In this paper we study partitions whose successive ranks belong to a given set. We enumerate such partitions while keeping track of the number of parts, the largest part, the side of the Durfee square, and the height of the Durfee rectangle. We also obtain a new bijective proof of a result of Andrews and Bressoud that the number of partitions of $N$ with all ranks at least $1-\ell$ equals the number of partitions of $N$ with no parts equal to $\ell+1$, for $\ell\ge0$, which allows us to refine it by the above statistics. Combining Foata's second fundamental transformation for words with Greene and Kleitman's mapping for subsets, interpreted in terms of lattice paths, we obtain enumeration formulas for partitions whose successive ranks satisfy certain constraints, such as being bounded by a constant.

In 1997 Bousquet-Mélou and Eriksson introduced lecture hall partitions as the inversion vectors of elements of the parabolic quotient $\widetilde{C}/C$. We provide a new view of their correspondence that allows results in one domain to be translated into the other. We determine the equivalence between combinatorial statistics in each domain and use this correspondence to translate certain generating function formulas on lecture hall partitions to new observations about $\widetilde{C}/C$.

Permutations that avoid given patterns have been studied in great depth for their connections to other fields of mathematics, computer science, and biology. From a combinatorial perspective, permutation patterns have served as a unifying interpretation that relates a vast array of combinatorial structures. In this paper, we introduce the notion of patterns in inversion sequences. A sequence $(e_1,e_2,\ldots,e_n)$ is an inversion sequence if $0 \leq e_i \pi_i \}|$. This correspondence makes it a natural extension to study patterns in inversion sequences much in the same way that patterns have been studied in permutations. This paper, the first of two on patterns in inversion sequences, focuses on the enumeration of inversion sequences that avoid words of length three. Our results connect patterns in inversion sequences to a number of well-known numerical sequences including Fibonacci numbers, Bell numbers, Schr\"oder numbers, and Euler up/down numbers.

We investigate the arithmetic-geometric structure of the lecture hall cone $L_n \ := \ \big\{\lambda\in \mathbb{R}^n: \, 0\leq \frac{\lambda_1}{1}\leq \frac{\lambda_2}{2}\leq \frac{\lambda_3}{3}\leq \cdots \leq \frac{\lambda_n}{n}\big\}$. We show that $L_n$ is isomorphic to the cone over the lattice pyramid of a reflexive simplex whose Ehrhart $h^*$-polynomial is given by the $(n-1)$st Eulerian polynomial and prove that lecture hall cones admit regular, flag, unimodular triangulations. After explicitly describing the Hilbert basis for $L_n$, we conclude with observations and a conjecture regarding the structure of unimodular triangulations of $L_n$, including connections between enumerative and algebraic properties of $L_n$ and cones over unit cubes.

Over the past twenty years, lecture hall partitions have emerged as fundamental combinatorial structures, leading to new generalizations and interpretations of classical theorems and new results. In recent years, geometric approaches to lecture hall partitions have used polyhedral geometry to discover further properties of these rich combinatorial objects. In this paper we give an overview of some of the surprising connections that have surfaced in the process of trying to understand the lecture hall partitions.

Given a graph G, the acyclic orientation graph of G, denoted AO(G), is the graph whose vertices are the acyclic orientations of G, and two acyclic orientations are joined by an edge in AO(G) iff they differ by the reversal of a single edge. A hamilton cycle in AO(G) gives a Gray code listing of the acyclic orientations of G. We prove that for certain graphs G, AO(G) is hamiltonian, and give explicit constructions of hamilton cycles or paths. This work includes Gray codes for listing the acyclic orientations of trees, complete graphs, odd cycles, chordal graphs, odd ladder graphs, and odd wheel graphs. We also give examples of graphs whose acyclic orientation graph is not hamiltonian. We show that the acyclic orientations of even cycles, some complete bipartite graphs, even ladder graphs, and even wheel graphs cannot be listed by the defined Gray code.

