Enumerative properties of Ferrers graphs

TL;DR

This paper introduces Ferrers graphs, a class of bipartite graphs linked to Ferrers diagrams, and derives formulas for their spanning trees, Hamiltonian paths, chromatic polynomial, and symmetric function, revealing combinatorial properties.

Contribution

It provides new explicit formulas for enumerative invariants of Ferrers graphs, connecting graph theory with combinatorial statistics.

Findings

Number of spanning trees expressed explicitly

Chromatic polynomial coefficient linked to excedance set statistic

Formulas for Hamiltonian paths and symmetric functions derived

Abstract

We define a class of bipartite graphs that correspond naturally with Ferrers diagrams. We give expressions for the number of spanning trees, the number of Hamiltonian paths when applicable, the chromatic polynomial, and the chromatic symmetric function. We show that the linear coefficient of the chromatic polynomial is given by the excedance set statistic.