Some families of trees arising in permutation analysis
Mathilde Bouvel, Marni Mishna, Cyril Nicaud
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26 pages; An extended abstract of this work appeared in the Proceedings of the International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC) 2013
We extend classical results on simple varieties of trees (asymptotic enumeration, average behavior of tree parameters) to trees counted by their number of leaves. Motivated by genome comparison of related species, we then apply these results to strong interval trees with a restriction on the arity of prime nodes. Doing so, we describe a filtration of the set of permutations based on their strong interval trees. This filtration is also studied from a purely analytical point of view, thus illustrating the convergence of analytic series towards a non-analytic limit at the level of the asymptotic behavior of their coefficients.
Permutation polynomials are of particular significance in several areas of applied mathematics, such as Coding theory and Cryptography. Many recent constructions are based on the Akbary-Ghioca-Wang (AGW) criterion. Along this line of research, we provide new classes of permutation trinomials in $\mathbb{F}_{q^2}$ of the form $f(x)=x^r h(x^{q-1})$, by studying permutations of the set of $(q+1)$-th roots of unity, which look like monomials on the sets of suitable partitions.
Ellis, Friedgut and Pilpel proved that for large enough $n$, a $t$-intersecting family of permutations contains at most $(n-t)!$ permutations. Their main theorem also states that equality holds only for $t$-cosets. We show that their proof of the characterization of extremal families is wrong. However, the characterization follows from a paper of Ellis, as mentioned already by Ellis, Friedgut and Pilpel.
For an arbitrary finite permutation group $G$, subgroup of the symmetric group $S_\ell$, we determine the permutations involving only members of $G$ as $\ell$-patterns, i.e., avoiding all patterns in the set $S_\ell \setminus G$. The set of all $n$-permutations with this property constitutes again a permutation group. We consequently refine and strengthen the classification of sets of permutations closed under pattern involvement and composition that is due to Atkinson and Beals.
Ever since the famous Erdős-Ko-Rado theorem initiated the study of intersecting families of subsets, extremal problems regarding intersecting properties of families of various combinatorial objects have been extensively investigated. Among them, studies about families of subsets, vector spaces and permutations are of particular concerns. Recently, the authors proposed a new quantitative intersection problem for families of subsets: For $\mathcal{F}\subseteq {[n]\choose k}$, define its \emph{total intersection number} as $\mathcal{I}(\mathcal{F})=\sum_{F_1,F_2\in \mathcal{F}}|F_1\cap F_2|$. The…
Permutation statistics $\wnm$ and $\rlm$ are both arising from permutation tableaux. $\wnm$ was introduced by Chen and Zhou, which was proved equally distributed with the number of unrestricted rows of a permutation tableau. While $\rlm$ is showed by Nadeau equally distributed with the number of $1$'s in the first row of a permutation tableau. In this paper, we investigate the joint distribution of $\wnm$ and $\rlm$. Statistic $(\rlm,\wnm,\rlmin,\des,(\underline{321}))$ is shown equally distributed with $(\rlm,\rlmin,\wnm,\des,(\underline{321}))$ on $S_n$. Then the generating function of $(\rl…
The Euler number $E_n$ (resp. Entringer number $E_{n,k}$) enumerates the alternating (down-up) permutations of $\{1,\dots,n\}$ (resp. starting with $k$). The Springer number $S_n$ (resp. Arnold number $S_{n,k}$) enumerates the type $B$ alternating permutations (resp. starting with $k$). In this paper, using bijections we first derive the counterparts in {\em André permutations} and {\em Simsun permutations} for the Entringer numbers $(E_{n,k})$, and then the counterparts in {\em signed André permutations} and {\em type $B$ increasing 1-2 trees} for the Arnold numbers $(S_{n,k})$.
We construct an intriguing bijection between $021$-avoiding inversion sequences and $(2413,4213)$-avoiding permutations, which proves a sextuple equidistribution involving double Eulerian statistics. Two interesting applications of this result are also presented. Moreover, this result inspires us to characterize all permutation classes that avoid two patterns of length $4$ whose descent polynomial equals that of separable permutations.
