Related papers: Generating functions for permutations which avoid …

For a permuton $\mu$ let $H_n(\mu)$ denote the Shannon entropy of the sampling distribution of $\mu$ on $n$ points. We investigate the asymptotic growth of $H_n(\mu)$ for a wide class of permutons. We prove that if $\mu$ has a non-vanishing…

Recently Richard Stanley initiated a study of the distribution of the length as(w) of the longest alternating subsequence in a random permutation w from the symmetric group $S_n$. Among other things he found an explicit formula for the…

Let $\lambda_1(n)$ denote the least invariant factor in the invariant factor decomposition of the multiplicative group $M_n = (\mathbb Z/n\mathbb Z)^\times$. We give an asymptotic formula, with order of magnitude $x/\sqrt{\log x}$, for the…

We improve the lower bound on the number of permutations of {1,2,...,n} in which no 3-term arithmetic progression occurs as a subsequence, and derive lower bounds on the upper and lower densities of subsets of the positive integers that can…

A set of permutations is called sign-balanced if the set contains the same number of even permutations as odd permutations. Let $S_n(\sigma_1, \sigma_2, \ldots, \sigma_r)$ be the set of permutations in the symmetric group $S_n$ which avoids…

We extend and generalize many of the enumerative results concerning West's stack-sorting map $s$. First, we prove a useful theorem that allows one to efficiently compute $|s^{-1}(\pi)|$ for any permutation $\pi$, answering a question of…

We study the generating function for the number of even (or odd) permutations on n letters containing exactly $r\gs0$ occurrences of 132. It is shown that finding this function for a given r amounts to a routine check of all permutations in…

We derive product and series representations of the gamma function using Newton interpolation series. Using these identities, a new formula for the coefficients in the Taylor series of the reciprocal gamma function is found. We also find…

Leo Creedon and Kieran Hughes in [18] studied derivations of a group ring $RG$ (of a group $G$ over a commutative unital ring $R$) in terms of generators and relators of group $G$. In this article, we do that for $(\sigma,…

Using a result of Gessel and Reutenauer, we find a simple formula for the number of cyclic permutations with a given descent set, by expressing it in terms of ordinary descent numbers (i.e., those counting all permutations with a given…

We revisit several hybrid multiplicative-to-additive type functions from a recent preprint article. These functions, $g(n)$ with Dirichlet generating function (DGF) $\zeta(s)^{-1} (1+P(s))^{-1}$ for $\Re(s) > 1$ where $P(s) = \sum_p p^{-s}$…

Recently is has been proved that if $\sigma\in GL_n(R)$ where $R$ is an commutative ring and $n\geq 3$, then each of the elementary transvections $t_{kl}(\sigma_{ij})~(i\neq j,k\neq l)$ is a product of eight $E_n(R)$-conjugates of $\sigma$…

Define the minimal excludant of an overpartition $\pi$, denoted $ \overline{\text{mex}}(\pi)$, to be the smallest positive integer that is not a part of the non-overlined parts of $\pi$. For a positive integer $n$, the function…

In this work we present a general and versatile algorithmic framework for exhaustively generating a large variety of different combinatorial objects, based on encoding them as permutations. This approach provides a unified view on many…

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…

In a recent paper, Backelin, West and Xin describe a map $\phi ^*$ that recursively replaces all occurrences of the pattern $k... 21$ in a permutation $\sigma$ by occurrences of the pattern $(k-1)... 21 k$. The resulting permutation…

Given a permutation $\pi$ chosen uniformly from $S_n$, we explore the joint distribution of $\pi(1)$ and the number of descents in $\pi$. We obtain a formula for the number of permutations with $\des(\pi)=d$ and $\pi(1)=k$, and use it to…

We develop a nonstandard approach to exploring polynomials associated with peaks and runs of permutations. With the aid of a context-free grammar, or a set of substitution rules, one can perform a symbolic calculus, and the computation…

A permutation whose any prefix has no more descents than ascents is called a ballot permutation. In this paper, we present a decomposition of ballot permutations that enables us to construct a bijection between ballot permutations and odd…

Iteration of the modular l-th power function f(x) = x^l (mod n) provides a common pseudorandom number generator (known as the Blum-Blum-Shub generator when l=2). The period of this pseudorandom number generator is closely related to…