Congruences for Consecutive Coefficients of Gaussian Polynomials with Crank Statistics

Abstract

In this paper, we establish infinite families of congruences in consecutive arithmetic progressions modulo any odd prime $\ell$ for the function $p\big(n,m,N\big)$, which enumerates the partitions of $n$ into at most $m$ parts with no part larger than $N$. We also treat the function $p\big(n,m,(a,b]\big)$, which bounds the largest part above and below, and obtain similar infinite families of congruences. For $m \leq 4$ and $\ell = 3$, simple combinatorial statistics called "cranks" witness these congruences. We prove this analytically for $m=4$, and then both analytically and combinatorially for $m = 3$. Our combinatorial proof relies upon explicit dissections of convex lattice polygons.   For $m \leq 4$ and $\ell = 3$, simple combinatorial statistics called ``cranks"  witness these congruences.  We prove this analytically for $m=4$, and then both analytically and combinatorially for $m = 3$.  Our combinatorial proof relies upon explicit dissections of convex lattice polygons.