Explicit congruences for k-marked Durfee symbols

Abstract

In 2007, Andrews introduced k-marked Durfee symbols to give a combinatorial interpretation of 2kth symmetrized moment η2k(n) of ranks of partitions of n. Let Dk(n) denote the number of k-marked Durfee symbols of n. Bringmann, Garvan and Mahlburg proved that there exist infinitely many arithmetic progressions An+B such that Dk(An+B)≡0(modpj), where j is a positive integer and p≥5 is a prime. In addition, Keith proved that D4(3n)≡0(mod3). Motivated by their works, we establish some explicit congruences modulo 2, 3, 4 and 8 of Dk(n) for some small k by employing some identities due to Garvan, and Lewis and Santa-Gadea on M(r,m,n) which counts the number of partitions of n with crank congruent to r modulo m.