  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.
