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Characters of symmetric groups in terms of free cumulants and Frobenius coordinates

Abstract : Free cumulants are nice and useful functionals of the shape of a Young diagram, in particular they give the asymptotics of normalized characters of symmetric groups $\mathfrak{S}(n)$ in the limit $n \to \infty$. We give an explicit combinatorial formula for normalized characters of the symmetric groups in terms of free cumulants. We also express characters in terms of Frobenius coordinates. Our formulas involve counting certain factorizations of a given permutation. The main tool are Stanley polynomials which give values of characters on multirectangular Young diagrams.
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Maciej Dolega, Valentin Féray, Piotr Sniady. Characters of symmetric groups in terms of free cumulants and Frobenius coordinates. 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009), 2009, Hagenberg, Austria. pp.337-348. ⟨hal-01185445⟩

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