Stanley's character polynomials and coloured factorizations in the symmetric group

Stanley 特征公式中的树与一个轮换的因子分解之间的双射

Bijection between trees in Stanley character formula and factorizations of a cycle

本文针对 Stanley 特征多项式最高次项系数对应的带装饰平面树，构造了其与轮换最小因子分解之间的明确双射。

Richard Stanley through a crystal lens and from a random angle

We review Stanley's seminal work on the number of reduced words of the longest element of the symmetric group and his Stanley symmetric functions. We shed new light on this by giving a crystal theoretic interpretation in terms of decreasing factorizations of permutations. Whereas crystal operators on tableaux are coplactic operators, the crystal operators on decreasing factorization intertwine with the Edelman-Greene insertion. We also view this from a random perspective and study a Markov chain on reduced words of the longest element in a finite Coxeter group, in particular the symmetric grou…

Additive induced-hereditary properties and unique factorization

We show that additive induced-hereditary properties of coloured hypergraphs can be uniquely factorised into irreducible factors. Our constructions and proofs are so general that they can be used for arbitrary concrete categories of combinatorial objects; we provide some examples of such combinatorial objects.

A Demazure crystal construction for Schubert polynomials

Stanley symmetric functions are the stable limits of Schubert polynomials. In this paper, we show that, conversely, Schubert polynomials are Demazure truncations of Stanley symmetric functions. This parallels the relationship between Schur functions and Demazure characters for the general linear group. We establish this connection by imposing a Demazure crystal structure on key tableaux, recently introduced by the first author in connection with Demazure characters and Schubert polynomials, and linking this to the type A crystal structure on reduced word factorizations, recently introduced by …

再论 Stanley 猜想

The Stanley Conjecture Revisited

本文基于 Stanley 猜想，提出 Jack Littlewood-Richardson 系数的窗口化性质，并将 Schur L-R 的分解性质推广到 Jack 情形，从而建立大量关系并实现一类情形的直接计算。

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

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 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. Résumé. Les cumulants libres sont des fo…