k-Marked Dyson Symbols and Congruences for Moments of Cranks

TL;DR

This paper introduces $k$-marked Dyson symbols to interpret symmetrized moments of cranks of partitions and establishes infinite families of congruences for these moments modulo prime powers.

Contribution

It provides a combinatorial interpretation of symmetrized moments of cranks using $k$-marked Dyson symbols and proves infinite congruence families for their crank functions.

Findings

Combinatorial interpretation of $\mu_{2k}(n)$ via $k$-marked Dyson symbols.

Existence of infinite congruence families for the crank function modulo prime powers.

Extension of Andrews' and Garvan's work on moments of ranks and cranks.

Abstract

By introducing $k$-marked Durfee symbols, Andrews found a combinatorial interpretation of $2k$-th symmetrized moment $\eta_{2k}(n)$ of ranks of partitions of $n$. Recently, Garvan introduced the $2k$-th symmetrized moment $\mu_{2k}(n)$ of cranks of partitions of $n$ in the study of the higher-order spt-function $spt_k(n)$. In this paper, we give a combinatorial interpretation of $\mu_{2k}(n)$. We introduce $k$-marked Dyson symbols based on a representation of ordinary partitions given by Dyson, and we show that $\mu_{2k}(n)$ 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 $p\geq 5$ and positive integers $r$ and $k\leq (p-1)/2$, there exist infinitely many non-nested arithmetic progressions $An+B$ such that $\mu_{2k}(An+B)\equiv 0\pmod{p^r}$.