Explicit combinatorial interpretation of Kerov character polynomials as numbers of permutation factorizations

Abstract

We find an explicit combinatorial interpretation of the coefficients of Kerov character polynomials which express the value of normalized irreducible characters of the symmetric groups S(n) in terms of free cumulants R_2,R_3,... of the corresponding Young diagram. Our interpretation is based on counting certain factorizations of a given permutation.

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Cited by in corpus (30)

• Gaussian fluctuations of Young diagrams and structure constants of Jack characters
• Jack polynomials and free cumulants
• Asymptotics of characters of symmetric groups related to Stanley character formula
• Jack polynomials and orientability generating series of maps
• Zonal polynomials via Stanley's coordinates and free cumulants
• On Kerov polynomials for Jack characters
• Gaussian fluctuations of Jack-deformed random Young diagrams
• Structure coefficients for Jack characters: approximate factorization property
• Asymptotics of Jack characters
• Random strict partitions and random shifted tableaux
• Shifted symmetric functions and multirectangular coordinates of Young diagrams
• Polynomial splitting measures and cohomology of the pure braid group
• Stanley character polynomials
• Asymptotic results for Representation Theory
• Asymptotics of characters of symmetric groups: structure of Kerov character polynomials
• Linear versus spin: representation theory of the symmetric groups
• A Spin Analogue of Kerov Polynomials
• New characterization of two-state normal distribution
• Combinatorics of asymptotic representation theory
• Random matrices associated to Young diagrams
• Bijection between oriented maps and weighted non-oriented maps
• Normalized characters of symmetric groups and Boolean cumulants via Khovanov's Heisenberg category
• Quasi-symmetric functions as polynomial functions on Young diagrams
• Characters of symmetric groups in terms of free cumulants and Frobenius coordinates
• Planar algebras for the Young graph and the Khovanov Heisenberg category
• Universality of global asymptotics of Jack-deformed random Young diagrams at varying temperatures
• Bijection between trees in Stanley character formula and factorizations of a cycle
• Modulus of continuity of Kerov transition measure for continual Young diagrams
• Asymptotyczna teoria reprezentacji grup permutacji (Asymptotic representation theory of symmetric groups)
• $\lambda$-shaped random matrices, $\lambda$-plane trees, and $\lambda$-Dyck paths