Applications of Symmetric Functions to Cycle and Increasing Subsequence Structure after Shuffles (Part 2)

Applications of Symmetric Functions to Cycle and Subsequence Structure after Shuffles

Using symmetric function theory, we study the cycle structure and increasing subsequence structure of permutations after various shuffling methods, emphasizing the role of Cauchy type identities and the Robinson-Schensted-Knuth correspondence. One consequence is a cycle index for riffle shuffles when one deals from the bottom of the deck; conclusions are drawn concerning the distribution of fixed points and the asymptotic distribution of cycle structure. In studying cycle structure of unimodal permutations and their generalizations we are led to an interesting variation of the RSK corresponden…

On subsequences of quiddity cycles and Nichols algebras

We provide a tool to obtain local descriptions of quiddity cycles. As an application, we classify rank two affine Nichols algebras of diagonal type.

随机洗牌的转置前2的极限分布

Limit profile for the transpose top-2 with random shuffle

本文利用非交换傅里叶分析的比较方法，研究交错群上由特定3-循环生成的随机洗牌游走，确定其极限分布并给出交错群图的完整谱。

关于所有重数均相等的词中的最长递增子序列

On longest increasing subsequences in words in which all multiplicities are equal

本文针对所有重数相等的多重集排列推广了Gessel的Bessel行列式公式，提出一个表示为Gessel型Toeplitz行列式与对称群轮换指标多项式乘积多重积分的生成函数，用于统计不含特定长度递增子序列的词。

An Application of the Whitehouse Module to Riffle Shuffles Followed by a Cut

Using representation theoretic work on the Whitehouse module, a formula is obtained for the cycle structure of a riffle shuffle followed by a cut. This result will be merged with the paper [F6].

The combinatorics of biased riffle shuffles

This paper studies biased riffle shuffles, first defined by Diaconis, Fill, and Pitman. These shuffles generalize the well-studied Gilbert-Shannon-Reeds shuffle and convolve nicely. An upper bound is given for the time for these shuffles to converge to the uniform distribution; this matches lower bounds of Lalley. A careful version of a bijection of Gessel leads to a generating function for cycle structure after one of these shuffles and gives new results about descents in random permutations. Results are also obtained about the inversion and descent structure of a permutation after one of the…