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

TL;DR

This paper provides new combinatorial interpretations of Bell numbers related to set partitions, connects these to random-to-top shuffles, and uses algebraic tools to analyze eigenvalues and asymptotics across Dynkin Types A, B, and D.

Contribution

It introduces novel combinatorial models for Bell numbers using shuffles and Young diagram moves, and extends eigenvalue results to more general shuffle types with algebraic proofs.

Findings

New combinatorial interpretations of Bell numbers and their variants.

Short proof of a generalized eigenvalue result for random-to-top shuffles.

Asymptotic formulas and generating functions for the studied quantities.

Abstract

Let $B_t(n)$ 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 $\{1,\ldots, t\}$ into at most $n$ parts such that no part contains both $1$ and~$t$ or both $i$ and $i+1$ for any $i \in \{1,\ldots,t-1\}$. We give two new combinatorial interpretations of the numbers $B_t(n)$ and $B'_t(n)$ using sequences of random-to-top shuffles, %that leave a deck of cards invariant, 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.