Partitions with Durfee triangles of fixed size

TL;DR

This paper investigates partitions with a fixed-sized Durfee triangle, deriving their generating functions and asymptotic behavior, extending classical results on Durfee squares to a new geometric partition statistic.

Contribution

It introduces and analyzes the enumeration of partitions with Durfee triangles of fixed size, providing explicit generating functions and asymptotic formulas.

Findings

Derived rational generating functions for partitions with fixed Durfee triangle size.

Established the leading asymptotic behavior of the number of such partitions.

Extended classical Durfee square results to Durfee triangles.

Abstract

A well-studied statistic of an integer partition is the size of its Durfee square. In particular, the number $D_k (n)$ of partitions of $n$ with Durfee square of fixed size $k$ has a well-known simple rational generating function. We study the number $R_k (n)$ of partitions of $n$ with Durfee triangle of size $k$ (the largest subpartition with parts $1, 2, \ldots, k$). We determine the corresponding generating functions which are rational functions of a similar form. Moreover, we explicitly determine the leading asymptotic of $R_k (n)$, as $n \rightarrow \infty$.