受限排列上的下降顶集与下降底集的分布

施布特多项式与初等对称多项式

Schubert Polynomials and Elementary Symmetric Products

本文研究了施布特多项式分解为初等对称多项式的问题，提出基于排列模式回避的猜想，并探讨了其在代数组合学中的意义。

New results on the peak algebra

The peak algebra is a unital subalgebra of the symmetric group algebra, linearly spanned by sums of permutations with a common set of peaks. By exploiting the combinatorics of sparse subsets of [n-1] (and of certain classes of compositions of n called almost-odd and thin), we construct three new linear bases of this algebra. We discuss two peak analogs of the first Eulerian idempotent and construct a basis of semi-idempotent elements. We use these bases to describe the Jacobson radical of the peak algebra and to characterize the elements of this algebra in terms of the canonical action of the …

A refinement of Wilf-equivalence for patterns of length 4

In their paper \cite{DokosDwyer:Permutat12}, Dokos et al. conjecture that the major index statistic is equidistributed among 1423-avoiding, 2413-avoiding, and 2314-avoiding permutations. In this paper we confirm this conjecture by constructing two major index preserving bijections, $Θ:S_n(1423)\to S_n(2413)$ and $Ω:S_n(2314)\to S_n(2413)$. In fact, we show that $Θ$ (respectively, $Ω$) preserves numerous other statistics including the descent set, right-to-left maximum (respectively, left-to-right minimum), and a new statistic we call top-steps (respectively, bottom-steps).

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…

Stack sorting with restricted stacks

The (classical) problem of characterizing and enumerating permutations that can be sorted using two stacks connected in series is still largely open. In the present paper we address a related problem, in which we impose restrictions both on the procedure and on the stacks. More precisely, we consider a greedy algorithm where we perform the rightmost legal operation (here "rightmost" refers to the usual representation of stack sorting problems). Moreover, the first stack is required to be $σ$-avoiding, for some permutation $σ$, meaning that, at each step, the elements maintained in the stack av…

最优选择博弈中的重选

Reselection in the game of best choice

本文研究Steck提出的对称群概率分布，给出其基于排列统计量的新组合公式，定义带过滤步骤的最优选择博弈，证明最优胜率渐近为1/e且最优策略与经典模型不同。

非对称搁架洗牌的截止现象

Cutoff for asymmetric shelf shuffle

研究非对称搁架洗牌（每张牌以概率p放在顶部或以1-p放在底部）产生的排列分布，证明下降数和谷数构成充分统计量，并给出总变差距离、分离距离和相对熵的截止现象及显式轮廓。