A positivity conjecture for Jack polynomials

  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.

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