For fixed n and k, we find a three-variable generating function for the set of sequences (lambda(1),...,lambda(n)) satisfyingk >= lambda(1)/a(1) >= lambda(2)/a(2) >= ... >= lambda(n)/a(n) >= 0,where a := (a(1),...,a(n)) = (1,2,...,n) or (n, n - 1,...,1). When k -> infinity we recover the refined anti-lecture hall and lecture hall theorems. When a = (1,2,...,n) and n -> infinity, we obtain a refinement of a recent result of Chen, Sang and Shi. The main tools are elementary combinatorics and Andrews' generalization of the Watson Whipple transformation. (C) 2015 Elsevier Inc. All rights reserved.

We study the roots of generalized Eulerian polynomials via a novel approach. We interpret Eulerian polynomials as the generating polynomials of a statistic over inversion sequences. Inversion sequences (also known as Lehmer codes or subexcedant functions) were recently generalized by Savage and Schuster, to arbitrary sequences s of positive integers, which they called s-inversion sequences.Our object of study is the generating polynomial of the ascent statistic over the set of s-inversion sequences of length n. Since this ascent statistic over inversion sequences is equidistributed with the descent statistic over permutations, we call this generalized polynomial the s-Eulerian polynomial. The main result of this paper is that, for any sequence s of positive integers, the s-Eulerian polynomial has only real roots.This result is first shown to generalize several existing results about the real-rootedness of various Eulerian polynomials. We then show that it can be used to settle a conjecture of Brenti, that Eulerian polynomials for all finite Coxeter groups have only real roots, and partially settle a conjecture of Dilks, Petersen, Stembridge on type B affine Eulerian polynomials. It is then extended to several q-analogs. We show that the MacMahon-Carlitz q-Eulerian polynomial has only real roots whenever q is a positive real number, confirming a conjecture of Chow and Gessel. The same holds true for the hyperoctahedral group and the wreath product groups, confirming further conjectures of Chow and Gessel, and Chow and Mansour, respectively.Our results have interesting geometric consequences as well.

It follows from work of Chung and Graham that for a certain family of polynomials T_n(x), derived from the descent statistic on permutations, the coefficient sequence of T_n-1(x) coincides with that of the polynomial T_n(x)/(1+x+⋯+x^n-1). We observed computationally that the inflated 𝐬-Eulerian polynomial Q_n^(𝐬)(x), which satisfies Q_n^(𝐬)(x) = T_n(x) when 𝐬=(1,2,…,n), also satisfies this property for many sequences 𝐬. In this work we characterize those sequences 𝐬 for which the coefficient sequence of Q_n-1^(𝐬)(x) coincides with that of the polynomial Q_n^(𝐬)(x)/(1+x+⋯+x^s_n-1). In particular, we show that all nondecreasing sequences satisfy this property. We also settle a conjecture of Pensyl and Savage by showing that the inflated 𝐬-Eulerian polynomials are unimodal for all choices of positive integer sequences s. In addition, we determine when these polynomials are palindromic and show our characterization is equivalent to another of Beck, Braun, Köppe, Savage, and Zafeirakopoulos.

In 1997, Bousquet-Mélou and Eriksson initiated the study of lecture hall partitions, a fascinating family of partitions that yield a finite version of Euler’s celebrated odd/distinct partition theorem. In subsequent work on s-lecture hall partitions, they considered the self-reciprocal property for various associated generating functions, with the goal of characterizing those sequences s that give rise to generating functions of the form \(((1-q^{e_{1}})(1-q^{e_{2}}) \cdots(1-q^{e_{n}}))^{-1}\).

We give an intrinsic proof of a conjecture of Brenti that all the roots of the Eulerian polynomial of type $D$ are real and a proof of a conjecture of Dilks, Petersen, and Stembridge that all the roots of the affine Eulerian polynomial of type $B$ are real, as well.

We show how combinatorial arguments involving a variety of statistics on words can produce nontrivial identities between hypergeometric series in two variables. We establish relationships to the Rogers-Fine identity, Heine’s second transformation, and mock theta functions. Finally, we show that any hypergeometric series of a certain form can be interpreted in terms of generalized statistics on words.

