Rational Noncrossing Partitions for all Coprime Pairs

2017/01/25 by Michelle Bodnar, Bodnar, Michelle
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #math.CO

paper · pdf · doi:10.48550/arxiv.1701.07198

43 pages, 14 figures, typos corrected, updated reference

arxiv created 2017/10/14 · arxiv updated 2017/10/17

Abstract

For coprime positive integers a<b, Armstrong, Rhoades, and Williams (2013) defined a set NC(a,b) of rational noncrossing partitions, a subset of the ordinary noncrossing partitions of \1, …, b-1\. Bodnar and Rhoades (2015) confirmed their conjecture that NC(a,b) is closed under rotation and proved an instance of the cyclic sieving phenomenon for this rotation action. We give a definition of NC(a,b) which works for all coprime a and b and prove closure under rotation and cyclic sieving in this more general setting. We also generalize noncrossing parking functions to all coprime a and b, and provide a character formula for the action of \mathfrakS_a × ℤ_b-1 on Park^NC(a,b).

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