Annals of Combinatorics最新文献

We introduce a class of polytopes that we call chainlink polytopes and show that they allow us to construct infinite families of pairs of non-isomorphic rational polytopes with the same Ehrhart quasipolynomial. Our construction is based on circular fence posets, a recently introduced class of posets, which admit a non-obvious and nontrivial symmetry in their rank sequences. We show that this symmetry can be lifted to the level of polyhedral models (which we call chainlink polytopes) for these posets. Along the way, we introduce the related class of chainlink posets and show that they exhibit analogous nontrivial symmetry properties. We further prove an outstanding conjecture on the unimodality of rank polynomials of circular fence posets.

In this work, we give the sharp upper bound for the number of cliques in graphs with bounded odd circumferences. This generalized Turán-type result is an extension of the celebrated Erdős and Gallai theorem and a strengthening of Luo’s recent result. The same bound for graphs with bounded even circumferences is a trivial application of the theorem of Li and Ning.

Stanley introduced and studied two lattice polytopes, the order polytope and chain polytope, associated with a finite poset. Recently, Ohsugi and Tsuchiya introduce an enriched version of them, called the enriched order polytope and enriched chain polytope. In this paper, we give a piecewise-linear bijection between these enriched poset polytopes, which is an enriched analogue of Stanley’s transfer map and bijectively proves that they have the same Ehrhart polynomials. Also, we construct explicitly unimodular triangulations of two enriched poset polytopes, which are the order complexes of graded posets.

The approximate uniform sampling of graph realizations with a given degree sequence is an everyday task in several social science, computer science, engineering etc. projects. One approach is using Markov chains. The best available current result about the well-studied switch Markov chain is that it is rapidly mixing on P-stable degree sequences (see DOI:10.1016/j.ejc.2021.103421). The switch Markov chain does not change any degree sequence. However, there are cases where degree intervals are specified rather than a single degree sequence. (A natural scenario where this problem arises is in hypothesis testing on social networks that are only partially observed.) Rechner, Strowick, and Müller–Hannemann introduced in 2018 the notion of degree interval Markov chain which uses three (separately well studied) local operations (switch, hinge-flip and toggle), and employing on degree sequence realizations where any two sequences under scrutiny have very small coordinate-wise distance. Recently, Amanatidis and Kleer published a beautiful paper (DOI:10.4230/LIPIcs.STACS.2023.7), showing that the degree interval Markov chain is rapidly mixing if the sequences are coming from a system of very thin intervals which are centered not far from a regular degree sequence. In this paper, we substantially extend their result, showing that the degree interval Markov chain is rapidly mixing if the intervals are centered at P-stable degree sequences.

Let P(n) be the number of polyominoes of n cells and (lambda ) be Klarner’s constant, that is, (lambda =lim _{nrightarrow infty } root n of {P(n)}). We show that there exist some positive numbers A, T, so that for every n

This is somewhat a step toward the well-known conjecture that there exist positive (C,theta ), so that (P(n)sim Cn^{-theta }lambda ^n) for every n. In fact, if we assume another popular conjecture that (P(n)/P(n-1)) is increasing, we can get rid of (log n) to have

Beside the above theoretical result, we also conjecture that the ratio of the number of some class of polyominoes, namely inconstructible polyominoes, over P(n) is decreasing, by observing this behavior for the available values. The conjecture opens a nice approach to bounding (lambda ) from above, since if it is the case, we can conclude that

which is quite close to the current best lower bound (lambda > 4.0025) and greatly improves the current best upper bound (lambda < 4.5252). The approach is merely analytically manipulating the known or likely properties of the function P(n), instead of giving new insights of the structure of polyominoes. The techniques can be applied to other lattice animals and self-avoiding polygons of a given area with almost no change.

We study labeled chip-firing on binary trees starting with (2^n-1) chips initially placed at the root. We prove a sorting property of terminal configurations of the process. We also analyze the end game moves poset and prove that this poset is a modular lattice.

We say that a Hamilton cycle (C=(x_1,ldots ,x_n)) in a graph G is k-symmetric, if the mapping (x_imapsto x_{i+n/k}) for all (i=1,ldots ,n), where indices are considered modulo n, is an automorphism of G. In other words, if we lay out the vertices (x_1,ldots ,x_n) equidistantly on a circle and draw the edges of G as straight lines, then the drawing of G has k-fold rotational symmetry, i.e., all information about the graph is compressed into a (360^circ /k) wedge of the drawing. The maximum k for which there exists a k-symmetric Hamilton cycle in G is referred to as the Hamilton compression of G. We investigate the Hamilton compression of four different families of vertex-transitive graphs, namely hypercubes, Johnson graphs, permutahedra and Cayley graphs of abelian groups. In several cases, we determine their Hamilton compression exactly, and in other cases, we provide close lower and upper bounds. The constructed cycles have a much higher compression than several classical Gray codes known from the literature. Our constructions also yield Gray codes for bitstrings, combinations and permutations that have few tracks and/or that are balanced.

Let (nge 2) be an integer. We prove the convexity of the so-called MacMahon q-Catalan polynomials (C_n(q)=frac{1}{[n+1]_q}left[ 2n atop n right] _q) viewed as functions of q over the entire set of reals. Along the way, several auxiliary properties of the q-Catalan polynomials and intermediate results in the form of inequalities are presented, with the aim to make the paper self-contained. We also include a commentary on the convexity of the generating function for the integer partitions.