Séminaire Lotharingien de Combinatoire, 93B.25 (2025), 12 pp.
Edward E. Allen, Kyle Celano, and Sarah K. Mason
A new proof of an inverse Kostka matrix problem posed by
Eğecioğlu and Remmel and related identities in Sym and NSym
Abstract.
Eğecioğlu and Remmel provide a combinatorial proof (using special rim hook tableaux) that the Kostka matrix times its inverse equals the identity matrix and pose the problem of proving the reverse identity (that the inverse Kostka matrix times the Kostka matrix equals the identity) combinatorially. Sagan and Lee prove a special case of this identity using overlapping special rim hook tableaux. Loehr and Mendes provide a full proof using bijective matrix algebra that relies on the Eğecioğlu-Remmel map. In this extended abstract, we solve the problem in full generality independent of the Eğecioğlu-Remmel bijection. To do this, we start by proving NSym versions of both Kostka matrix identities using the tunnel hook coverings recently introduced by the first and third authors. Then we modify our sign-reversing involutions to reduce to Sym.
Received: November 15, 2024.
Accepted: February 15, 2025.
Final version: April 1, 2025.
The following versions are available: