• Korean Journal of Mathematics
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• v.18 no.3
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• pp.289-298
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• 2010

E$\breve{g}$ecio$\breve{g}$lu and Remmel [1] gave a combinatorial interpretation for the entries of the inverse Kostka matrix $K^{-1}$. Using this interpretation Sagan and Lee [8] constructed a sign reversing involution on special rim hook tableaux. In this paper we generalize Sagan and Lee's algorithm on special rim hook tableaux to give a combinatorial partial proof of $K^{-1}K=I$.

• Korean Journal of Mathematics
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• v.16 no.3
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• pp.425-437
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• 2008

Let P be a poset and G = G(P) be the incomparability graph of P. Stanley [7] defined the chromatic symmetric function $X_{G(P)}$ which generalizes the chromatic polynomial ${\chi}_G$ of G, and showed all coefficients are nonnegative in the e-expansion of $X_{G(P)}$ for a poset P of rank 1. In this paper, we construct a sign reversing involution on the set of special rim hook P-tableaux with some conditions. It gives a combinatorial proof for (3+1)-free conjecture of a poset P of rank 1.