相关论文: Permutation statistics on involutions

The Goulden$\unicode{x2013}$Jackson cluster method, adapted to permutations by Elizalde and Noy, reduces the problem of counting permutations by occurrences of a prescribed consecutive pattern to that of counting clusters, which are special…

The convolution product of two conjugacy classes of the unitary group $U_n$ is described by a probability distribution on the space of central measures. Relating this convolution to the quantum cohomology of Grassmannians and using recent…

We deal with the study of the evolution of the allelic frequencies, at a single locus, for a population distributed continuously over a bounded habitat. We consider evolution which occurs under the joint action of selection and arbitrary…

Motivated by recent results on quasi-Stirling permutations, which are permutations of the multiset $\{1,1,2,2,\dots,n,n\}$ that avoid the "crossing" patterns 1212 and 2121, we consider nonnesting permutations, defined as those that avoid…

We say that permutations $\pi_1,\dots, \pi_r \in \mathcal{S}_n$ invariably generate $\mathcal{S}_n$ if, no matter how one chooses conjugates $\pi'_1,\dots,\pi'_r$ of these permutations, $\pi'_1,\dots,\pi'_r$ generate $\mathcal{S}_n$. We…

The paper is devoted to the study of some well-knonw combinatorial functions on the symmetric group $\sn$ --- the major index $\maj$, the descent number $\des$, and the inversion number $\inv$ --- from the representation-theoretic point of…

The volume and the number of lattice points of the permutohedron P_n are given by certain multivariate polynomials that have remarkable combinatorial properties. We give several different formulas for these polynomials. We also study a more…

We suggest the method for group classification of evolution equations admitting nonlocal symmetries which are associated with a given evolution equation possessing nontrivial Lie symmetry. We apply this method to second-order evolution…

We present exponential generating function analogues to two classical identities involving the ordinary generating function of the complete homogeneous symmetric functions. After a suitable specialization the new identities reduce to…

We consider a particular class of n-dimensional homogeneous diffusions all of which have an identity diffusion matrix and a drift function that is piecewise constant and scale invariant. Abstract stochastic calculus immediately gives us…

Let $G=\Dc_{n}$ be the dicyclic group of order $4n$. Let $\varphi$ be an automorphism of $G$ of order $k$. We describe $\varphi$ and the generalized symmetric space $Q$ of $G$ associated with $\varphi$. When $\varphi$ is an involution, we…

Recently, Gross et al. posed the LLC conjecture for the locally log-concavity of the genus distribution of every graph, and provided an equivalent combinatorial version, the CLLC conjecture, on the log-concavity of the generating function…

We study algebraic dynamical systems (and, more generally, $\sigma$-varieties) $\Phi:{\mathbb A}^n_{\mathbb C} \to {\mathbb A}^n_{\mathbb C}$ given by coordinatewise univariate polynomials by refining a theorem of Ritt. More precisely, we…

The systematic study of inversion sequences avoiding triples of relations was initiated by Martinez and Savage. For a triple $(\rho_1,\rho_2,\rho_3)\in\{<,>,\leq,\geq,=,\neq,-\}^3$, they introduced $\I_n(\rho_1,\rho_2,\rho_3)$ as the set of…

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…

We study $I(T)$, the number of inversions in a tree $T$ with its vertices labeled uniformly at random, which is a generalization of inversions in permutations. We first show that the cumulants of $I(T)$ have explicit formulas involving the…

Taking transposes of Standard Young Tableaux defines a natural involution on the set $I(n)$ of involutions of length $n$ via the the Robinson-Schensted correspondence. In some cases, this involution can be defined without resorting to the…

It is known that, when $n$ is even, the number of permutations of $\{1,2,\dots,n\}$ all of whose cycles have odd length equals the number of those all of whose cycles have even length. Adin, Heged\H{u}s and Roichman recently found a…

We consider the binomial distribution with parameters $n$ and $x$, and show that the sum of the squared probabilities is a log-convex function of $x$. This completes the proof of a conjecture formulated in 2014. Applications to R\'{e}nyi…

Building on the work of Grinberg and Stanley, we begin a systematic study of permutations with a prescribed $X$-descent set. In particular, for a set $X \subseteq \mathbb{N}^2$, and $I \subseteq [n-1]$, we study the permutations $\pi \in…