School of Mathematical & Statistical Sciences Faculty Publications and Presentations

Supercranks for partitions with a fixed number of parts

Document Type

Article

Publication Date

7-15-2025

Abstract

Dyson’s rank of a partition witnesses Ramanujan’s first two congruences for p(n): p(5n+4)≡0(mod5) and p(7n+5)≡0(mod7). However, the 5- and 7-divisibility outside of these arithmetic progressions is not always witnessed by the rank. In this paper, we find statistics witnessing every instance of m-divisibility of P(n, d), the number of partitions of n into exactly d parts. We call these statistics “supercranks," and we give several examples. We conjecture that these examples, along with just one other known supercrank, are the only supercranks for P(n, d) modulo any m.

Comments

https://rdcu.be/eADY6

Publication Title

Journal of Algebraic Combinatorics

DOI

10.1007/s10801-025-01413-7

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