Counting set systems by weight

Counting dominating sets and related structures in graphs

We consider some problems concerning the maximum number of (strong) dominating sets in a regular graph, and their weighted analogues. Our primary tool is Shearer's entropy lemma. These techniques extend to a reasonably broad class of graph parameters enumerating vertex colorings satisfying conditions on the multiset of colors appearing in (closed) neighborhoods. We also generalize further to enumeration problems for what we call existence homomorphisms. Here our results are substantially less complete, though we do solve some natural problems.

Random Weighting, Asymptotic Counting, and Inverse Isoperimetry

For a family X of k-subsets of the set 1,...,n, let |X| be the cardinality of X and let Gamma(X,mu) be the expected maximum weight of a subset from X when the weights of 1,...,n are chosen independently at random from a symmetric probability distribution mu on R. We consider the inverse isoperimetric problem of finding mu for which Gamma(X,mu) gives the best estimate of ln|X|. We prove that the optimal choice of mu is the logistic distribution, in which case Gamma(X,mu) provides an asymptotically tight estimate of ln|X| as k^{-1}ln|X| grows. Since in many important cases Gamma(X,mu) can be eas…

Counting SET-free sets

We consider the following counting problem related to the card game SET: How many $k$-element SET-free sets are there in an $n$-dimensional SET deck? Through a series of algebraic reformulations and reinterpretations, we show the answer to this question satisfies two polynomiality conditions.

Counting polygon spaces, Boolean functions and majority games

We explain why numbers occurring in the classification of polygon spaces coincide with numbers of self-dual equivalence classes of threshold functions, or of regular Boolean functions, or of decisive weighted majority games.

M-partitions: Optimal partitions of weight for one scale pan

An M-partition of a positive integer m is a partition with as few parts as possible such that any positive integer less than m has a partition made up of parts taken from that partition of m. This is equivalent to partitioning a weight m so as to be able to weigh any integer weight l < m with as few weights as possible and only one scale pan. We show that the number of parts of an M-partition is a log-linear function of m and the M-partitions of m correspond to lattice points in a polytope. We exhibit a recurrence relation for counting the number of M-partitions of m and, for ``half'' of the p…

On computing the total displacement number via weighted Motzkin paths

Counting the number of permutations of a given total displacement is equivalent to counting weighted Motzkin paths of a given area (Guay-Paquet and Petersen, 2014). The former combinatorial problem is still open. In this work, we show that this connection allows to construct efficient algorithms for counting and for sampling such permutations. These algorithms provide a tool to better understand the original combinatorial problem. A by-product of our approach is a different way of counting based on certain building sequences for Motzkin paths, which may be of independent interest.

基于路径计数的完美图着色多项式时间算法

A Polynomial-Time Algorithm for Coloring Perfect Graphs Based on Walk Counting

该研究提出基于图论操作的多项式时间算法，核心是通过迭代计算加权路径判定完美图是否含给定大小团，权重先统一初始化再依上次迭代路径计数更新，用于对完美图进行最优着色。

加权双色平面树的计数及其几何应用

Counting Weighted Bi-Colored Plane Trees and Their Geometric Applications

本文解决了具有指定黑白顶点数和顶点边权重的加权双色平面树的计数问题，并将其应用于计算三分支点球面间的强Hurwitz数和单锥奇点HCMU球面模空间连通分支数。

Counting Paths in Graphs

We give a simple combinatorial proof of a formula that extends a result by Grigorchuk (rediscovered by Cohen) relating cogrowth and spectral radius of random walks. Our main result is an explicit equation determining the number of `bumps' on paths in a graph: in a $d$-regular (not necessarily transitive) non-oriented graph let the series $G(t)$ count all paths between two fixed points weighted by their length $t^{length}$, and $F(u,t)$ count the same paths, weighted as $u^{number of bumps}t^{length}$. Then one has $$F(1-u,t)/(1-u^2t^2) = G(t/(1+u(d-u)t^2))/(1+u(d-u)t^2).$$ We then derive the c…

Dimension reduction in vertex-weighted exponential random graphs

We investigate the behavior of vertex-weighted exponential random graphs. We show that vertex-weighted exponential random graphs with edge weights induced by products of independent vertex weights are approximate mixtures of graphs whose vertex weight vector is a near fixed point of a certain vector equation. For graphs with Hamiltonians counting cliques, it is demonstrated that, under appropriate conditions, every solution to this equation is close to a block vector with a small number of communities. We prove that for the cases of positive weights and small weights in the Hamiltonian in part…