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^{ -1}K=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.
Categories: math.CO