q-integrals over order polytopes.We need some of the P -partition theory in [9, Chapter 3].
) for all i ∈ Des(π).Then for any function f : [n] → N, there is a unique π ∈ S n for which f is π-compatible.
Let P be a poset with n elements.A labeling of P is a bijection ω : A linear extension of P is an arrangement (t 1 , t 2 , . . ., t n ) of the elements in P such that if t i < P t j then i < j.The Jordan-Hölder set L(P, ω) is the set of permutations of the form ω(t 1 )ω(t 2 ) • • • ω(t n ) for some linear extension (t 1 , t 2 , . . ., t n ) of P .
Let A(P, ω) denote the set of (P, ω)-partitions.For a permutation π ∈ S n , we denote by S π (P, ω) the set of functions σ : P → N such that σ • ω −1 is π-compatible.Notice that in the definition of S π (P, ω) we only need the underlying set of P .Thus we can consider S π (P, ω) when P is a set with n elements and ω : P → [n] is a bijection.We will use the following facts [9, Lemma 3.15.3,Theorem 3.15.7]: σ∈A(P,ω) Lemma 2.1 can be restated as follows.

Changing posets
In this section we consider two posets P and Q where Q is obtained from P by adding a chain.The results in this section will be used for the next two sections.Lemma 4.1.Let ρ ∈ S m .Let P be a poset on {x 1 , . . ., x n } with x s ≤ P x t .Define Q to be the poset on {x 1 , . . ., x n , y 1 , y 2 , . . ., y m } with relations x i ≤ Q x j if and only if x i ≤ P x j , and Then, we have Moreover, (9) holds if the order of integration is obtained from d q x 1 • • • d q x n by inserting d q y 1 • • • d q y m anywhere between x s and x t .
Similarly we can prove the following lemma.

Examples
In this section we will compute the q-beta integral and a q-analog of Dirichlet's integral using our results.We will then find a connection with the general q-beta integral due to Andrews and Askey [2].

The q-beta integral
The following is the well known integral called the q-beta integral.We can prove this using our methods.
Corollary 5.1.We have Proof.By (10) and (11) we have By Corollary 2.3, we get the q-beta integral formula.
Corollary 5.2.Let π be a permutation on [n].Let T be the poset obtained from the chain {π 1 < π 2 < • • • < π n } by adding k i elements covered by π i .Then T becomes a tree and for each element v we define the hook length h v of v to be the number of elements u with u ≤ T v. Let maj(T ) = i∈Des(π) h πi .Then Proof.This can be shown by applying (11) to each factor x ki i .

A q-analog of Dirichlet integral
We now consider the simplex Dirichlet's integral is the following, see [1, Theorem 1.8.1]: By introducing new variables We obtain a q-analog of (13) as follows.
Corollary 5.3.For nonnegative integers k 1 , . . ., k n , we have Proof.By applying (11) to the factor y k1 1 and ( 9) to the factors (qy i−1 /y i ; q) ki y ki i for 2 ≤ i ≤ n, the left hand side is equal to where By Corollary 2.3 we obtain the right hand side.

The general q-beta integral of Andrews and Askey
In this subsection, we will show that Theorem 2.2 is related to the following result of Andrews and Askey [2] on a generalization of the q-beta integral: Let π ∈ S n .We will compute the integral in Theorem 2.2 in a different way.First we decompose π into π = σnτ using the largest integer n.Suppose that σ and τ have r and s letters respectively and where y 1 , . . ., y r and z 1 , . . ., z s are obtained by rearranging x σ1 , . . ., x σr and x τ1 , . . ., x τs respectively so that subscripts are increasing.By applying Theorem 2.2 to the two inside integrals, the above is equal to b a x r q maj(σ) [r] q !(aq −k1 /x; q) r b s q maj(τ ) [s] q !(xq −k2 /b; q) s d q x.
6 q-Selberg integrals In this section we will find a combinatorial interpretation for a q-Selberg integral.
For a set of variables x = (x 1 , . . ., x n ) we denote There are many generalizations of the Selberg integral, see [4].In this section we consider the following two q-Selberg integrals: where ∆ n,m (x) = 1≤i<j≤n x m j q 1−m x i /x j ; q m x m j (x i /x j ; q) m = ∆(x) It is easy to check that ∆ n,m (x) is symmetric in the variables x 1 , x 2 , . . ., x n .Askey [3] conjectured that which has been proved by Habsieger [5] and Kadell [6] independently.Kadell [7, Eq. (4.11)] showed that Since the integrand of KS n (α, β, m) is symmetric under any permutation of x 1 , . . ., x n and zero whenever x i = x j , we have Thus we have By ( 16) and (17), we have We define the Selberg poset P (n, r, s, m) to be the poset in which the elements are x i , y i , z i , w with i = j, and the covering relations are as follows: For an example of P (n, r, s, m), see Figure 1.
The following theorem implies that the q-Selberg integral is the q-volume of an order polytope up to a certain factor.Theorem 6.1.We have x r i (qx i ; q) s 1≤i<j≤n x m j q 1−m x i /x j ; q m x m j (x i /x j ; q) m d q x 1 • • • d q x n = q −( m 2 )( n 2 ) ([r] q !) n ([s] q !) n ([m] q !) 2( n 2 ) O(P (n,r,s,m)) where the order of integration d q W d q Y d q Xd q Z is given by In the order of integration (19), the order of d q w (k) i,j and d q w (k) i ,j for (i, j) = (i , j ) does not matter.The following corollary gives a combinatorial interpretation for the q-Selberg integral in terms of linear extensions.

Lemma
w