给定一个置换$π\in \Sn\_n$，如果（i）$i<j$和$π(i)<π(j)$且（ii）不存在$k$使得$i < k < j$和$π(i)<π(k)<π(j)$，则通过将$i$连接到$j$来构造顶点集$\{1,2, ..., n\}$上的图$G\_π$。如果$G\_π$是一个森林，我们说$π$是类森林的。我们首先根据模式避免来描述森林样排列，然后通过某种线性映射来描述。由于最近的结果吴和勇，这表明，森林般的排列特征舒伯特品种是当地的阶乘。因此，类森林置换推广了光滑置换（对应于光滑舒伯特簇）。我们计算了类森林排列的生成函数。在光滑的情况下，它是代数的。然后，我们调整我们的方法来计算排列，其中$G\_π$是一棵树，或路径，并恢复已知的生成函数的顺利排列。
英文摘要：
Given a permutation $π\in \Sn\_n$, construct a graph $G\_π$ on the vertex set $\{1,2, ..., n\}$ by joining $i$ to $j$ if (i) $i<j$ and $π(i)<π(j)$ and (ii) there is no $k$ such that $i < k < j$ and $π(i)<π(k)<π(j)$. We say that $π$ is forest-like if $G\_π$ is a forest. We first characterize forest-like permutations in terms of pattern avoidance, and then by a certain linear map being onto. Thanks to recent results of Woo and Yong, this shows that forest-like permutations characterize Schubert varieties which are locally factorial. Thus forest-like permutations generalize smooth permutations (corresponding to smooth Schubert varieties). We compute the generating function of forest-like permutations. As in the smooth case, it turns out to be algebraic. We then adapt our method to count permutations for which $G\_π$ is a tree, or a path, and recover the known generating function of smooth permutations.
We say that an unordered rooted labeled forest avoids the pattern $π\in\mathcal{S}_n$ if the sequence obtained from the labels along the path from the root to any vertex does not contain a subsequence that is in the same relative order as $π$. We enumerate several classes of forests that avoid certain sets of permutations, including the set of unimodal forests, via bijections with set partitions with certain properties. We also define and investigate an analog of Wilf-equivalence for forests.
The classes of tree permutations and forest permutations were defined by Acan and Hitczenko (2016). We study random permutations of a given length from these classes, and in particular the number of occurrences of a fixed pattern in one of these random permutations. The main results show that the distributions of these numbers are asymptotically normal. The proof uses representations of random tree and forest permutations that enable us to express the number of occurrences of a pattern by a type of $U$-statistics; we then use general limit theorems for the latter.
Consider the following partial "sorting algorithm" on permutations: take the first entry of the permutation in one-line notation and insert it into the position of its own value. Continue until the first entry is 1. This process imposes a forest structure on the set of all permutations of size $n$, where the roots are the permutations starting with 1 and the leaves are derangements. Viewing the process in the opposite direction towards the leaves, one picks a fixed point and moves it to the beginning. Despite its simplicity, this "fixed point forest" exhibits a rich structure. In this paper, w…
We give an explicit combinatorial proof of a weighted version of strong log-concavity for the generating polynomial of increasing spanning forests of a finite simple graph equipped with a total ordering of the vertices. In contrast to similar proofs in the literature, our injection is local in the sense that it proceeds by moving a single edge from one forest to the other. In the particular case of the complete graph, this gives a new combinatorial proof of log-concavity of unsigned Stirling numbers of the first kind where a pair of permutations is transformed into a new pair by breaking a sin…
We define a new family of generalized Stirling permutations that can be interpreted in terms of ordered trees and forests. We prove that the number of generalized Stirling permutations with a fixed number of ascents is given by a natural three-parameter generalization of the well-known Eulerian numbers. We give the generating function for this new class of numbers and, in the simplest cases, we find closed formulas for them and the corresponding row polynomials. By using a non-trivial involution our generalized Eulerian numbers can be mapped onto a family of generalized Ward numbers, forming a…
In this paper, we confirm a conjecture of Laborde-Zubieta on the enumeration of corners in tree-like tableaux. Our proof is based on Aval, Boussicault and Nadeau's bijection between tree-like tableaux and permutation tableaux, and Corteel and Nadeau's bijection between permutation tableaux and permutations. This last bijection sends a corner in permutation tableaux to an ascent followed by a descent in permutations, this enables us to enumerate the number of corners in permutation tableaux, and thus to completely solve L.-Z.'s conjecture. Moreover, we give a bijection between corners and runs …
Given a graph, we can form a spanning forest by first sorting the edges in some order, and then only keep edges incident to a vertex which is not incident to any previous edge. The resulting forest is dependent on the ordering of the edges, and so we can ask, for example, how likely is it for the process to produce a graph with $k$ trees. We look at all graphs which can produce at most two trees in this process and determine the probabilities of having either one or two trees. From this we construct infinite families of graphs which are non-isomorphic but produce the same probabilities.
Tree-like tableaux are objects in bijection with alternative or permutation tableaux. They have been the subject of a fruitful combinatorial study for the past few years. In the present work, we define and study a new subclass of tree-like tableaux enumerated by Baxter numbers. We exhibit simple bijective links between these objects and three other combinatorial classes: (packed or mosaic) floorplans, twisted Baxter permutations and triples of non-intersecting lattice paths. From several (and unrelated) works, these last objects are already known to be enumerated by Baxter numbers, and our mai…
Inspired by the definition of color-dressed amplitudes in string theory, we define analogous {\it color-dressed permutations} replacing the color-ordered string amplitudes by their corresponding permutations. Decomposing the color traces into symmetrized traces and structure constants, the color-dressed permutations define {\it BRST-invariant permutations}, which we show are elements of the inverse Solomon descent algebra and we find a closed formula for them. We then present evidence that these permutations encode KK-like relations among the different $α'$ corrections to disk amplitudes refin…
All doubly stochastic solutions to $AXA = XAX$ when $A$ is a permutation matrix
Mathematical Institute of the Serbian Academy of Sciences and Arts（塞尔维亚科学院数学研究所）；University of Niš, Faculty of Sciences and Mathematics（尼什大学理学院）；University of Sorbonne（索邦大学）
