Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
권호 표지

GENERALIZATION OF THE SIGN REVERSING INVOLUTION ON THE SPECIAL RIM HOOK TABLEAUX

  • Received : 2010.08.16
  • Accepted : 2010.09.10
  • Published : 2010.09.30
Download PDF
⟨ Previous Next ⟩

Abstract

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 $K^{-1}K=I$.

Keywords

References

  1. O. Egecioglu and J. Remmel, A combinatorial interpretation of the inverse Kostka matrix, Linear Multilinear Algebra 26(1990), 59-84. https://doi.org/10.1080/03081089008817966
  2. W. Fulton, "Young Tableaux", London Mathematical Society Student Texts 35, Cambridge University Press, Cambridge, 1999.
  3. I. Gessel and G. Viennot, Binomial determinants, paths, and hook length for- mulae, Adv. Math. 58 (1985), 300-321. https://doi.org/10.1016/0001-8708(85)90121-5
  4. I. Gessel and G. Viennot, Determinants, paths, and plane partitions, in prepa-ration.
  5. B. Lindstrom, On the vector representation of induced matroids, Bull. Lond. Math. Soc. 5 (1973), 85-90. https://doi.org/10.1112/blms/5.1.85
  6. I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd edition, Oxford University Press, Oxford, 1995.
  7. B. Sagan, The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions, 2nd edition, Springer-Verlag, New York, 2001.
  8. B. Sagan and Jaejin Lee, An algorithmic sign-reversing involution for special rim-hook tableaux, J. Algorithms 59(2006), 149-161. https://doi.org/10.1016/j.jalgor.2004.06.003
  9. R. P. Stanley, Enumerative Combinatorics, Volume 2, Cambridge University Press, Cambridge, 1999.
  10. D. Stanton and D. White, A Schensted correspondence for rim hook tableaux, J. Combin. Theory Ser. A40 (1985), 211-247.
  11. D. White, A bijection proving orthogonality of the characters of $S_n$, Adv. Math. 50 (1983), 160-186. https://doi.org/10.1016/0001-8708(83)90038-5