Andrews-Beck Type Congruences for Overpartitions

Eunmi Kim

doi:10.37236/10456

Abstract

We prove Andrews-Beck type congruences for overpartitions concerning the $D$-rank and $M_2$-rank. To prove congruences, we establish the generating function for weighted $D$-rank (respectively, $M_2$-rank) moment of overpartitions and find a connection with the second $D$-rank (respectively, $M_2$-rank) moment for overpartitions.

Highlights

Ramanujan’s congruences for the partition function p(n) are one of remarkable results in the theory of partitions: p(5n + 4) ≡ 0, p(7n + 5) ≡ 0, p(11n + 6) ≡ 0, Dyson [8] defined the rank of a partition, which is defined as the largest part minus the number of parts, conjectured combinatorial explanations for the Ramanujan congruences modulo 5 and 7, and conjectured the existence of a crank function for partitions that could provide a combinatorial proof of Ramanujan’s congruences modulo 11
We prove Andrews-Beck type congruence on N T (b, k, n) and N T 2(b, k, n) modulo 4 and 8 as follows
We discover more congruences on N T (b, k, n)

Summary

Introduction

Ramanujan’s congruences for the partition function p(n) are one of remarkable results in the theory of partitions: p(5n + 4) ≡ 0 (mod 5), p(7n + 5) ≡ 0 (mod 7), p(11n + 6) ≡ 0 (mod 11), Dyson [8] defined the rank of a partition, which is defined as the largest part minus the number of parts, conjectured combinatorial explanations for the Ramanujan congruences modulo 5 and 7, and conjectured the existence of a crank function for partitions that could provide a combinatorial proof of Ramanujan’s congruences modulo 11. Let N T (b, k, n) be the total number of parts in the partitions of n with rank congruent to b modulo k. If Mω(b, k, n) counts the total number of ones in the partitions of n with crank congruent to b modulo k, we have, for all integers n 0, Mω(1, 5, 5n + 4) + 2Mω(2, 5, 5n + 4) − 2Mω(3, 5, 5n + 4) − Mω(4, 5, 5n + 4) ≡ 0 (mod 5). We prove Andrews-Beck type congruence on N T (b, k, n) and N T 2(b, k, n) modulo 4 and 8 as follows.

Weighted D-rank moments of overpartitions

Weighted M2-rank moments of overpartitions

Proof of Theorem 3