On Two Related Questions of Wilf Concerning Standard Young Tableaux
Miklos Bona
TL;DR
The paper addresses Wilf's questions on obtaining a purely combinatorial proof of the even-$k$ identity $ {2n\choose n} u_k(n) = \sum_{r=0}^{2n} {2n\choose r} (-1)^r y_k(r)y_k(2n-r)$ and determining the correct analogue for odd $k$, using the Robinson-Schensted correspondence and Beissinger's fixed-point result. It develops a sign-canceling, involution-based framework for the case $k=2n$, and extends the ideas to odd $k$ via fixed-point-free involutions constrained by decreasing subsequence bounds, yielding a nonnegative-summand reformulation. The paper then derives a complete closed form in the special case $k=3$, linking to Motzkin and Catalan numbers and producing the explicit identity $\sum_{r=0}^{2n} (-1)^r {2n\choose r} y_{3}(r)y_{3}(2n-r) = y_4(2n) = C_{n}C_{n+1}$. Overall, it clarifies when and why the summands can be taken nonnegative and connects tableau counts to classical combinatorial sequences, enriching the combinatorial understanding of Wilf's results.
Abstract
We consider two questions of Wilf related to Standard Young Tableaux. We provide a partial answer to one question, and that will lead us to a more general answer to the other question. Our answers are purely combinatorial.