k-Marked Dyson Symbols and Congruences for Moments of Cranks
William Y.C. Chen, Kathy Q. Ji and Erin Y.Y. Shen
Abstract: By introducing k-marked Durfee symbols, Andrews found a combinatorial interpretation of 2k-th symmetrized moment η2k(n) of ranks of partitions of n. Recently, Garvan introduced the 2k-th symmetrized moment η2k(n) of cranks of partitions of n in the study of the higher-order spt-function sptk(n). In this paper, we give a combinatorial interpretation of η2k(n). We introduce k-marked Dyson symbols based on a representation of ordinary partitions given by Dyson, and we show that η2k(n) equals the number of (k + 1)-marked Dyson symbols of n. We then introduce the full crank of a k-marked Dyson symbol and show that there exist an infinite family of congruences for the full crank function of k-marked Dyson symbols which implies that for fixed prime p ≥ 5 and positive integers r and k ≤ (p - 1)/2, there exist infinitely many non-nested arithmetic progressions An + B such that η2k(An + B) ≡ 0 (mod pr).
AMS Classification: 05A17, 11P83, 05A30
Keywords: crank of a partition, moments of cranks, congruence