The f-vector of the descent polytope

For a positive integer n and a subset S of [n-1], the descent polytope DP_S is the set of points x_1, ..., x_n in the n-dimensional unit cube [0,1]^n such that x_i >= x_{i+1} for i in S and x_i <= x_{i+1} otherwise. First, we express the f-vector of DP_S as a sum over all subsets of [n-1]. Second, we use certain factorizations of the associated word over a two-letter alphabet to describe the f-vector. We show that the f-vector is maximized when the set S is the alternating set {1,3,5, ...}. We derive a generating function for the f-polynomial F_S(t) of DP_S, written as a formal power series in two non-commuting variables with coefficients in Z[t]. We also obtain the generating function for the Ehrhart polynomials of the descent polytopes.Comment: 14 pages; to appear in Discrete & Computational Geometr

Bijections for Entringer families

Andr\'e proved that the number of alternating permutations on $\{1, 2, \dots, n\}$ is equal to the Euler number $E_n$. A refinement of Andr\'e's result was given by Entringer, who proved that counting alternating permutations according to the first element gives rise to Seidel's triangle $(E_{n,k})$ for computing the Euler numbers. In a series of papers, using generating function method and induction, Poupard gave several further combinatorial interpretations for $E_{n,k}$ both in alternating permutations and increasing trees. Kuznetsov, Pak, and Postnikov have given more combinatorial interpretations of $E_{n,k}$ in the model of trees. The aim of this paper is to provide bijections between the different models for $E_{n,k}$ as well as some new interpretations. In particular, we give the first explicit one-to-one correspondence between Entringer's alternating permutation model and Poupard's increasing tree model.Comment: 19 page

Parallelogram polyominoes, the sandpile model on a complete bipartite graph, and a q,t-Narayana polynomial

We classify recurrent configurations of the sandpile model on the complete bipartite graph K_{m,n} in which one designated vertex is a sink. We present a bijection from these recurrent configurations to decorated parallelogram polyominoes whose bounding box is a m*n rectangle. Several special types of recurrent configurations and their properties via this bijection are examined. For example, recurrent configurations whose sum of heights is minimal are shown to correspond to polyominoes of least area. Two other classes of recurrent configurations are shown to be related to bicomposition matrices, a matrix analogue of set partitions, and (2+2)-free partially ordered sets. A canonical toppling process for recurrent configurations gives rise to a path within the associated parallelogram polyominoes. This path bounces off the external edges of the polyomino, and is reminiscent of Haglund's well-known bounce statistic for Dyck paths. We define a collection of polynomials that we call q,t-Narayana polynomials, defined to be the generating function of the bistatistic (area,parabounce) on the set of parallelogram polyominoes, akin to the (area,hagbounce) bistatistic defined on Dyck paths in Haglund (2003). In doing so, we have extended a bistatistic of Egge, Haglund, Kremer and Killpatrick (2003) to the set of parallelogram polyominoes. This is one answer to their question concerning extensions to other combinatorial objects. We conjecture the q,t-Narayana polynomials to be symmetric and prove this conjecture for numerous special cases. We also show a relationship between Haglund's (area,hagbounce) statistic on Dyck paths, and our bistatistic (area,parabounce) on a sub-collection of those parallelogram polyominoes living in a (n+1)*n rectangle

Polytopes, generating functions, and new statistics related to descents and inversions in permutations

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2008.Includes bibliographical references (p. 75-76).We study new statistics on permutations that are variations on the descent and the inversion statistics. In particular, we consider the alternating descent set of a permutation [sigma] = [sigma] 1 [sigma] 2 an defined as the set of indices i such that either i is odd and ai > ui+l, or i is even and au < au+l. We show that this statistic is equidistributed with the 3-descent set statistic on permutations [sigma] = [sigma] 1 [sigma] 2 ... [sigma] n+1 with al = 1, defined to be the set of indices i such that the triple [sigma] i [sigma] i + [sigma] i +2 forms an odd permutation of size 3. We then introduce Mahonian inversion statistics corresponding to the two new variations of descents and show that the joint distributions of the resulting descent-inversion pairs are the same. We examine the generating functions involving alternating Eulerian polynomials, defined by analogy with the classical Eulerian polynomials ... using alternating descents. By looking at the number of alternating inversions in alternating (down-up) permutations, we obtain a new qanalog of the Euler number En and show how it emerges in a q-analog of an identity expressing E, as a weighted sum of Dyck paths. Other parts of this thesis are devoted to polytopes relevant to the descent statistic. One such polytope is a "signed" version of the Pitman-Stanley parking function polytope, which can be viewed as a generalization of the chain polytope of the zigzag poset. We also discuss the family of descent polytopes, also known as order polytopes of ribbon posets, giving ways to compute their f-vectors and looking further into their combinatorial structure.by Denis Chebikin.Ph.D

