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Let $B_t(n)$ be the number of set partitions of a set of size~$t$ into atmost $n$ 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 newcombinatorial interpretations of the numbers $B_t(n)$ and $B'_t(n)$ usingsequences of random-to-top shuffles, %that leave a deck of cards invariant, andsequences of box moves on the Young diagrams of partitions. Using these ideaswe obtain a very short proof of a generalization of a result of Phatarfod onthe eigenvalues of the random-to-top shuffle. We also prove analogous resultsfor random-to-top shuffles that may flip certain cards. The proofs use theSolomon descent algebras of Types A, B and~D. We give generating functions andasymptotic results for all the combinatorial quantities studied in this paper.

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