Explicit formulae for Kerov polynomials
P. Petrullo
and D. Senato2
2P. Petrullo
DOI: 10.1007/s10801-010-0239-3
Abstract
We prove two formulae expressing the Kerov polynomial Σ k as a weighted sum over the set of noncrossing partitions of the set {1,\cdots , k+1}. We also give a combinatorial description of a family of symmetric functions specializing in the coefficients of Σ k .
Pages: 141–151
Keywords: keywords kerov polynomials; noncrossing partitions; symmetric group; normalized characters; symmetric functions
Full Text: PDF
References
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2. Biane, P.: Some properties of crossings and partitions. Discrete Math. 175, 41-53 (1997)
3. Biane, P.: Representations of symmetric groups and free probability. Adv. Math. 138(1), 126-181 (1998)
4. Biane, P.: Characters of the Symmetric Group and Free Cumulants. Lecture Notes in Math., vol. 1815, pp. 185-200. Springer, Berlin (2003)
5. Biane, P.: On the formula of Goulden and Rattan for Kerov Polynomials. Sémin. Lothar. Comb. 55 (2006)
6. Doł\?ega, M., Féray, V., Śniady, P.: Explicit combinatorial interpretation of Kerov character polynomials as number of permutation factorizations. Adv. Math. (2010). doi:
7. Di Nardo, E., Petrullo, P., Senato, D.: Cumulants and convolutions via Abel polynomials. Eur. J. Combin. (2010).
8. Féray, V.: Combinatorial interpretation and positivity of Kerov's character polynomials. J. Algebr. Comb. 29, 473-507 (2009)
9. Féray, V.: Fonctions sur l'ensemble des diagrammes de Young: caractères du groupe symétrique et polynômes de Kerov. Ph.D. thesis (2009). Available at
10. Goulden, I.P., Rattan, A.: An explicit form for Kerov's character polynomials. Trans. Am. Math. Soc. 359, 3669-3685 (2007)
11. Kerov, S.V.: Talk at IHP Conference (2000)
12. Lassalle, M.: Two positive conjectures for Kerov polynomials. Adv. Appl. Math. 41, 407-422 (2008)
13. Lehner, F.: Free cumulants and enumeration of connected partitions. Eur. J. Comb. 22, 1025-1031 (2002)
14. Macdonald, I.G.: Symmetric Functions and Hall Polynomials, 2nd edn. Oxford University Press, London (1995)
15. Nica, A., Speicher, R.: Lectures on the Combinatorics of Free Probability. Cambridge University Press, Cambridge (2006)
16. Speicher, R.: Multiplicative functions on the lattice on nocrossing partitions and free convolution.
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