Related papers: Limit Densities of Patterns in Permutation Inflati…
We develop a theoretical approach to compute the conditioned spectral density of $N \times N$ non-invariant random matrices in the limit $N \rightarrow \infty$. This large deviation observable, defined as the eigenvalue distribution…
The NP-complete Permutation Pattern Matching problem asks whether a $k$-permutation $P$ is contained in a $n$-permutation $T$ as a pattern. This is the case if there exists an order-preserving embedding of $P$ into $T$. In this paper, we…
We consider random permutation matrices following a one-parameter family of deformations of the uniform distribution, called Ewens' measures, and modifications of these matrices where the entries equal to one are replaced by i.i.d uniform…
We study the expansions of permutation statistics in the basis of functions counting occurrences of a fixed pattern in a permutation. We show the finiteness of these pattern expansions for a class of permutation statistics including the…
New bounds on the number of similar or directly similar copies of a pattern within a finite subset of the line or the plane are proved. The number of equilateral triangles whose vertices all lie within an $n$-point subset of the plane is…
A well known argument in cosmology gives that the power spectrum (or structure function) $P(k)$ of mass density fluctuations produced from a uniform initial state by physics which is causal (i.e. moves matter and momentum only up to a…
In this paper, we provide an upper bound for the number of one-element commutation classes of a permutation, that is, the number of reduced words in which no commutation can be applied. Using this upper bound, we prove a conjecture that…
We explore how the asymptotic structure of a random permutation of $[n]$ with $m$ inversions evolves, as $m$ increases, establishing thresholds for the appearance and disappearance of any classical, consecutive or vincular pattern. The…
A collection $B$ of patterns is called inversion monotone if $\mathrm{av}_n^k(B)$, the number of $B$-avoiding permutations of length $n$ with $k$ inversions, is weakly increasing in $n$ for any fixed $k$. In 2012, Claesson, Jel\'inek and…
An $(n,k)$ sequence covering array is a set of permutations of $[n]$ such that each sequence of $k$ distinct elements of $[n]$ is a subsequence of at least one of the permutations. An $(n,k)$ sequence covering array is perfect if there is a…
It is natural to expect that there are only three possible types of scaling limits for the collection of all percolation interfaces in the plane: (1) a trivial one, consisting of no curves at all, (2) a critical one, in which all points of…
We prove that the set of growth rates of permutation classes includes an infinite sequence of intervals whose infimum is $\theta_B\approx2.35526$, and that it also contains every value at least $\lambda_B\approx2.35698$. These results…
Let $\mathbb{K}$ be a field of characteristic $0$. For each choice of distinct $a_1, \ldots, a_n\in \mathbb{K}$ and distinct $b_1, \ldots, b_n\in \mathbb{K}$, consider the sum $S=\sum_{i=1}^n a_i b_{\pi(i)}$ as $\pi$ ranges over the…
An encoder observes a point pattern---a finite number of points in the interval $[0,T]$---which is to be described to a reconstructor using bits. Based on these bits, the reconstructor wishes to select a subset of $[0,T]$ that contains all…
We obtain the asymptotic behaviour of the longest increasing/non-decreasing subsequences in a random uniform multiset permutation in which each element in {1,...,n} occurs k times, where k may depend on n. This generalizes the famous…
In this paper, we compute the natural density of the set of k x n integer matrices that can be extended to an invertible n x n matrix over the integers. As a corollary, we find the density of rectangular matrices with Hermite normal form [O…
Let $G_{k,n}$ be a group of permutations of $kn$ objects which permutes things independently in disjoint blocks of size $k$ and then permutes the blocks. We investigate the probabilistic and/or enumerative aspects of random elements of…
We prove limit theorems for the number of fixed points occurring in a random pattern-avoiding permutation distributed according to a one-parameter family of biased distributions. The bias parameter exponentially tilts the distribution…
Most models of inflation have small parameters, either to guarantee sufficient inflation or the correct magnitude of the density perturbations. In this paper we show that, in supersymmetric theories with weak scale supersymmetry breaking,…
In his Ph.D. thesis, Ira Gessel proved a reciprocity formula for noncommutative symmetric functions which enables one to count words and permutations with restrictions on the lengths of their increasing runs. We generalize Gessel's theorem…