Coherent random permutations with record statistics

记录统计量与Schubert积的前向稳定性

The record statistic and forward stability of Schubert products

本文通过排列的记录统计量研究Schubert多项式积的前向稳定性，分析了均匀排列、Grassmannian排列和布尔排列的记录概率，并推导了其渐进行为。

记录偏置排列及其排列极限

Record-biased permutations and their permuton limit

本文研究了一种基于记录数的非均匀排列分布，探讨了其生成过程及高效采样方法，并分析了经典排列统计量的期望及其极限分布，最终在记录数线性增长时证明了排列趋于确定性排列极限。

Coherent permutations with descent statistic and the boundary problem for the graph of zigzag diagrams

The graph of zigzag diagrams is a close relative of Young's lattice. The boundary problem for this graph amounts to describing coherent random permutations with descent-set statistic, and is also related to certain positive characters on the algebra of quasi-symmetric functions. We establish connections to some further relatives of Young's lattice and solve the boundary problem by reducing it to the classification of spreadable total orders on integers, as recently obtained by Jacka and Warren.

The height of record-biased trees

Given a permutation $σ$, its corresponding binary search tree is obtained by recursively inserting the values $σ(1),\ldots,σ(n)$ into a binary tree so that the label of each node is larger than the labels of its left subtree and smaller than the labels of its right subtree. In 1986, Devroye proved that the height of such trees when $σ$ is a random uniform permutation is of order $(c^*+o_\mathbb{P}(1))\log n$ as $n$ tends to infinity, where $c^*$ is the only solution to $c\log(2e/c)=1$ with $c\geq2$. In this paper, we study the height of binary search trees drawn from the record-biased model of…

Record-dependent measures on the symmetric groups

A probability measure $P_n$ on the symmetric group ${\mathfrak S}_n$ is said to be record-dependent if $P_n(σ)$ depends only on the set of records of a permutation $σ\in{\mathfrak S}_n$. A sequence $P=(P_n)_{n\in{\mathbb N}}$ of consistent record-dependent measures determines a random order on $\mathbb N$. In this paper we describe the extreme elements of the convex set of such $P$. This problem turns out to be related to the study of asymptotic behavior of permutation-valued growth processes, to random extensions of partial orders, and to the measures on the Young-Fibonacci lattice.

Square permutations are typically rectangular

We describe the limit (for two topologies) of large uniform random square permutations, i.e., permutations where every point is a record. The starting point for all our results is a sampling procedure for asymptotically uniform square permutations. Building on that, we first describe the global behavior by showing that these permutations have a permuton limit which can be described by a random rectangle. We also explore fluctuations about this random rectangle, which we can describe through coupled Brownian motions. Second, we consider the limiting behavior of the neighborhood of a point in th…