Polynomial functions on Young diagrams arising from bipartite graphs

TL;DR

This paper investigates functions on Young diagrams derived from bipartite graph embeddings, providing criteria for polynomiality using combinatorial properties and developing a differential calculus framework.

Contribution

It introduces a criterion to identify polynomial functions on Young diagrams based on bipartite graph embeddings and develops a differential calculus approach.

Findings

Established a criterion for polynomiality of functions on Young diagrams

Connected combinatorial properties of bipartite graphs to polynomial functions

Developed a differential calculus framework for functions on Young diagrams

Abstract

We study the class of functions on the set of (generalized) Young diagrams arising as the number of embeddings of bipartite graphs. We give a criterion for checking when such a function is a polynomial function on Young diagrams (in the sense of Kerov and Olshanski) in terms of combinatorial properties of the corresponding bipartite graphs. Our method involves development of a differential calculus of functions on the set of generalized Young diagrams.