Let $\mathfrak{S}_n$ be the permutation group on $[n]$. Given the pattern $\sigma=k(k-1)\cdots321$, let $I_n(\sigma)$ be the number of involutions in $\mathfrak{S}_n$ that avoid the pattern $\sigma$. Amitai Regev proved (see this paper and references therein) the case $k=4$: $$I_n((4321))=\sum_{k\geq0}\binom{n}{2k}C_k$$ which are the Motzkin numbers. Here $C_k$ are the Catalan numbers.
Given an integer partition $\lambda$, draw the Young diagram and fill in hook-lengths of each cell. If a number $t$ is not among these hook-lengths then $\lambda$ is called a $t$-core. If it misses $a, b, c$ then call it an $(a,b,c)$-core partition.
Let $N(n,n+1,n+2)$ be the number of all partitions that are $(n,n+1,n+2)$-core partitions. Then (see this paper for definitions and this result) $$N(n,n+1,n+2)=\sum_{k\geq0}\binom{n}{2k}C_k.$$
QUESTION. Is there a direct bijection between the above $(4321)$-avoiding involutions in $\mathfrak{S}_n$ and $(n,n+1,n+2)$-core partitions?
EDIT. Fedor's comment/question below alerted me to correct a mistake: we are only counting involutions and not all permutations.
