Counting Bipartite, k-Colored and Directed Acyclic Multi Graphs Through F-nomial coefficients

TL;DR

This paper explores the combinatorial interpretation of F-nomial coefficients related to bipartite, k-colored, and directed acyclic multigraphs, establishing new relations and counting methods for these graph classes.

Contribution

It introduces a novel interpretation of F-nomial coefficients via bipartite multigraphs and relates them to multi N(a)-nomial coefficients and acyclic multigraph counts.

Findings

F-nomial coefficients count labeled bipartite multigraphs

Explicit relation between k-colored multigraphs and N(a)-nomial coefficients

Number of directed acyclic multigraphs corresponds to the first row of the inversion matrix

Abstract

F-nomial coefficients encompass among others well-known binomial coefficients or Gaussian coefficients that count subsets of finite set and subspaces of finite vector space respectively. Here, the so called F-cobweb tiling sequences N(a) are considered. For such specific sequences a new interpretation with respect to Kwasniewski general combinatorial interpretation of F-nomial coefficients is unearhed. Namely, for tiling sequences F = N(a)$ the F-nomial coefficients are equal to the number of labeled special bipartite multigraphs denoted here as a-multigraphs G(a,n,k). An explicit relation between the number of k-colored a-multigraphs and multi N(a)-nomial coefficients is established. We also prove that the unsigned values of the first row of inversion matrix for N(a) -nomial coefficients considered here are equal to the numbers of directed acyclic a-multigraphs with n nodes.