Related papers: Combinatorial Statistics on Pattern-avoiding Permu…
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…
A classical result of MacMahon shows that the length function and the major index are equi-distributed over the symmetric group. Foata and Sch\"utzenberger gave a remarkable refinement and proved that these parameters are equi-distributed…
A great deal of inference in statistics is based on making the approximation that a statistic is normally distributed. The error in doing so is generally $O(n^{-1/2})$ and can be very considerable when the distribution is heavily biased or…
We investigate extreme values of Mahonian and Eulerian distributions arising from counting inversions and descents of random elements of finite Coxeter groups. To this end, we construct a triangular array of either distribution from a…
The standard algorithm for generating a random permutation gives rise to an obvious permutation statistic $\stat$ that is readily seen to be Mahonian. We give evidence showing that it is not equal to any previously published statistic. Nor…
Statistical modeling of multivariate and spatial extreme events has attracted broad attention in various areas of science. Max-stable distributions and processes are the natural class of models for this purpose, and many parametric families…
Let $T_n$ be the set of 321-avoiding permutations of order $n$. Two properties of $T_n$ are proved: (1) The {\em last descent} and {\em last index minus one} statistics are equidistributed over $T_n$, and also over subsets of permutations…
We use various combinatorial and probabilistic techniques to study growth rates for the probability that a random permutation from the Mallows distribution avoids consecutive patterns. The Mallows distribution behaves like a $q$-analogue of…
Babson and Steingr\'{\i}msson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. Claesson presented a complete solution for the number of…
In [GM] Guibert and Mansour studied involutions on n letters avoiding (or containing exactly once) 132 and avoiding (or containing exactly once) an arbitrary pattern on k letters. They also established a bijection between 132-avoiding…
In a uniform random permutation \Pi of [n] := {1,2,...,n}, the set of elements k in [n-1] such that \Pi(k+1) = \Pi(k) + 1 has the same distribution as the set of fixed points of \Pi that lie in [n-1]. We give three different proofs of this…
As suggested by Currie, we apply the probabilistic method to problems regarding pattern avoidance. Using techniques from analytic combinatorics, we calculate asymptotic pattern occurrence statistics and use them in conjunction with the…
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…
A variety of descent and major-index statistics have been defined for symmetric groups, hyperoctahedral groups, and their generalizations. Typically associated to pairs of such statistics is an Euler--Mahonian distribution, a bivariate…
We show that the bistatistic of right nestings and right crossings in matchings without left nestings is equidistributed with the number of occurrences of two certain patterns in permutations, and furthermore that this equidistribution…
Recently, we proved the equidistribution of the pairs of permutation statistics $(r\textsf{des},r\textsf{maj})$ and $(r\textsf{exc},r\textsf{den})$. Any pair of permutation statistics that is equidistributed with these pairs is said to be…
A systematic study of avoidance of mesh patterns of length 2 was conducted by Hilmarsson et al. in 2015. In a recent paper Kitaev and Zhang examined the distribution of the aforementioned patterns. The aim of this paper is to prove more…
We present some combinatorial interpretations for coefficients appearing in series partitioning the permutations avoiding 132 along marked mesh patterns. We identify for patterns in which only one parameter is non zero the combinatorial…
In this paper, we construct bijections between Dyck paths, noncrossing partitions, and 231-avoiding permutations, which send the area statistic on Dyck paths to the inversion number on noncrossing partitions and on 231-avoiding…
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…