Related papers: Distribution of statistics on separable permutatio…
The stable roommates problem does not necessarily have a solution, i.e. a stable matching. We had found that, for the uniformly random instance, the expected number of solutions converges to $e^{1/2}$ as $n$, the number of members, grows,…
We show that for any permutation $\pi$ there exists an integer $k_{\pi}$ such that every permutation avoiding $\pi$ as a pattern is a product of at most $k_{\pi}$ separable permutations. In other words, every strict class $\mathcal C$ of…
This is a study on pattern Hopf algebras in combinatorial structures. We introduce the notion of combinatorial presheaf, by adapting the algebraic framework of species to the study of substructures in combinatorics. Afterwards, we consider…
In this undergraduate thesis, we expand on the study of statistics on restricted growth functions avoiding patterns initiated by Campbell, et. al. Restricted growth functions are of interest because they are in bijection with set…
We consider the enumeration of ordered set partitions avoiding a permutation pattern, as introduced by Godbole, Goyt, Herdan and Pudwell. Let $\op_{n,k}(p)$ be the number of ordered set partitions of $\{1,2,\ldots,n\}$ into $k$ blocks that…
A striking result of [Acharya et al. 2017] showed that to estimate symmetric properties of discrete distributions, plugging in the distribution that maximizes the likelihood of observed multiset of frequencies, also known as the profile…
The maximally clustered permutations are characterized by avoiding the classical permutation patterns 3421, 4312, and 4321. This class contains the freely-braided permutations and the fully-commutative permutations. In this work, we show…
In this paper, we compute the distributions of the statistic number of crossings over permutations avoiding one of the pairs $\{321,231\}$, $\{123,132\}$ and $\{123,213\}$. The obtained results are new combinatorial interpretations of two…
A ballot permutation is a permutation $\pi$ such that in any prefix of $\pi$ the descent number is not more than the ascent number. By using a reversal concatenation map, we give a formula for the joint distribution (pk, des) of the peak…
In various disordered systems or non-equilibrium dynamical models, the large deviations of some observables have been found to display different scalings for rare values bigger or smaller than the typical value. In the present paper, we…
We propose a mean functional which exists for any probability distributions, and which characterizes the Pareto distribution within the set of distributions with finite left endpoint. This is in sharp contrast to the mean excess plot which…
This paper is a continuation of the systematic study of the distributions of quadrant marked mesh patterns initiated in [6]. Given a permutation $\sg = \sg_1 ... \sg_n$ in the symmetric group $S_n$, we say that $\sg_i$ matches the quadrant…
Bimodal truncated count distributions are frequently observed in aggregate survey data and in user ratings when respondents are mixed in their opinion. They also arise in censored count data, where the highest category might create an…
Recently, Hong and Li launched a systematic study of length-four pattern avoidance in inversion sequences, and in particular, they conjectured that the number of $0021$-avoiding inversion sequences can be enumerated by the OEIS entry…
In many applications involving binary variables, only pairwise dependence measures, such as correlations, are available. However, for multi-way tables involving more than two variables, these quantities do not uniquely determine the joint…
The study of joint equidistributions of mesh patterns 123 and 132 with the same symmetric shadings was recently initiated by Kitaev and Lv, where 75 of 80 potential joint equidistributions were proven. In this paper, we prove 112 out of 126…
We introduce a complex-plane generalization of the consecutive level-spacing distribution, used to distinguish regular from chaotic quantum spectra. Our approach features the distribution of complex-valued ratios between nearest- and…
Any permutation statistic $f:\sym\to\CC$ may be represented uniquely as a, possibly infinite, linear combination of (classical) permutation patterns: $f= \Sigma_\tau\lambda_f(\tau)\tau$. To provide explicit expansions for certain…
Given a permutation statistic $\operatorname{st}$, define its inverse statistic $\operatorname{ist}$ by $\operatorname{ist}(\pi):=\operatorname{st}(\pi^{-1})$. We give a general approach, based on the theory of symmetric functions, for…
An invariant ensemble of $N\times N$ random matrices can be characterised by a joint distribution for eigenvalues $P(\lambda_1,\cdots,\lambda_N)$. The study of the distribution of linear statistics, i.e. of quantities of the form…