A new proof of an Eğecioğlu--Remmel inverse Kostka matrix problem via a Garsia--Milne involution involving Sym and NSym

Abstract

Eğecioğlu and Remmel provide a combinatorial proof (using special rim hook tableaux) that the product of the Kostka matrix $K$ and its inverse $K^{-1}$ equals the identity matrix $I$. They then pose the problem of proving the reverse identity $K^{-1}K =I$ 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 article, 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 sign-reversing involutions involving the tunnel hook coverings recently introduced by the first and third authors. Then we modify our sign-reversing involutions to reduce to Sym. Finally, we show that our bijection is different than the Loehr and Mendes result by constructing an injective map between special rim tableaux and the symmetric group $S_n.$