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While working on some computations on Hilbert schemes, I came across the following combinatorial problem. Let $D(k,n)$ be the weighted number of binary trees (children are left/right) with $n$ ...

Let $ c_{n,k} $ be the Simsun permutations$^1$ defined by the following relations: $\displaystyle c_{n,0} = 1, \hspace{0.1cm} (n \ge 1);$ $$ c_{n,k} = (k+1) c_{n-1,k} +(n-2k+1) c_{n-1,k-1}, \hspace{0....