Partitions with Durfee triangles of fixed size
N. Guru Sharan, Armin Straub

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 of partitions of with Durfee square of fixed size has a well-known simple rational generating function. We study the number of partitions of with Durfee triangle of size (the largest subpartition with parts ). We determine the corresponding generating functions which are rational functions of a similar form. Moreover, we explicitly determine the leading asymptotic of , as .
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Taxonomy
TopicsAdvanced Mathematical Identities · Advanced Combinatorial Mathematics · Analytic Number Theory Research
