On the Andrews-Stanley refinement of Ramanujan’s partition congruence modulo 5 and generalizations

Abstract

In a recent study of sign-balanced, labelled posets, Stanley introduced a new integral partition statistic s r a n k ( π ) = O ( π ) − O ( π ′ ) , \begin{equation*} \mathrm {srank}(\pi ) = {\mathcal O}(\pi ) - {\mathcal O}(\pi ’), \end{equation*} where O ( π ) {\mathcal O}(\pi ) denotes the number of odd parts of the partition π \pi and π ′ \pi ’ is the conjugate of π \pi . In a forthcoming paper, Andrews proved the following refinement of Ramanujan’s partition congruence mod 5 5 : p 0 ( 5 n + 4 ) a m p ; ≡ p 2 ( 5 n + 4 ) ≡ 0 ( mod 5 ) , p ( n ) a m p ; = p 0 ( n ) + p 2 ( n ) , \begin{align*} p_0(5n+4) &\equiv p_2(5n+4) \equiv 0 \pmod {5}, p(n) &= p_0(n) + p_2(n), \end{align*} where p i ( n ) p_i(n) ( i = 0 , 2 i=0,2 ) denotes the number of partitions of n n with s r a n k ≡ i ( mod 4 ) \mathrm {srank}\equiv i\pmod {4} and p ( n ) p(n) is the number of unrestricted partitions of n n . Andrews asked for a partition statistic that would divide the partitions enumerated by p i ( 5 n + 4 ) p_i(5n+4) ( i = 0 , 2 i=0,2 ) into five equinumerous classes. In this paper we discuss three such statistics: the ST-crank, the 2 2 -quotient-rank and the 5 5 -core-crank. The first one, while new, is intimately related to the Andrews-Garvan (1988) crank. The second one is in terms of the 2 2 -quotient of a partition. The third one was introduced by Garvan, Kim and Stanton in 1990. We use it in our combinatorial proof of the Andrews refinement. Remarkably, the Andrews result is a simple consequence of a stronger refinement of Ramanujan’s congruence mod 5 5 . This more general refinement uses a new partition statistic which we term the BG-rank. We employ the BG-rank to prove new partition congruences modulo 5 5 . Finally, we discuss some new formulas for partitions that are 5 5 -cores and discuss an intriguing relation between 3 3 -cores and the Andrews-Garvan crank.