We study a family of equivalence relations on $S_n$, the group of permutations on $n$ letters, created in a manner similar to that of the Knuth relation and the forgotten relation. For our purposes, two permutations are in the same equivalence class if one can be reached from the other through a series of pattern-replacements using patterns whose order permutations are in the same part of a predetermined partition of $S_c$. In particular, we are interested in the number of classes created in $S_n$ by each relation and in characterizing these classes. Imposing the condition that the partition o…
Generalized Stirling permutations, families of increasing trees and urn models
Bona [2007+]研究了Gessel和Stanley [1978]引入的Stirling排列类中上升、平稳和下降的分布。最近，Janson [2008+]证明了Stirling置换和平面递归树之间的联系，并证明了Bona考虑的参数的联合正规律。这里我们将考虑广义Stirling置换，推广了Bona和Janson的早期结果，并将它们与某些广义平面递归树族以及$(k+1)$ -元增树联系起来。我们还给出了两个不同的双射之间的某些家庭的增长树，这两个作为一个特殊情况的双射之间的三元增长树和平面递归树。为了描述感兴趣的参数的（渐近）行为，我们研究了三个（广义）Polya瓮模型使用各种方法。
A family of permutations $A \subset S_n$ is said to be \emph{$t$-set-intersecting} if for any two permutations $σ, π\in A$, there exists a $t$-set $x$ whose image is the same under both permutations, i.e. $σ(x)=π(x)$. We prove that if $n$ is sufficiently large depending on $t$, the largest $t$-set-intersecting families of permutations in $S_n$ are cosets of stabilizers of $t$-sets. The $t=2$ case of this was conjectured by János Körner. It can be seen as a variant of the Deza-Frankl conjecture, proved in [4]. Our proof uses similar techniques to those of [4], namely, eigenvalue methods, togeth…
For the calculation of Springer numbers (of root systems) of type $B_n$ and $D_n$, Arnold introduced a signed analogue of alternating permutations, called $β_n$-snakes, and derived recurrence relations for enumerating the $β_n$-snakes starting with $k$. The results are presented in the form of double triangular arrays ($v_{n,k}$) of integers, $1\le |k|\le n$. An Arnold family is a sequence of sets of such objects as $β_n$-snakes that are counted by $(v_{n,k})$. As a refinement of Arnold's result, we give analogous arrays of polynomials, defined by recurrence, for the calculation of the polynom…
We consider uniform random permutations drawn from a family enumerated through generating trees. We develop a new general technique to establish a central limit theorem for the number of consecutive occurrences of a fixed pattern in such permutations. We propose a technique to sample uniform permutations in such families as conditioned random colored walks. Building on that, we derive the behavior of the consecutive patterns in random permutations studying properties of the consecutive increments in the corresponding random walks. The method applies to families of permutations with a one-dimen…
Erdős-Ko-Rado-type problem for hypergraph matchings
School of Mathematics and Statistics, Beijing Jiaotong University（北京交通大学数学与统计学院）；Hebei Provincial Key Laboratory of Mathematical Theory and Analysis for Network and Data Science, Beijing Jiaotong University（北京交通大学网络与数据科学数学理论与分析河北省重点实验室）；School of Mathematics and Statistics, Ningbo University（宁波大学数学与统计学院）；Institute of Mathematics and Interdisciplinary Sciences, Xidian University（西安电子科技大学数学与交叉科学研究院）
Baxter permutations are a class of permutations which are in bijection with a class of floorplans that arise in chip design called mosaic floorplans. We study a subclass of mosaic floorplans called $HFO_k$ defined from mosaic floorplans by placing certain geometric restrictions. This naturally leads to studying a subclass of Baxter permutations. This subclass of Baxter permutations are characterized by pattern avoidance. We establish a bijection, between the subclass of floorplans we study and a subclass of Baxter permutations, based on the analogy between decomposition of a floorplan into sma…
The maximum $t$-diversity of $t$-intersecting families of permutations
School of Mathematical Sciences, Capital Normal University（首都师范大学数学科学学院）；School of Mathematics, Sichuan University（四川大学数学学院）；School of Mathematical Sciences, University of Science and Technology of China（中国科学技术大学数学科学学院）
A set of permutations $I \subset S_n$ is said to be {\em k-intersecting} if any two permutations in $I$ agree on at least $k$ points. We show that for any $k \in \mathbb{N}$, if $n$ is sufficiently large depending on $k$, then the largest $k$-intersecting subsets of $S_n$ are cosets of stabilizers of $k$ points, proving a conjecture of Deza and Frankl. We also prove a similar result concerning $k$-cross-intersecting subsets. Our proofs are based on eigenvalue techniques and the representation theory of the symmetric group.
