On the generating function for intervals in Young's lattice

TL;DR

This paper investigates generating functions for partitions in Young's lattice, proving they satisfy rational recursions, and explores their asymptotic behavior and geometric interpretations related to Grassmannians.

Contribution

It establishes that a family of generating functions for partitions in Young's lattice are rational functions, providing new recursive formulas and geometric insights.

Findings

Generating functions satisfy rational recursions.

Asymptotic behavior of lower order ideals is characterized.

Homological interpretation links to Grassmannians.

Abstract

In this paper, we study a family of generating functions whose coefficients are polynomials that enumerate partitions in lower order ideals of Young's lattice. Our main result is that this family satisfies a rational recursion and are therefore rational functions. As an application, we calculate the asymptotic behavior of the cardinality of lower order ideals for the ``average" partition of fixed length and give a homological interpretation of this result in relation to Grassmannians and their Schubert varieties.