Staircase skew Schur functions are Schur
Federico Ardila
and Luis G. Serrano
Department of Mathematics, San Francisco State University, San Francisco, USA
DOI: 10.1007/s10801-012-0342-8
Abstract
We prove Stanley's conjecture that, if δ n is the staircase shape, then the skew Schur functions $s_{\delta_{n} / μ}$ are non-negative sums of Schur P-functions. We prove that the coefficients in this sum count certain fillings of shifted shapes. In particular, for the skew Schur function $s_{\delta_{n} / δ_{n-2}}$ , we discuss connections with Eulerian numbers and alternating permutations.
Pages: 409–423
Keywords: Schur functions; Schur P-functions; shifted tableaux; Eulerian numbers; alternating permutations
Full Text: PDF
References
Carlitz, L.: Enumeration of up-down sequences. Discrete Math. 4, 273-286 (1973) CrossRef Dewitt, E.: Identities relating Schur s-functions and Q-functions. Ph.D. Thesis, University of Michigan (2012) Foulkes, H.O.: Enumeration of permutations with prescribed up-down and inversion sequences. Discrete Math. 15, 235-252 (1976) CrossRef Fulton, W.: Young Tableaux. London Mathematical Society Student Texts, vol.
35. Cambridge University Press, Cambridge (1997) Gessel, I.M.: Generating functions and enumeration of sequences. Ph.D. Thesis, Massachusetts Institute of Technology (1977) Gessel, I.M.: Symmetric functions and P-recursiveness. J. Comb. Theory, Ser. A 53, 257-285 (1990) CrossRef Goulden, I.P., Jackson, D.M.: Combinatorial Enumeration. Wiley, New York (1983) (Dover Reprint, 2004) Haiman, M.D.: On mixed insertion, symmetry, and shifted Young tableaux. J. Comb. Theory, Ser. A 50, 196-225 (1989) CrossRef Haiman, M.D.: Dual equivalence with applications, including a conjecture of Proctor. Discrete Math. 99, 79-113 (1992) CrossRef Hoffman, P.N., Humphreys, J.F.: Projective Representations of the Symmetric Groups. Oxford University Press, London (1992) Józefiak, T.: Schur Q-functions and cohomology of isotropic Grassmannians. Math. Proc. Camb. Philos. Soc. 109, 471-478 (1991) CrossRef Sagan, B.: Shifted tableaux, Schur Q-functions, and a conjecture of R. Stanley. J. Comb. Theory, Ser. A 45, 62-103 (1987) CrossRef Shimozono, M.: Multiplying Schur Q-functions. J. Comb. Theory, Ser. A $87(1)$, 198-232 (1999) CrossRef Stanley, R.P.: Enumerative Combinatorics, vol.
2. Cambridge University Press, Cambridge (1999) CrossRef Stanley, R.P.: Personal communication (2001) Stembridge, J.: Shifted tableaux and the projective representations of symmetric groups. Adv. Math. 74, 87-134 (1989) CrossRef
35. Cambridge University Press, Cambridge (1997) Gessel, I.M.: Generating functions and enumeration of sequences. Ph.D. Thesis, Massachusetts Institute of Technology (1977) Gessel, I.M.: Symmetric functions and P-recursiveness. J. Comb. Theory, Ser. A 53, 257-285 (1990) CrossRef Goulden, I.P., Jackson, D.M.: Combinatorial Enumeration. Wiley, New York (1983) (Dover Reprint, 2004) Haiman, M.D.: On mixed insertion, symmetry, and shifted Young tableaux. J. Comb. Theory, Ser. A 50, 196-225 (1989) CrossRef Haiman, M.D.: Dual equivalence with applications, including a conjecture of Proctor. Discrete Math. 99, 79-113 (1992) CrossRef Hoffman, P.N., Humphreys, J.F.: Projective Representations of the Symmetric Groups. Oxford University Press, London (1992) Józefiak, T.: Schur Q-functions and cohomology of isotropic Grassmannians. Math. Proc. Camb. Philos. Soc. 109, 471-478 (1991) CrossRef Sagan, B.: Shifted tableaux, Schur Q-functions, and a conjecture of R. Stanley. J. Comb. Theory, Ser. A 45, 62-103 (1987) CrossRef Shimozono, M.: Multiplying Schur Q-functions. J. Comb. Theory, Ser. A $87(1)$, 198-232 (1999) CrossRef Stanley, R.P.: Enumerative Combinatorics, vol.
2. Cambridge University Press, Cambridge (1999) CrossRef Stanley, R.P.: Personal communication (2001) Stembridge, J.: Shifted tableaux and the projective representations of symmetric groups. Adv. Math. 74, 87-134 (1989) CrossRef