A family of bijections between G-parking functions and spanning trees

For a directed graph G on vertices {0,1,...,n}, a G-parking function is an n-tuple (b_1,...,b_n) of non-negative integers such that, for every non-empty subset U of {1,...,n}, there exists a vertex j in U for which there are more than b_j edges going from j to G-U. We construct a family of bijective maps between the set P_G of G-parking functions and the set T_G of spanning trees of G rooted at 0, thus providing a combinatorial proof of |P_G| = |T_G|.Comment: 11 pages, 4 figures; a family of bijections containing the two original bijections is presented; submitted to J. Combinatorial Theory, Series

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

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

Cyclotomic factors of the descent set polynomial

We introduce the notion of the descent set polynomial as an alternative way of encoding the sizes of descent classes of permutations. Descent set polynomials exhibit interesting factorization patterns. We explore the question of when particular cyclotomic factors divide these polynomials. As an instance we deduce that the proportion of odd entries in the descent set statistics in the symmetric group S_n only depends on the number on 1's in the binary expansion of n. We observe similar properties for the signed descent set statistics.Comment: 21 pages, revised the proof of the opening result and cleaned up notatio

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