Eǧecioǧlu and Remmel [Linear Multilinear Algebra 26 (1990) 59-84] 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 an algorithmic 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 [preprint, math.CO/9712230, 1997] we combine our involution with a result of Gasharov [Discrete Math. 157 (1996) 193-197] to give a combinatorial proof of a special case of the (3 + 1)-free Conjecture of Stanley and Stembridge [J. Combin. Theory Set. A 62 (1993) 261-279].
['Bruce E. Sagan', 'Jaejin Lee']
An algorithmic sign-reversing involution for special rim-hook tableaux