Congruences for Consecutive Coefficients of Gaussian Polynomials with Crank Statistics

Authors

Dennis Eichhorn
Lydia Engle
Brandt Kronholm, The University of Texas Rio Grande Valley

Document Type

Article

Publication Date

12-2-2022

Abstract

In this paper, we establish infinite families of congruences in consecutive arithmetic progressions modulo any odd prime ℓ for the function p ( n , m , N ) , which enumerates the partitions of n into at most m parts with no part larger than N . We also treat the function p ( n , m , ( a , b ] ) , which bounds the largest part above and below, and obtain similar infinite families of congruences.

For m ≤ 4 and ℓ = 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 ≤ 4 and ℓ = 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.

Comments

©The authors. Released under the CC BY-ND license (International 4.0).

Recommended Citation

Eichhorn, Dennis, Lydia Engle, and Brandt Kronholm. "Congruences for Consecutive Coefficients of Gaussian Polynomials with Crank Statistics." The Electronic Journal of Combinatorics (2022): P4-38. https://doi.org/10.37236/10493

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

Publication Title

Electronic Journal of Combinatorics

DOI

10.37236/10493