Automating the Enumeration of Sequences Defined by Digraphs

Abstract

We consider sequences of nonnegative integers $S=(s_1,s_2,ldots, s_n)$ defined by systems of constraints represented as weighted directed graphs in which edge $(s_a,s_b)$ of weight $w$ indicates constraint $s_a~geq~s_b~+~w$. We propose a set of seven rules and a decomposition technique for obtaining multivariate and single-variable generating functions for families of such graphs. Our method compares to existing techniques by offering an elegant and intuitive approach to obtaining generating functions and recurrences, albeit only for a subset of the partition and composition enumeration problems addressed by other techniques. The decomposition technique we propose remains relevant, nevertheless, to a wide range of applications, including several well-known ones. Moreover, our objective is to obtain recurrences for generating functions so as to assist the formulation and proof of their closed-form solutions. For integer sequences defined by directed graphs with $w in 0,1]$, we prove that our technique holds sufficient.

We describe the formulation of finite-variable generating function recurrences from multi-variable ones and provide a set of rules to determine the variables chosen. The construction tree is introduced as the tree representation of the construction of a generating function from the decomposition of a weighted directed graph. Given such a construction tree, automation of the process of building the multi-variable and finite-variable recurrences is possible, and we implement it as a computer program.

Finally, we apply our techniques and tools to a wide range of famous problems, including 2-rowed plane partitions, up-down compositions and hexagonal plane partitions, as well as some new problems, and obtain recurrences to each. We find that our methods are not only effective but also easy and simple to use.