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

John R. Britnell, Mark Wildon

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Let Bt(n) be the number of set partitions of a set of size t into at most n parts and let B't (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 B't (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.
Original languageEnglish
Pages (from-to)116-144
Number of pages29
JournalJournal of Combinatorial Theory, Series A
Early online date27 Dec 2016
Publication statusPublished - May 2017

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