Congruences for Andrews' Smallest Parts Partition Function and New Congruences for Dyson's Rank

TL;DR

This paper generalizes Ramanujan-type congruences for the smallest parts function spt(n) and constructs explicit examples of congruences for Dyson’s rank, advancing understanding of partition-related functions.

Contribution

It extends known congruences for spt(n) to new primes and explicitly constructs examples of Dyson’s rank congruences using elementary methods.

Findings

Explicit Ramanujan-type congruences for spt(n) mod 11, 17, 19, 29, 31, 37.

Construction of explicit nontrivial examples of Dyson's rank congruences mod 11.

Generalization of previous congruences using relations between rank and crank moments.

Abstract

Let spt(n) denote the total number of appearances of smallest parts in the partitions of n. Recently, Andrews showed how spt(n) is related to the second rank moment, and proved some surprising Ramanujan-type congruences mod 5, 7 and 13. We prove a generalization of these congruences using known relations between rank and crank moments. We obtain explicit Ramanujan-type congruences for spt(n) mod p for p = 11, 17, 19, 29, 31 and 37. Recently, Bringmann and Ono proved that Dyson's rank function has infinitely many Ramanujan-type congruences. Their proof is non-constructive and utilizes the theory of weak Maass forms. We construct two explicit nontrivial examples mod 11 using elementary congruences between rank moments and half-integer weight Hecke eigenforms.