Some Schubert shenanigans

Abstract

We give a conjectured evaluation of the determinant of a certain matrix $\tilde{D}(n,k)$. The entries of $\tilde{D}(n,k)$ are either 0 or specializations ${\mathfrak{S}}_w(1,\dots ,1)$ of Schubert polynomials. The conjecture implies that the weak order of the symmetric group $S_n$ has the strong Sperner property. A number of peripheral results and problems are also discussed.

8 pages

Cited by in corpus (5)

• Derivatives of Schubert polynomials and proof of a determinant conjecture of Stanley
• A combinatorial duality between the weak and strong Bruhat orders
• Principal specializations of Schubert polynomials and pattern containment
• Differential operators on Schur and Schubert polynomials
• An efficient algorithm for deciding vanishing of Schubert polynomial coefficients