Related papers: Maximizing the Edelman-Greene statistic

The article presents the results of experiments in computation of statistical values related to Young diagrams, including the estimates on maximum and average (by Plancherel distribution) dimension of irreducible representation of symmetric…

We introduce a new statistic on the hyperoctahedral groups (Coxeter groups of type B), and give a conjectural formula for its signed distributions over arbitrary descent classes. The statistic is analogous to the classical Coxeter length…

Richard Stanley proved that the centralized/normalized version of the random variable "length of largest up-down subsequence" in a random permutation of length n is asymptotically normal. We go beyond and present a more refined asymptotic…

In this article we consider the cycle structure of compositions of pairs of involutions in the symmetric group S_n chosen uniformly at random. These can be modeled as modified 2-regular graphs, giving rise to exponential generating…

The length is(w) of the longest increasing subsequence of a permutation w in the symmetric group S_n has been the object of much investigation. We develop comparable results for the length as(w) of the longest alternating subsequence of w,…

We study Hilbert-Samuel multiplicity for points of Schubert varieties in the complete flag variety, by Groebner degenerations of the Kazhdan-Lusztig ideal. In the covexillary case, we give a positive combinatorial rule for multiplicity by…

Edgeworth expansions for random walks on covering graphs with groups of polynomial volume growths are obtained under a few natural assumptions. The coefficients appearing in this expansion depends on not only geometric features of the…

A contractive $n$-tuple $A=(A_1,...,A_n)$ has a minimal joint isometric dilation $S=(S_1,...,S_n)$ where the $S_i$'s are isometries with pairwise orthogonal ranges. This determines a representation of the Cuntz-Toeplitz algebra. When $A$…

We prove Stanley's conjecture that, if delta_n is the staircase shape, then the skew Schur functions s_{delta_n / mu} are non-negative sums of Schur P-functions. We prove that the coefficients in this sum count certain fillings of shifted…

The development of the theories of the second-order Eulerian polynomials began with the works of Buckholtz and Carlitz in their studies of an asymptotic expansion. Gessel-Stanley introduced Stirling permutations and presented combinatorial…

We prove an upper bound for characters of the symmetric groups. Namely, we show that there exists a constant a>0 with a property that for every Young diagram \lambda with n boxes, r(\lambda) rows and c(\lambda) columns |Tr \rho^\lambda(\pi)…

We exploit Krattenthaler's bijection between the set $S_n(3\textrm{-}1\textrm{-}2)$ of permutations in $S_n$ avoiding the classical pattern $3\textrm{-}1\textrm{-}2$ and Dyck $n$-paths to study the distribution of every consecutive pattern…

A permutation $\tau$ in the symmetric group $S_j$ is minimally overlapping if any two consecutive occurrences of $\tau$ in a permutation $\sigma$ can share at most one element. B\'ona \cite{B} showed that the proportion of minimal…

We show that the poset of shuffles introduced by Greene in 1988 is flag-symmetric, and we describe a "local" permutation action of the symmetric group on the maximal chains which is closely related to the flag symmetric function of the…

This paper develops an analogy between the cycle structure of, on the one hand, random permutations with cycle lengths restricted to lie in an infinite set $S$ with asymptotic density $\sigma$ and, on the other hand, permutations selected…

We prove a formula for the number of permutations in $S_n$ such that their first $n-k$ entries are increasing and their longest increasing subsequence has length $n-k$. This formula first appeared as a consequence of character polynomial…

For a Coxeter group $W$ with length function $\ell$, the excess zero graph $\mathcal{E}_0(W)$ has vertex set the non-identity involutions of $W$, with two involutions $x$ and $y$ adjacent whenever $\ell(xy)=\ell(x)+\ell(y)$. Properties of…

We calculate the spectra and spectral measures associated to random walks on restricted wreath products of finite groups with the infinite cyclic group, by calculating the Kesten-von Neumann-Serre spectral measures for the random walks on…

Whiston proved that the maximum size of an irredundant generating set in the symmetric group $S_n$ is $n-1$, and Cameron and Cara characterized all irredundant generating sets of $S_n$ that achieve this size. Our goal is to extend their…

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, the authors of a recent paper showed that the expected number of…