Let $\Delta_n$ be the staircase-shaped partition $(n-1,n-2,\dots,1)$. Are there any non-obvious combinatorial objects that index $s_{\Delta_n}^2$? Here, $s_\lambda$ is the Schur function indexed by the partition $\lambda$. I would also be interested in objects corresponding to $s_{\Delta_n} s_{\Delta_{n+1}}$.

By non-obvious, I mean to avoid simple simple characterizations that avoid of any particular structure of $\Delta_n$ such as:

  • The monomial generating function over semi-standard Young tableaux of skew-shape $\Delta_{2n} \setminus (n)^n$ where $(n)^n$ is the square partition.

  • The generating function over Gessel's fundamental quasisymmetric functions indexed by descent sets of the standard Young tableau of the same skew shape. Essentially these are pairs of standard Young tableaux of shape $\Delta_n$ sharing the index set $\{1,,2,\dots,2n\}$. There are many equivalent objects using standard bijections such as RSK and Edelman-Greene.

Note the number of standard Young tableaux of skew-shape $\Delta_{2n} \setminus (n)^n$ is $$ f^{\Delta_{2n} \setminus (n)^n} = {2 {n \choose 2} \choose {n\choose 2}} (f^{\Delta_n})^2 $$ where $f^{\lambda}$ is the number of standard Young tableau of shape $\lambda$. The first five terms of this sequence are 1, 2, 80, 236544, 108973522944, unless I screwed something up. Note this does not appear in OEIS.

While my interest is primarily in combinatorial objects, I would also be quite interested in cases where $s_{\Delta_n}^2$ arises from geometry or representation theory in a non-obvious way.

Edit: I have since learned that $s^2_{\Delta_n} = P_{(2n,2n-2, \dots,2)}$ where $P_\lambda$ is the Schur-$P$ function of shape $\lambda$. This is a result of Dewitt, in her thesis (Theorem V.3). Similarly, $s_{\Delta_n} s_{\Delta_{n+1}} = P_{(2n+1,2n-1, \dots, 1)}$. These shapes are trapezoids. Since Schur-$P$ functions are generating functions of shifted standard Young tableaux with some off-diagonal entries circled (see e.g. Sagan), this provides a new family of objects indexing $s^2_{\Delta_n}$. Rather than posting this as an answer, let us now consider these objects "obvious" as well for the purpose of the question.

As an aside, reduced words of the longest element in Type B correspond to shifted standard Young tableaux of the latter trapezoid shape via Kraskiewicz insertion. However, in this setting no entries are circled.

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    $\begingroup$ While this might be not what you are looking for, Carre and Leclerc article on splitting the square of a Schur function might be useful. Domino tableaux come into play there. $\endgroup$ – user61318 Mar 3 '15 at 20:22
  • $\begingroup$ maybe of interest, maybe also obvious: expand the square into a sum of Schur functions (using LR) and look at the sum in representation space: replace $s_\lambda$ by $\chi_\lambda$ in $S_{n(n+1)}$ and check that the non-zero classes are indexed by partitions $\mu$ of n(n+1) with odd parts <= 2n-1. $\endgroup$ – Wouter M. Jul 18 '15 at 20:22

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