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Britnell, Mark Wildon","submitted_at":"2015-07-16T23:55:33Z","abstract_excerpt":"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. 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