David Bevan
We introduce a new natural notion of convergence for permutations at any specified scale, in terms of the density of patterns of restricted width. In this setting we prove that limits may be chosen independently at a countably infinite number of scales.
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Asymptotic free independence and entry permutations for Gaussian random matrices. Part II: Infinitesimal freeness
本文研究了具有置换元素的GUE矩阵的渐近无穷小分布，证明了随机均匀置换下GUE矩阵的零无穷小分布及不同置换间的无穷小自由性，并指出GUE矩阵与其转置非无穷小自由。
The paper presents conditions on entry permutations that induce asymptotic freeness when acting on Gaussian random matrices. The class of permutations described includes the matrix transpose, as well as entry permutations relevant in Quantum Information Theory and Quantum Physics.
Voiculescu's notion of asymptotic free independence is known for a large class of random matrices including independent unitary invariant matrices. This notion is extended for independent random matrices invariant in law by conjugation by permutation matrices. This fact leads naturally to an extension of free probability, formalized under the notions of traffic probability. We first establish this construction for random matrices. We define the traffic distribution of random matrices, which is richer than the *-distribution of free probability. The knowledge of the individual traffic distribut…
Random permutation matrix models for graph products
本文利用随机置换矩阵的共轭运算，构造了在矩阵上合并的图独立随机矩阵模型，从而为 sofic 群的图积仍是 sofic 的提供了新的概率证明。
In this paper, we revisit the application of generating trees to the pattern avoidance problem for permutations. In particular, we study this problem for certain general sets of patterns and propose a new procedure leveraging the FinLabel algorithm and exploiting the subregularities in the associated generating trees. We consider some general kinds of generating trees for which the FinLabel algorithm fails to determine in a finite number of iterations the generating function that enumerates the underlying class of permutations. Our procedure provides a unified approach in these cases leading t…
Permutations in competing growth processes and balls-in-bins
研究N个独立递增过程在随机等待时间下的竞争增长，证明在无领导者时所有排名排列几乎必然无限频繁出现，并验证了Spencer关于带反馈球入箱模型的猜想。
We prove that it is decidable if a finitely based permutation class contains infinitely many simple permutations, and establish an unavoidable substructure result for simple permutations: every sufficiently long simple permutation contains an alternation or oscillation of length k.
Asymptotic independence for random permutations from surface groups
本文研究亏格至少为2的可定向双曲曲面群到对称群随机同态中元素不动点数的渐近行为，证明多个不动点变量的乘积在无显然障碍时渐近独立，并给出极限分布及短循环统计量的对应结果。
Permutation--invariant Niven numbers
本文提出了一种新的尼文数类，即排列不变的尼文数，并证明其存在无限多个，同时给出了系统搜索方法。
Generating trees are a useful technique in the enumeration of various combinatorial objects, particularly restricted permutations. Quite often the generating tree for the set of permutations avoiding a set of patterns requires infinitely many labels. Sometimes, however, this generating tree needs only finitely many labels. We characterize the finite sets of patterns for which this phenomenon occurs. We also present an algorithm - in fact, a special case of an algorithm of Zeilberger - that is guaranteed to find such a generating tree if it exists.
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 …
A permutation sigma in Sn is a k-derangement if for any subset X = {a1, . . ., ak} \subseteq [n], {sigma(a1), . . ., sigma(ak)} is not equal to X. One can form the k-derangement graph on the set of permutations of Sn by connecting two permutations sigma and tau if sigma(tau)^-1 is a k-derangement. We characterize when such a graph is connected or Eulerian. For n an odd prime power, we determine the independence, clique and chromatic number of the 2-derangement graph.
We introduce "fertility Wilf equivalence," "strong fertility Wilf equivalence," and "postorder Wilf equivalence," three variants of Wilf equivalence for permutation classes that formalize some phenomena that have appeared in the study of West's stack-sorting map. We introduce "sliding operators" and show that they induce useful bijections among sets of valid hook configurations. Combining these maps with natural decompositions of valid hook configurations, we give infinitely many examples of fertility, strong fertility, and postorder Wilf equivalences. As a consequence, we obtain infinitely ma…
The goal of this paper is to accurately describe the maximal zero-free region of the independence polynomial for graphs of bounded degree, for large degree bounds. In previous work with de Boer, Guerini and Regts it was demonstrated that this zero-free region coincides with the normality region of the related occupation ratios. These ratios form a discrete semi-group that is in a certain sense generated by finitely many rational maps. We will show that as the degree bound converges to infinity, the properly rescaled normality regions converge to a limit domain, which can be described as the ma…
This paper is a contribution to the study of hereditary classes of finite graphs. We classify these classes according to the number of prime structures they contain. We consider such classes that are \emph{minimal prime}: classes that contain infinitely many primes but every proper hereditary subclass contains only finitely many primes. We give a complete characterization of such classes. In fact, each one of these classes is a well quasi ordered age and there are uncountably many of them. Eleven of these ages remain well quasi ordered when labels in a well quasi ordering are added. Among the …
This paper is a contribution to the study of hereditary classes of finite graphs. We classify these classes according to the number of prime structures they contain. We consider such classes that are \emph{minimal prime}: classes that contain infinitely many primes but every proper hereditary subclass contains only finitely many primes. We give a complete description of such classes. In fact, each one of these classes is a well-quasi-ordered (w.q.o) age and there are uncountably many of them. Eleven of these ages are almost multichainable; they remain w.q.o when labels in a w.q.o are added, he…
A Sturmian word is a map W from the natural numbers into {0,1} for which the set of {0,1}-vectors F_n(W):={(W(i),W(i+1),...,W(i+n-1))^T : i \ge 0} has cardinality exactly n+1 for each positive integer n. Our main result is that the volume of the simplex whose n+1 vertices are the n+1 points in F_n(W) does not depend on W. Our proof of this motivates studying algebraic properties of the permutation $π$ (depending on an irrational x and a positive integer n) that orders the fractional parts {1 x}, {2 x}, ..., {n x}, i.e., 0 < {π(1) x} < {π(2) x} < ... < {π(n) x} < 1. We give a formula for the si…
Lower bounds on the independence number of a graph in terms of degrees
研究图的独立数与度数关系，提出基于度数的下界公式，并证明其在多项式时间内可构造独立集。
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