Fourientations and the Tutte polynomial

A fourientation of a graph is a choice for each edge of the graph whether to orient that edge in either direction, leave it unoriented, or biorient it. Fixing a total order on the edges and a reference orientation of the graph, we investigate properties of cuts and cycles in fourientations which give trivariate generating functions that are generalized Tutte polynomial evaluations of the form (k + m)[superscript n−1](k + l)[superscript gT](αk + βl + m/k + m , γ k + l + δm/ k + l) for α, γ ∈ {0, 1, 2} and β, δ ∈ {0, 1}. We introduce an intersection lattice of 64 cut–cycle fourientation classes enumerated by generalized Tutte polynomial evaluations of this form. We prove these enumerations using a single deletion–contraction argument and classify axiomatically the set of fourientation classes to which our deletion–contraction argument applies. This work unifies and extends earlier results for fourientations due to Gessel and Sagan (Electron J Combin 3(2):Research Paper 9, 1996), results for partial orientations due to Backman (Adv Appl Math, forthcoming, 2014. arXiv:1408.3962), and Hopkins and Perkinson (Trans Am Math Soc 368(1):709–725, 2016), as well as results for total orientations due to Stanley (Discrete Math 5:171–178, 1973; Higher combinatorics (Proceedings of NATO Advanced Study Institute, Berlin, 1976). NATO Advanced Study Institute series, series C: mathematical and physical sciences, vol 31, Reidel, Dordrecht, pp 51–62, 1977), Las Vergnas (Progress in graph theory (Proceedings, Waterloo silver jubilee conference 1982), Academic Press, New York, pp 367–380, 1984), Greene and Zaslavsky (Trans Am Math Soc 280(1):97–126, 1983), and Gioan (Eur J Combin 28(4):1351–1366, 2007), which were previously unified by Gioan (2007), Bernardi (Electron J Combin 15(1):Research Paper 109, 2008), and Las Vergnas (Tutte polynomial of a morphism of matroids 6. A multi-faceted counting formula for hyperplane regions and acyclic orientations, 2012. arXiv:1205.5424). We conclude by describing how these classes of fourientations relate to geometric, combinatorial, and algebraic objects including bigraphical arrangements, cycle–cocycle reversal systems, graphic Lawrence ideals, Riemann–Roch theory for graphs, zonotopal algebra, and the reliability polynomial. Keywords: Partial graph orientations, Tutte polynomial, Deletion–contraction, Hyperplane arrangements, Cycle–cocycle reversal system, Chip-firing, G-parking functions, Abelian sandpile model, Riemann–Roch theory for graphs, Lawrence ideals, Zonotopal algebra, Reliability polynomialNational Science Foundation (U.S.) (Grant 1122374

Graph powers and k-ordered Hamiltonicity

AbstractIt is known that if G is a connected simple graph, then G3 is Hamiltonian (in fact, Hamilton-connected). A simple graph is k-ordered Hamiltonian if for any sequence v1, v2,…,vk of k vertices there is a Hamiltonian cycle containing these vertices in the given order. In this paper, we prove that if k⩾4, then G⌊3k/2⌋-2 is k-ordered Hamiltonian for every connected graph G on at least k vertices. By considering the case of the path graph Pn, we show that this result is sharp. We also give a lower bound on the power of the cycle Cn that guarantees k-ordered Hamiltonicity

Variations on Descents and Inversions in Permutations

We study new statistics on permutations that are variations on the descent and the inversion statistics. In particular, we consider the alternating descent set of a permutation $\sigma = \sigma_1\sigma_2\cdots\sigma_n$ defined as the set of indices $i$ such that either $i$ is odd and \sigma_i > \sigma_{i+1}, or $i$ is even and \sigma_i < \sigma_{i+1}. We show that this statistic is equidistributed with the odd $3$-factor set statistic on permutations $\tilde{\sigma} = \sigma_1\sigma_2\cdots\sigma_{n+1}$ with $\sigma_1=1$, defined to be the set of indices $i$ such that the triple $\sigma_i \sigma_{i+1} \sigma_{i+2}$ forms an odd permutation of size $3$. We then introduce Mahonian inversion statistics corresponding to the two new variations of descents and show that the joint distributions of the resulting descent-inversion pairs are the same, establishing a connection to two classical Mahonian statistics, maj and stat, along the way. We examine the generating functions involving alternating Eulerian polynomials, defined by analogy with the classical Eulerian polynomials $\sum_{\sigma\in\mathcal{S}_n} t^{{\rm des}(\sigma)+1}$ using alternating descents. For the alternating descent set statistic, we define the generating polynomial in two non-commutative variables by analogy with the $ab$-index of the Boolean algebra $B_n$, providing a link to permutations without consecutive descents. By looking at the number of alternating inversions, which we define in the paper, in alternating (down-up) permutations, we obtain a new $q$-analog of the Euler number $E_n$ and show how it emerges in a $q$-analog of an identity expressing $E_n$ as a weighted sum of Dyck paths.</jats:p