The (1/k)-Eulerian Polynomials

We use the theory of lecture hall partitions to de ne a generalization of the Eulerian polynomials, for each positive integer k. We show that these 1=k-Eulerian polynomials have a simple combinatorial interpretation in terms of a single statistic on generalized inversion sequences. The theory provides a geometric realization of the polynomials as the h -polynomials of k-lecture hall polytopes. Many of the de ning relations of the Eulerian polynomials have natural 1=k-generalizations. In fact, these properties extend to a bivariate generalization obtained by replacing 1=k by a continuous variable. The bivariate polynomials have appeared in the work of Carlitz, Dillon, and Roselle on Eulerian numbers of higher order and, more recently, in the theory of rook polynomials. 1 Overview The Eulerian polynomials, An(x), can be de ned, for n 0, by any of the following relations: An(x) = X 2Sn x ; (1) X t 0 (t+ 1)x = An(x) (1 x)n+1 ; (2) X n 0 An(x) z n! = (1 x) ez(x 1) x : (3) the electronic journal of combinatorics 19 (2012), #P9 1 In (1), Sn is the set of permutations : f1; 2; : : : ; ng ! f1; 2; : : : ; ng and des( ) is the number of i such that (i) > (i+ 1). Referring to Foata's survey [9] on the history of the Eulerian polynomials, (2) and (3) are due to Euler [8] and (1) is due to Riordan [16]. In this paper, for positive integers k, we de ne the 1=k-Eulerian polynomial combinatorially, as the distribution of a certain statistic \asc" over a set of \k-inversion sequences", In;k, speci ed in the next subsection. These polynomials arise naturally in the theory of lecture hall partitions, via an associated \k-lecture hall polytope". We will show that the Ehrhart polynomial of the k-lecture hall polytope can be computed explicitly. Consequently, the exponential generating of the 1=k-Eulerian polynomials can be derived to establish the following relations analogous to (1) (3): The 1=k-Eulerian polynomials, A (k) n (x), can be de ned for n 0 by any of the following relations: A n (x) = X e2In;k x; (4) X t 0 t 1 + 1 k t (kt+ 1)x = A (k) n (x) (1 x) 1 k ; (5) X n 0 A n (x) z n! = 1 x ekz(x 1) x 1 k : (6) Their name is derived from (6) where their exponential generating function is the 1=k-th power of a k-generalization of (3). Our main contribution is to show that the 1=k-Eulerian polynomials have a simple combinatorial interpretation in terms of inversion sequences and a geometric realization in terms of lecture hall polytopes. 1.1 Inversion sequences and ascents In eq. (4), the sum is over the set In;k of k-inversion sequences de ned by In;k = fe 2 Z n j 0 ei (i 1)kg: (7) For e 2 In;k, asc(e) is the number of ascents of e, de ned as asc(e) = # i : 1 i n 1 ei (i 1)k + 1 < ei+1 ik + 1 : Note the somewhat unusual de nition of \ascent". See Figure 1 for an example of the computation of A (2) 3 (x) from (4) using these de nitions. the electronic journal of combinatorics 19 (2012), #P9 2 1.2 The 1=k-Eulerian numbers De ne the 1=k-Eulerian numbers a (k) n;j by a (k) n;j = #fe 2 In;k j asc(e) = jg. The triangle of 1=2-Eulerian numbers is shown below, for 1 n 7. 1 1 2 1 10 4 1 36 60 8 1 116 516 296 16 1 358 3508 5168 1328 32 1 1086 21120 64240 42960 5664 64 The third row corresponds to the polynomial in Figure 1. Of course, from the de nition, Pn 1 j=0 a (k) n;j = Qn 1 i=0 (ik + 1). 1.3 Lecture hall polytopes It is a bit surprising that the polynomials A (k) n (x) de ned by (6) have the simple combinatorial interpretation (4). This interpretation has its roots in the theory of lecture hall partitions [1, 2]. In Section 2, the equivalence of (4) and (5) is established using recent work of Savage and Schuster relating lecture hall polytopes to statistics on inversion sequences [17]. It follows from Theorem 5 in [17] that A (k) n (x), de ned by (4) has a geometric interpretation as the h -vector of the k-lecture hall polytope, Pn;k, de ned by Pn;k = 2 R j 0 1 1 2 k + 1 3 2k + 1 : : : n (n 1)k + 1 1 : (8) A key part of the proof of the equivalence of (4) and (5) in Section 2 is a rather involved, explicit computation of the Ehrhart polynomial of Pn;k. All de nitions and background are provided in Section 2. 1.4 Generalizing properties of the Eulerian polynomials In addition to properties (1) (3), the Eulerian polynomials satisfy many relations, including: a recursive de nition as an n-term sum; a two-term di erential recurrence; a di erential operator de nition; a recurrence for the coe cients; an explicit formula for the coe cients; and a Worpitzky identity (See [9]). All of these properties can be generalized to the 1=k-Eulerian polynomials. the electronic journal of combinatorics 19 (2012), #P9 3 e ascents asc(e) e ascents asc(e) e ascents asc(e) 000 fg 0 010 f1g 1 020 f1g 1 001 f2g 1 011 f1g 1 021 f1g 1 002 f2g 1 012 f1; 2g 2 022 f1g 1 003 f2g 1 013 f1; 2g 2 023 f1g 1 004 f2g 1 014 f1; 2g 2 024 f1; 2g 2 Figure 1: Computation of A (2) 3 (x) = 1 + 10x+ 4x 2 using (4). In order to do so, in Section 3, we view them in a more general setting. In the process, we will uncover a connection between the 1=k-Eulerian polynomials and previous work. To this end, for n 0, de ne the bivariate polynomial Fn(x; y) by X t 0 t+ y 1 t (t+ y)x = Fn(x; y) (1 x)n+y : (9) Then from (5), the relationship between A (k) n (x) and Fn(x; y) is given by A n (x) = k Fn(x; 1=k): (10) This provides further motivation for the term \1=k-Eulerian polynomials". In Section 3, we prove that all of the aforementioned properties of the Eulerian polynomials generalize to Fn(x; y) and thereby to the 1=k-Eulerian polynomials. Most of these identities appear in some form in earlier work and we make the connections in Section 4. However, it is unexpected that the non-integral case of y = 1=k should be so interesting. 