关于Kerov多项式的Goulden公式和Rynn公式

Quadratic coefficients of Goulden-Rattan character polynomials

Goulden-Rattan polynomials give the exact value of the subdominant part of the normalized characters of the symmetric groups in terms of certain quantities ($C_i$) which describe the macroscopic shape of the Young diagram. The Goulden-Rattan positivity conjecture states that the coefficients of these polynomials are positive rational numbers with small denominators. We prove a special case of this conjecture for the coefficient of the quadratic term $C_2^2$ by applying certain bijections involving maps (i.e., graphs drawn on surfaces).

部分Jucys-Murphy元与星因子分解

Partial Jucys-Murphy elements and star factorizations

在这篇文章中，我们研究了一个给定的置换分解成星型置换的个数。特别是，我们给出了一个隐藏的对称性的自然解释，回答了I. P. Goulden和D.M. Jackson.我们也有一个新的证明他们的显式公式。另一个结果是由P. Biane引入的对称群代数的某些中心元素的正规类展开。为了得到这个结果，我们使用V. Ivanov和S.凯洛夫我们研究了它们的性质，并使用A. Lascoux和J.Y. Thibon给出了它们的幂和在不变子代数的自然基上的展开式。