On the generating function for intervals in Young's lattice

2021/07/19 by Faqruddin Azam, Azam, Faqruddin, Edward Richmond +1 · 1 citation
Mathematics · #05A17 #14M15 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2107.09149

openalex publication_date 2021/07/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Cited by

• The Generalized Interval Polynomial of a Poset

Related

• Some comments on "On the generating function for intervals in Young's lattice" by Azam and Richmond
• Order ideals in weak subposets of Young's lattice and associated unimodality conjectures
• Polynomials associated with Partitions: Polynomials associated with Partitions: Their Asymptotics and Zeros
• A unified treatment of families of partition functions
• Hall-Littlewood polynomials, affine Schubert series, and lattice enumeration