Page of Michel Lassalle
List of publications
Some preprints
Tables for characters of the symmetric group
Symmetric functions and JucysMurphy elements
Tables for Jack polynomials :
Jack polynomials and alphacontents
A conjecture for Jack polynomials


This page gives new data for Jack polynomials. Our results have been published there.
Being given some parameter &alpha and an arbitrary partition &lambda, we consider the Jack polynomial associated to &lambda, and its development in terms of the power sum symmetric functions, i.e. we write
For &mu with no part 1 and weight k, we give the explicit expression of the coefficients
in terms of the free cumulants of the anisotropic diagram of &lambda.
These coefficients
are known to be polynomials in the free cumulants. We list them
 for any partition &mu, with weight(&mu)  length(&mu) < 9,
 when &mu is a hook (r,1,...,1), for r from 2 to 20,
 when &mu=(r,s) has length 2, for r+s from 4 to 18.
Our data support the following conjectures :
 These coefficients are polynomials in &alpha and &beta = 1 &alpha, with integer coefficients.
 When &mu is a hook, their integer coefficients are nonnegative.
 When &mu is not a hook, their integer coefficients may be negative but an appropriately modified polynomial has still nonnegative coefficients.
These conjectures extend the KerovBiane exconjecture for characters of the symmetric group, recently proved by Feray.
