 # Cumulants of Jack symmetric functions and the b-conjecture

Dołęga, Maciej; Féray, Valentin (2017). Cumulants of Jack symmetric functions and the b-conjecture. Transactions of the American Mathematical Society, 369(12):9015-9039.

## Abstract

Goulden and Jackson (1996) introduced, using Jack symmetric functions, some multivariate generating series ψ(x, y, z; t,1 + β) that might be interpreted as a continuous deformation of the generating series of rooted hypermaps. They made the following conjecture: the coefficients of ψ(x, y, z; t,1 + β) in the power-sum basis are polynomials in β with nonnegative integer coefficients (by construction, these coefficients are rational functions in β). We partially prove this conjecture, nowadays called the b-conjecture, by showing that coefficients of ψ(x, y, z; t, 1+β) are polynomials in β with rational coefficients. A key step of the proof is a strong factorization property of Jack polynomials when the Jack-deformation parameter α tends to 0, which may be of independent interest. © 2017 American Mathematical Society.

## Abstract

Goulden and Jackson (1996) introduced, using Jack symmetric functions, some multivariate generating series ψ(x, y, z; t,1 + β) that might be interpreted as a continuous deformation of the generating series of rooted hypermaps. They made the following conjecture: the coefficients of ψ(x, y, z; t,1 + β) in the power-sum basis are polynomials in β with nonnegative integer coefficients (by construction, these coefficients are rational functions in β). We partially prove this conjecture, nowadays called the b-conjecture, by showing that coefficients of ψ(x, y, z; t, 1+β) are polynomials in β with rational coefficients. A key step of the proof is a strong factorization property of Jack polynomials when the Jack-deformation parameter α tends to 0, which may be of independent interest. © 2017 American Mathematical Society.

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