Define a partition move to be the removal and then addition of a box of the Young diagram of a partition. Thus there are unique partition moves from $(2,1)$ to $(3)$ and from $(2,1)$ to $(1,1,1)$, and two moves (corresponding to the two removable boxes) from $(2,1)$ to $(2,1)$. Let $M_t(n)$ be the number of sequences of $t$ partition moves from $(n)$ to $(n)$. For example, $M_3(2) = 4$.

In a joint paper with John Britnell we used the descent algebra of the symmetric group to show that $M_t(n)$ is the number of set partitions of $\{1,\ldots,t\}$ into at most $n$ subsets.

We do not have a bijective proof of this identity. There is a natural candidate for a bijection, using the RSK-correspondence; rather unexpectedly, this bijection works for all $t \le 7$ and $n\le 12$, but fails for $t = 8$ and $n=5$. (See Proposition 7.2 in the paper.)