1.5 The combinatorics of Fn(x; y) In concluding this overview, we highlight one particular outcome of Section 3. It turns out that Fn(x; y) has a simple interpretation in terms of the statistics \excedance" and \number of cycles" on permutations. The excedance of a permutation 2 Sn is de ned by exc( ) = #fi j (i) > ig: Recall that every 2 Sn can be decomposed uniquely as the product of disjoint cycles. The number of such cycles is denoted by #cyc( ). The last relation we prove in Section 3 is that Fn(x; y) = X 2Sn x y : (11) As discussed in Section 4, this relationship has appeared in various forms elsewhere in the literature. However, the following two consequences of (11) are relevant here. First, the electronic journal of combinatorics 19 (2012), #P9 4 combining (11), (10) and (4), we have X e2In;k x = X 2Sn x k #cyc( : This gives further evidence that the k-inversion sequences and their associated ascent statistic are encoding something of combinatorial signi cance. Secondly, our results provide a geometric interpretation of the joint distribution (11) in terms of the k-lecture hall polytope de ned by (8), in the special case that y is the reciprocal of an integer. The variable y that tracks the number of cycles in a permutation is related to the angles at which the faces of the polytope meet. In Section 4 we discuss connections between Section 3 and other work in the literature and suggest some further directions for inquiry. 2 The geometry of the 1=k-Eulerian polynomials 2.1 Lecture hall polytopes and inversion sequences For a sequence s = fsigi 1 of positive integers, the s-lecture hall polytope P (s) n is de ned by P n = 2 R 0 1 s1 2 s2 n sn 1 : These polytopes, introduced in [17], were named after the lecture hall partitions [1, 2] of Bousquet-Melou and Eriksson. Let tP (s) n = ft j 2 P (s) n g denote the t-th dilation of P (s) n . De ne i(P (s) n ; t) by i(P n ; t) = jtP (s) n \ Z j: Since all vertices of P (s) n have integer coordinates, i(P (s) n ; t) is a rational polynomial in t, known as the Ehrhart polynomial of P (s) n [6, 7]. There is a relationship between the Ehrhart polynomial of P (s) n and the distribution of a certain statistic on s-inversion sequences, as we now describe. For a sequence s = fsigi 1 of positive integers, de ne the set I (s) n of s-inversion sequences of length n by I n = f(e1; : : : ; en) 2 Z n j 0 ei < si for 1 i ng : When s = (1; 2; : : : ; n), I (s) n is the familiar set of inversion sequences in bijection with Sn. For e 2 I (s) n , an ascent of e is a position i such that 1 i < n and ei si < ei+1 si+1 : the electronic journal of combinatorics 19 (2012), #P9 5 In addition, if e1 > 0 then 0 is an ascent of e. Let asc(e) be the number of ascents of e. For example, if s = (5; 3; 7), then e = (3; 2; 3) is an s-inversion sequence with ascents in positions 0 and 1, but not in position 2. As another example, if s = (1; 4; 7), then e = (0; 3; 4), as an s-inversion sequence, has an ascent only in position 1. In fact, no (1; 4; 7)-inversion sequence will have 0 as as ascent. The following relationship between the s-inversion sequences and the s-lecture hall polytopes was established in [17]. Theorem 1. [17] Let s be any sequence of positive integers. For integer n 0, X t 0 i(P n ; t) x t = P e2I (s) n x (1 x)n+1 : (12) Stanley [18] has shown that for any convex lattice polytope, P , of dimension n, there is a polynomial, h n(x), with nonnegative integer coe cients satisfying X t 0 i(P ; t)x = h n(x) (1 x)n+1 : The s-lecture hall polytopes are all convex, in fact they are simplices. So Theorem 1 says, in other words: The h -polynomial of the s-lecture hall polytope P (s) n is the ascent polynomial of the set of s-inversion sequences I (s) n . In this paper, our focus is on sequences s of the form s = (1; k + 1; 2k + 1; : : : ; (n 1)k + 1); (13) where k is a positive integer. Recalling In;k and Pn;k from (7) and (8) in Section 1, we have In;k = I (s) n and Pn;k = P (s) n ; where s is de ned by (13), giving the following corollary. Corollary 1. For integers k 1 and n 0, X t 0 i(Pn;k; t)x t = P e2In;k x (1 x)n+1 : For completeness, we include a proof of Corollary 1 in the Appendix. the electronic journal of combinatorics 19 (2012), #P9 6 2.2 The main result We will show in Section 2.3 that when the sequence s has the special form (13), the Ehrhart polynomial of the s-lecture hall polytope has the following closed form. Theorem 2. For integers k 1 and n; t 0, i(Pn;k; t) = ( 1) t t X p=0 1 k 1 t p 1=k p (kp+ 1); where i(Pn;k; t) = # 2 Z j 0 1 1 2 k + 1 3 2k + 1 : : : n (n 1)k + 1 t : Our main result, Theorem 3 below, is a consequence of Theorem 2 and Corollary 1. Theorem 3. For integers k 1 and n 0, let A n (x) = X e2In;k x: