Related papers: The Brownian limit of separable permutations

Uniform large deviation principles for positive functionals of all equivalent types of infinite dimensional Brownian motions acting together with a Poisson random measure are established. The core of our approach is a variational…

We study a one-dimensional random walk among random conductances, with unbounded jumps. Assuming the ergodicity of the collection of conductances and a few other technical conditions (uniform ellipticity and polynomial bounds on the tails…

A random dense countable set is characterized (in distribution) by independence and stationarity. Two examples are `Brownian local minima' and `unordered infinite sample'. They are identically distributed. A framework for such concepts,…

Fractional Brownian motion is a Gaussian stochastic process with stationary, long-time correlated increments and is frequently used to model anomalous diffusion processes. We study numerically fractional Brownian motion confined to a finite…

The poset of permutations of [n] under Bruhat ordering is studied. We give nontrivial upper and lower bounds for the number of comparable pairs of permutations in both the weak and strong versions of this order. In light of numerical…

Let $\pi_n$ be a uniformly chosen random permutation on $[n]$. Using an analysis of the probability that two overlapping consecutive $k$-permutations are order isomorphic, we show that the expected number of distinct consecutive patterns in…

We compare two examples of random dense countable sets, `Brownian local minima' and `unordered uniform infinite sample'. They appear to be identically distributed. A framework for such notions is proposed. In addition, random elements of…

A permutation is called Grassmannian if it has at most one descent. In this paper, we investigate pattern avoidance and parity restrictions for such permutations. As our main result, we derive formulas for the enumeration of Grassmannian…

In this paper we prove that among the permutations of length n with i fixed points and j excedances, the number of 321-avoiding ones equals the number of 132-avoiding ones, for all given i,j<=n. We use a new technique involving diagonals of…

For uniform random permutations conditioned to have no long cycles, we prove that the total number of cycles satisfies a central limit theorem. Under additional assumptions on the asymptotic behavior of the set of allowed cycle lengths, we…

As a first step toward a characterization of the limiting extremal process of branching Brownian motion, we proved in a recent work [Comm. Pure Appl. Math. 64 (2011) 1647-1676] that, in the limit of large time $t$, extremal particles…

We compute the expected number of commutations appearing in a reduced word for the longest element in the symmetric group. The asymptotic behavior of this value is analyzed and shown to approach the length of the permutation, meaning that…

Comtet introduced the notion of indecomposable permutations in 1972. A permutation is indecomposable if and only if it has no proper prefix which is itself a permutation. Indecomposable permutations were studied in the literature in various…

Machines whose main purpose is to permute and sort data are studied. The sets of permutations that can arise are analysed by means of finite automata and avoided pattern techniques. Conditions are given for these sets being enumerated by…

Consider $q_n$ a random pointed quadrangulation chosen equally likely among the pointed quadrangulations with $n$ faces. In this paper we show that, when $n$ goes to $+\infty$, $q_n$ suitably normalized converges weakly in a certain sense…

Let $S_n$ denote the set of permutations of $[n]:=\{1,\cdots, n\}$, and denote a permutation $\sigma\in S_n$ by $\sigma=\sigma_1\sigma_2\cdots \sigma_n$. For $l\ge2$ an integer, let $A^{(n)}_{l;k}\subset S_n$ denote the event that the set…

Let $\pi_n$ be a uniformly chosen random permutation on $[n]$. The authors of [2] showed that the expected number of distinct consecutive patterns of all lengths $k\in\{1,2,\ldots,n\}$ in $\pi_n$ was $\frac{n^2}{2}(1-o(1))$ as $n\to\infty$,…

We review how the monotone pattern compares to other patterns in terms of enumerative results on pattern avoiding permutations. We consider three natural definitions of pattern avoidance, give an overview of classic and recent formulas, and…

We introduce and study the writhe of a permutation, a circular variant of the well-known inversion number. This simple permutation statistics has several interpretations, which lead to some interesting properties. For a permutation sampled…

We elaborate on the theorem saying that as permeability coefficients of snapping-out Brownian motions tend to infinity in such a way that their ratio remains constant, these processes converge to a skew Brownian motion. In particular,…