Plane recursive trees, Stirling permutations and an urn model

Generalized Stirling permutations, families of increasing trees and urn models

Bona [2007+] studied the distribution of ascents, plateaux and descents in the class of Stirling permutations, introduced by Gessel and Stanley [1978]. Recently, Janson [2008+] showed the connection between Stirling permutations and plane recursive trees and proved a joint normal law for the parameters considered by Bona. Here we will consider generalized Stirling permutations extending the earlier results of Bona and Janson, and relate them with certain families of generalized plane recursive trees, and also $(k+1)$-ary increasing trees. We also give two different bijections between certain f…

关于Pólya-Young罐模型与增长过程

On Pólya-Young urn models and growth processes

本文推广了Pólya-Young罐模型，利用噪声增强Bessel过程局部时给出极限律，通过鞅结构得到中心极限定理和重对数律，并建立其与递增树、Stirling排列推广及带竞争中国餐馆过程的联系。

Symmetric generating functions and Euler-Stirling statistics on permutations

We present (bi-)symmetric generating functions for the joint distributions of Euler-Stirling statistics on permutations, including the number of descents ($\mathsf{des}$), inverse descents ($\mathsf{ides}$), the number of left-to-right maxima ($\mathsf{lmax}$), the number of right-to-left maxima ($\mathsf{rmax}$) and the number of left-to-right minima ($\mathsf{lmin}$). We also show how they recover the classical symmetric generating function of permutations due to Carlitz, Roselle and Scoville (1966). Our proofs exploit three different recursive constructions of inversion sequences, bijection…

随机递归树与优先连接树的广播诱导着色

Broadcasting induced colourings of random recursive trees and preferential attachment trees

本文针对随机线性优先连接树的广播诱导二着色问题，结合Pólya瓮模型与解析组合学方法，证明了多类相关统计量经重标度后的极限分布，拓展了随机优先连接树的键渗流研究。

Recursions for Excedance number in some permutations groups

The excedance number for S_n is known to have an Eulerian distribution. Nevertheless, the classical proof uses descents rather than excedances. We present a direct recursive proof which seems to be folklore and extend it to the colored permutation groups G_r,n. The generalized recursion yields some interesting connection to Stirling numbers of the second kind. We also show some logconcavity result concerning a variant of the excedance number. Finally, we show that the generating function of the excedance number defined on G_r,n is symmetric.

Context-free Grammars for Permutations and Increasing Trees

In this paper, we introduce the notion of a grammatical labeling to describe a recursive process of generating combinatorial objects based on a context-free grammar. For example, by labeling the ascents and descents of a Stirling permutation, we obtain a grammar for the second-order Eulerian polynomials. By using the grammar for $0$-$1$-$2$ increasing trees given by Dumont, we obtain a grammatical derivation of the generating function of the André polynomials obtained by Foata and Schützenberger, without solving a differential equation. We also find a grammar for the number $T(n,k)$ of permuta…