## A positivity conjecture for Jack polynomials

Page of Michel Lassalle

List of publications

Some preprints

Tables for
characters of the
symmetric group

Symmetric functions
and
Jucys-Murphy elements

Tables for
Jack polynomials :

Jack polynomials
and alpha-contents

Jack polynomials
and free cumulants

This page gives data supporting the following conjecture for Jack polynomials, which generalizes Stanley's ex-conjecture for normalized characters of the symmetric group.

Being given some parameter &alpha and an arbitrary partition &lambda, we consider the Jack polynomial associated with &lambda, and its development in terms of the power sum symmetric functions, i.e. we write

In a recent paper we conjecture that, when &lambda is formed of m rectangular blocks, i.e. has the shape

and &mu has no part 1 and weight k, the coefficients

are polynomials in the variables

with nonnegative integer coefficients.

Here we list these coefficients for m=3, and for any partition &mu without any part 1, such that weight(&mu) - length(&mu) < 6.

Table giving the coefficients for weight(&mu) - length(&mu) from 1 to 5
(1.7 Mo).

Actually we have computed these quantities for weight(&mu) - length(&mu) < 9. They are available upon request.

Our results are in Maple format. They should be read as follows.
• The letter b stands for the parameter (&alpha - 1).
• The partition &lambda is formed of three rectangular blocks (P1,Q1), (P2,Q2) and (P3,Q3).
• &mu denotes a partition without any part 1.
• The table gives first the partition &mu, then

as a polynomial in b, P1, -Q1, P2, -Q2, P3, -Q3.