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On the sign-imbalance of partition shapes

Jonas Sjöstrand
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Abstract

Let the sign of a standard Young tableau be the sign of the permutation you get by reading it row by row from left to right, like a book. A conjecture by Richard Stanley says that the sum of the signs of all SYTs with n squares is 2^[n/2]. We present a stronger theorem with a purely combinatorial proof using the Robinson-Schensted correspondence and a new concept called chess tableaux. We also prove a sharpening of another conjecture by Stanley concerning weighted sums of squares of sign-imbalances. The proof is built on a remarkably simple relation between the sign of a permutation and the signs of its RS-corresponding tableaux.

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Jonas Sjöstrand. 2005-09-30. On the sign-imbalance of partition shapes. https://arxiv.org/abs/math/0309231

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