Annals of Combinatorics 2 (1998) 145-163

Some Combinatorial Conjectures for Shifted Jack Polynomials

Michel Lassalle

Ecole Polytechnique, 91128 Palaiseau Cedex, France

Received May 18, 1998

AMS Subject Classification: 33C50, 33C80, 22E30, 05A10

Abstract. We present several combinatorial conjectures related to Jack generalized binomial coefficients, or equivalently to shifted Jack polynomials. We prove these conjectures when the degree of these polynomials is ≤ 5.

Keywords: Jack polynomials, shifted Jack polynmials, generalized and ordinary binomial coefficients, Stirling numbers of the first kind


1.  C. Bingham, An identity involving partitional generalized binomial coefficients, J. Multivariate Analysis 4 (1974) 210–223.

2.  J. Kaneko, Selberg integrals and hypergeometric functions associated with Jack polynomials, SIAM J. Math. Anal. 24 (1993) 1086–1110.

3.  S. Kerov, A. kounkov, and G. Olshanski, The boundary of Young graph with Jack edge multiplicities, Internat. Math. Res. Notices (to appear).

4.  M. Lassalle, Une formule du binôme gènèralisèe pour les polynòmes de Jack, C. R. Acad. Sci. Paris 310 (1990) 253–256.

5.  M. Lassalle, Coefficients binomiaux gènèralisès et polynômes de Macdonald, J. Funct. Anal. (to appear).

6.  M. Lassalle, Some combinatorial conjectures for Jack polynomials, Ann. Combin. 2 (1998) 61–83.

7.  M. Lassalle, Quelques conjectures combinatoires relatives à la formule classique de Chu-Vandermonde, Adv. Appl. Math. (to appear).

8.  M. Lassalle, Explicitation des polynômes de Jack (in preparation).

9.  I.G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd Ed., Clarendon Press, Oxford, 1995.

10.  A. Okounkov, (Shifted) Macdonald polynomials: q-integral representation and combinatorial formula, Compositio Math. 112 (1998) 147–182.

11.  A. Okounkov and G. Olshanski, Shifted Jack polynomials, binomial formula and applications, Math. Res. Lett. 4 (1997) 69–78.

12.  R.P. Stanley, Some combinatorial properties of Jack symmetric functions, Adv. Math. 77 (1989) 76–115.

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