For a sequence of positive integers s=(s 1,…,s n ), we define the rational lecture hall polytope \(\mathbf{R}_{n}^{(\mathbf{s})}\). We prove that its h ∗-polynomial, \(Q_{n}^{(\mathbf{s})}(x)\), has nonnegative integer coefficients that count certain statistics on s-inversion sequences. The polynomial \(Q_{n}^{(\mathbf{s})}(x)\) can be viewed as an inflated version of the s-Eulerian polynomial, \(A_{n}^{(\mathbf{s})}(x)\), associated with the integral lecture hall polytope, \(\mathbf {P}_{n}^{(\mathbf{s})}\), introduced by Savage and Schuster. The result is applied in three ways: (1) in the theory of s-lecture hall partitions, introduced by Bousquet-Mélou and Eriksson, the generating function, refined to include the size of the last part, now has an explicit description in terms of the inflated s-Eulerian polynomial, \(Q_{n}^{(\mathbf{s})}(x)\); (2) for special sequences, s, we get an explicit formula for \(Q_{n}^{(\mathbf{s})}(x)\) by computing the Ehrhart quasi-polynomial of \(\mathbf{R}_{n}^{(\mathbf {s})}\); and (3) for many sequences, s, the coefficients the inflated s-Eulerian polynomial form a symmetric unimodal sequence, even when the coefficients of the (uninflated) s-Eulerian polynomial, \(A_{n}^{(\mathbf {s})}(x)\), do not.

We show that a recent identity of Beck–Gessel–Lee–Savage on the generating function of symmetrically constrained compositions of integers generalizes naturally to a family of convex polyhedral cones that are invariant under the action of a finite reflection group. We obtain general expressions for the multivariate generating functions of such cones, and work out their general form more specifically for all symmetry groups of type A (previously known) and types B and D (new). We obtain several applications of these expressions in type B, including identities involving permutation statistics and lecture hall partitions.

We use the theory of lecture hall partitions to define 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 defining 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.

We introduce the notion of a Mahonian pair. Consider the set, P^*, of all words having the positive integers as alphabet. Given finite subsets S,T of P^*, we say that (S,T) is a Mahonian pair if the distribution of the major index, maj, over S is the same as the distribution of the inversion number, inv, over T. So the well-known fact that maj and inv are equidistributed over the symmetric group, S_n, can be expressed by saying that (S_n,S_n) is a Mahonian pair. We investigate various Mahonian pairs (S,T) with S different from T. Our principal tool is Foata's fundamental bijection f: P^* -> P^* since it has the property that maj w = inv f(w) for any word w. We consider various families of words associated with Catalan and Fibonacci numbers. We show that, when restricted to words in {1,2}^*, f transforms familiar statistics on words into natural statistics on integer partitions such as the size of the Durfee square. The Rogers-Ramanujan identities, the Catalan triangle, and various q-analogues also make an appearance. We generalize the definition of Mahonian pairs to infinite sets and use this as a tool to connect a partition bijection of Corteel-Savage-Venkatraman with the Greene-Kleitman decomposition of a Boolean algebra into symmetric chains. We close with comments about future work and open problems.

We show that an identity of Gessel and Stanton [I. Gessel, D. Stanton, Applications of q-Lagrange inversion to basic hypergeometric series, Trans. Amer. Math. Soc. 277 (1983) 197, Eq. (7.24)] can be viewed as a symmetric version of a recent analytic variation of the little Gollnitz identities. This is significant, since the Gollnitz-Gordon identities are considered the usual symmetric counterpart to little Gollnitz theorems. Is it possible, then, that the Gessel-Stanton identity is part of an infinite family of identities like those of Gollnitz-Gordon?Toward this end, we derive partners and generalizations of the Gessel-Stanton identity. We show that the new little Gollnitz identities enumerate partitions into distinct parts in which even-indexed (resp. odd-indexed) parts are even, and derive a refinement of the Gessel-Stanton identity that suggests a similar interpretation is possible. We study an associated system of q-difference equations to show that the Gessel-Stanton identity and its partner are actually two members of a three-element family. (C) 2010 Elsevier Inc. All rights reserved.

We use generalized lecture hall partitions to discover a new pair of q-series identities. These identities are unusual in that they involve partitions into parts from asymmetric residue classes, much like the little Göllnitz partition theorems. We derive a two-parameter generalization of our identities that, surprisingly, gives new analytic counterparts of the little Göllnitz theorems. Finally, we show that the little Göllnitz theorems also involve “lecture hall sequences,” that is, sequences constrained by the ratio of consecutive parts.