Brändén and Claesson introduced mesh patterns to provide explicit expansions for certain permutation statistics as linear combinations of (classical) permutation patterns. The first systematic study of avoidance of mesh patterns was conducted by Hilmarsson et al., while the first systematic study of the distribution of mesh patterns was conducted by the first two authors. In this paper, we provide far-reaching generalizations for 8 known distribution results and 5 known avoidance results related to mesh patterns by giving distribution or avoidance formulas for certain infinite families of mesh…
Let ${\mathscr L}=(X,\preceq)$ be a lattice. For ${\cal P}\subseteq X$ we say that ${\cal P}$ is $t$-{\it intersecting} if ${\sf rank}(x\wedge y)\ge t$ for all $x,y\in{\cal P}$. The seminal theorem of Erdős, Ko and Rado describes the maximum intersecting ${\cal P}$ in the lattice of subsets of a finite set with the additional condition that ${\cal P}$ is contained within a level of the lattice. The Erdős-Ko-Rado theorem has been extensively studied and generalized to other objects and lattices. In this paper, we focus on intersecting families of permutations as defined with respect to the weak…
Strongly refining results by Regev, Regev and Zeilberger, we prove surprising coincidences between characters to 2-part partitions of size n and characters to hooks of size n+2 on two related families obtained by extending 2-regular conjugacy classes.
Given two positive integers $n\geq 3$ and $t\leq n$, the permutations $σ,π\in \operatorname{Sym}(n)$ are $t$-setwise intersecting if they agree (setwise) on a $t$-subset of $\{1,2,\ldots,n\}$. A family $\mathcal{F} \subset \operatorname{Sym}(n)$ is $t$-setwise intersecting if any two permutations of $\mathcal{F}$ are $t$-setwise intersecting. Ellis [Journal of Combinatorial Theory, Series A, 119(4), 825--849, 2012] conjectured that if $t\leq n$ and $\mathcal{F} \subset \operatorname{Sym}(n)$ is a $t$-setwise intersecting family, then $|\mathcal{F}|\leq t!(n-t)!$ and equality holds only if $\ma…
We define two new families of polynomials that generalize permanents and prove upper and lower bounds on their determinantal complexities comparable to the known bounds for permanents. One of these families is obtained by replacing permutations by signed permutations, and the other by replacing permutations by surjective functions with preimages of prescribed sizes.
The main objects of the present paper are (i) Hibi rings (toric rings arising from order polytopes of posets), (ii) stable set rings (toric rings arising from stable set polytopes of perfect graphs), and (iii) edge rings (toric rings arising from edge polytopes of graphs satisfying the odd cycle condition). The goal of the present paper is to analyze those three toric rings and to discuss their structures in the case where their class groups have small rank. We prove that the class groups of (i), (ii) and (iii) are torsionfree. More precisely, we give descriptions of their class groups. Moreov…
Baxter permutations, plane bipolar orientations, and a specific family of walks in the non-negative quadrant are well-known to be related to each other through several bijections. We introduce a further new family of discrete objects, called coalescent-walk processes, that are fundamental for our results. We relate these new objects with the other previously mentioned families introducing some new bijections. We prove joint Benjamini--Schramm convergence (both in the annealed and quenched sense) for uniform objects in the four families. Furthermore, we explicitly construct a new fractal random…
Let $(σ_N^{(i)})_{i \in I}$ be a family of symmetric permutations of the entries of a Wigner matrix $\mathbf{W}_N$. We characterize the limiting traffic distribution of the corresponding family of dependent Wigner matrices $(\mathbf{W}_N^{σ_N^{(i)}})_{i \in I}$ in terms of the geometry of the permutations. We also consider the analogous problem for the limiting joint distribution of $(\mathbf{W}_N^{σ_N^{(i)}})_{i \in I}$. In particular, we obtain a description in terms of semicircular families with general covariance structures. As a special case, we derive necessary and sufficient conditions …
