Polyhedral geometry, supercranks, and combinatorial witnesses of congruences for partitions into three parts. (October 2017)

• Record Type: Journal Article
• Authors: Breuer, Felix; Eichhorn, Dennis; Kronholm, Brandt
• Is Part Of: European journal of combinatorics. Volume 65(2017)
• Page Start: 230
• Page End: 252
• Publication Date: 2017-10
• Subjects: Combinatorial analysis -- Periodicals; Analyse combinatoire -- Périodiques; Combinatorial analysis; Periodicals; Electronic journals; 511.6
• DOI: 10.1016/j.ejc.2017.06.002
• Languages: English
• ISSNs: 0195-6698

Abstract

Abstract: In this paper, we use a branch of polyhedral geometry, Ehrhart theory, to expand our combinatorial understanding of congruences for partition functions. Ehrhart theory allows us to give a new decomposition of partitions, which in turn allows us to define statistics called supercranks that combinatorially witness every instance of divisibility of p ( n, 3 ) by any prime m ≡ − 1 ( mod 6 ), where p ( n, 3 ) is the number of partitions of n into three parts. A rearrangement of lattice points allows us to demonstrate with explicit bijections how to divide these sets of partitions into m equinumerous classes. The behavior for primes m ′ ≡ 1 ( mod 6 ) is also discussed.