Universität zu Köln

Bringmann, Kathrin ORCID: 0000-0001-7126-1409, Gomez, Kevin ORCID: 0000-0001-8297-3209, Rolen, Larry and Tripp, Zack (2022). Infinite families of crank functions, Stanton-type conjectures, and unimodality. Res. Math. Sci., 9 (3). CHAM: SPRINGER INT PUBL AG. ISSN 2197-9847

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Abstract

Dyson's rank function and the Andrews-Garvan crank function famously give combinatorial witnesses for Ramanujan's partition function congruences modulo 5,7, and 11. While these functions can be used to show that the corresponding sets of partitions split into 5,7, or 11 equally sized sets, one may ask how to make the resulting bijections between partitions organized by rank or crank combinatorially explicit. Stanton recently made conjectures which aim to uncover a deeper combinatorial structure along these lines, where it turns out that minor modifications of the rank and crank are required. Here, we prove two of these conjectures. We also provide abstract criteria for quotients of polynomials by certain cyclotomic polynomials to have non-negative coefficients based on unimodality and symmetry. Furthermore, we extend Stanton's conjecture to an infinite family of cranks. This suggests further applications to other combinatorial objects. We also discuss numerical evidence for our conjectures, connections with other analytic conjectures such as the distribution of partition ranks.

Item Type:	Journal Article
Creators:	Creators Email ORCID ORCID Put Code Bringmann, Kathrin UNSPECIFIED orcid.org/0000-0001-7126-1409 UNSPECIFIED Gomez, Kevin UNSPECIFIED orcid.org/0000-0001-8297-3209 UNSPECIFIED Rolen, Larry UNSPECIFIED UNSPECIFIED UNSPECIFIED Tripp, Zack UNSPECIFIED UNSPECIFIED UNSPECIFIED
URN:	urn:nbn:de:hbz:38-668413
DOI:	10.1007/s40687-022-00333-3
Journal or Publication Title:	Res. Math. Sci.
Volume:	9
Number:	3
Date:	2022
Publisher:	SPRINGER INT PUBL AG
Place of Publication:	CHAM
ISSN:	2197-9847
Language:	English
Faculty:	Unspecified
Divisions:	Unspecified
Subjects:	no entry
Uncontrolled Keywords:	Keywords Language PARTITION; CONGRUENCES; POWERS Multiple languages Mathematics Multiple languages
URI:	http://kups.ub.uni-koeln.de/id/eprint/66841