Congruences for consecutive coefficients of Gaussian polynomials with crank statistics

Dennis Eichhorn¹ ; Lydia Engle² ; Brandt Kronholm

¹University of California, Irvine
²University of Minnesota

The electronic journal of combinatorics, Tome 29 (2022) no. 4

Résumé

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.

Détail

DOI : 10.37236/10493

Classification : 11P83, 05A17, 11P82
Mots-clés : Gaussian polynomials, crank statistics, arithmetic progressions