A sign-reversing involution for rooted special rim-hook tableaux
Bruce E. Sagan, Jaejin Lee
Abstract
Egecioglu and Remmel gave an interpretation for the entries of the inverse Kostka matrix K-1 in terms of special rim-hook tableaux. They were able to use this interpretation to give a combinatorial proof that KK-1=I but were unable to do the same for the equation K-1K=I. We define a sign-reversing involution on rooted special rim-hook tableaux which can be used to prove that the last column of this second product is correct. In addition, following a suggestion of Chow we combine our involution with a result of Gasharov to give a combinatorial proof of a special case of the (3+1)-free Conjecture of Stanley and Stembridge.
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