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Bell numbers, partition moves and the eigenvalues of the random-to-top shuffle in Dynkin Types A, B and D

Lookup NU author(s): Dr John Britnell

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Abstract

© 2016 Elsevier Inc.Let Bt(n) be the number of set partitions of a set of size t into at most n parts and let Bt ′(n) be the number of set partitions of {1,…,t} into at most n parts such that no part contains both 1 and t or both i and i+1 for any i∈{1,…,t−1}. We give two new combinatorial interpretations of the numbers Bt(n) and Bt ′(n) using sequences of random-to-top shuffles, and sequences of box moves on the Young diagrams of partitions. Using these ideas we obtain a very short proof of a generalization of a result of Phatarfod on the eigenvalues of the random-to-top shuffle. We also prove analogous results for random-to-top shuffles that may flip certain cards. The proofs use the Solomon descent algebras of Types A, B and D. We give generating functions and asymptotic results for all the combinatorial quantities studied in this paper.

Publication metadata

Author(s): Britnell JR, Wildon M

Publication type: Article

Publication status: Published

Journal: Journal of Combinatorial Theory. Series A

Year: 2017

Volume: 148

Pages: 116-144

Print publication date: 01/05/2017

Online publication date: 27/12/2016

Acceptance date: 02/04/2014

ISSN (print): 0097-3165

ISSN (electronic): 1096-0899

Publisher: Academic Press Inc.

URL: https://doi.org/10.1016/j.jcta.2016.12.003

DOI: 10.1016/j.jcta.2016.12.003

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