A polyhedral model of partitions with bounded differences and a bijective proof of a theorem of Andrews, Beck, and Robbins

The main result of this paper is a bijective proof showing that the generating function for partitions with bounded differences between largest and smallest part is a rational function. This result is similar to the closely related case of partitions with fixed differences between largest and smallest parts which has recently been studied through analytic methods by Andrews, Beck, and Robbins. Our approach is geometric: We model partitions with bounded differences as lattice points in an infinite union of polyhedral cones. Surprisingly, this infinite union tiles a single simplicial cone. This construction then leads to a bijection that can be interpreted on a purely combinatorial level.Comment: 12 pages, 5 figure

Congruences for Consecutive Coefficients of Gaussian Polynomials with Crank Statistics

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

Cranks for partitions with bounded largest part

For 80 years, Dyson’s rank has been known as the partition statistic that witnesses the first two of Ramanujan’s celebrated congruences for the ordinary partition function. In this paper, we show that Dyson’s rank actually witnesses families of partition congruences modulo every prime . This comes from an in-depth study of when a “multiplicity-based statistic” is a crank witnessing congruences for the function p ` n, m˘ , which enumerates partitions of n with parts of size at most m. We also show that as the modulus increases, there is an ever-growing collection of distinct multiplicity-based cranks witnessing these same families