Submit progress

on Tableaux problem 26

The statement

26.[Stan1999a]*Ch. 7 Exer. 38(b) Fix $0\leq k\leq\binom{n}{2}$ , and let $\ell(w)$ denote the number of inversions of the permutation $w\in\mathfrak{S}_{n}$ . Let $\lambda$ and $\mu$ b partitions of length at most $n$ , with $\mu\subset\lambda$ . Define the symmetric function

t_{\lambda/\mu,k}=(-1)^{k}\sum_{\overset{w\in\mathfrak{S}_{n}}{\ell(w)\geq k}}\varepsilon_{w}h_{\lambda+\delta-w(\mu+\delta)}.

Thus $t_{\lambda/\mu,k}$ is a “truncation” of the Jacobi–Trudi expansion [Stan1999a]*Ch. 7 eq. 7.69 of $s_{\lambda/\mu}$ . Is there a “nice” combinatorial interpretation of the scalar product $\left<t_{\lambda/\mu,k},s_{\nu}\right>$ ?

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.