Related papers: The Expected Shape of Random Doubly Alternating Ba…

It is a classical result that the parity-balance of the number of weak excedances of all permutations (derangements, respectively) of length $n$ is the Euler number $E_n$, alternating in sign, if $n$ is odd (even, respectively).…

We give a new expression for the expected number of inversions in the product of n random adjacent transpositions in the symmetric group S_{m+1}. We then derive from this expression the asymptotic behaviour of this number when n scales with…

We prove a conjecture of Cohn and Propp, which refines a conjecture of Bosley and Fidkowski about the symmetry of the set of alternating sign matrices (ASMs). We examine data arising from the representation of an ASM as a collection of…

We count permutations avoiding a nonconsecutive instance of a two- or three-letter pattern, that is, the pattern may occur but only as consecutive entries in the permutation. Two-letter patterns give rise to the Fibonacci numbers. The…

Inspired by a recent note of Zeilberger (arXiv:1110.4379), Alejandro Morales asked whether one can count alternating (i.e., up-down) permutations that contain the pattern 123 or 321 exactly once. In this note we answer the question in the…

Let $\pi$ be a cyclic permutation that can be expressed in its one-line form as $\pi = \pi_1\pi_2 \cdot\cdot\cdot \pi_n$ and in its standard cycle form as $\pi = (c_1,c_2, ..., c_n)$ where $c_1=1$. Archer et al. introduced the notion of…

Let $\pi_n$ be a uniformly chosen random permutation on $[n]$. The authors of [2] showed that the expected number of distinct consecutive patterns of all lengths $k\in\{1,2,\ldots,n\}$ in $\pi_n$ was $\frac{n^2}{2}(1-o(1))$ as $n\to\infty$,…

Let $S_n$ be the symmetric group of all permutations of $\{1, \cdots, n\}$ with two generators: the transposition switching $1$ with $2$ and the cyclic permutation sending $k$ to $k+1$ for $1\leq k\leq n-1$ and $n$ to $1$ (denoted by…

We prove a conjecture dating back to a 1978 paper of D.R.\ Musser~\cite{musserirred}, namely that four random permutations in the symmetric group $\mathcal{S}_n$ generate a transitive subgroup with probability $p_n > \epsilon$ for some…

A permutation of size $n$ can be identified to its diagram in which there is exactly one point per row and column in the grid $[n]^2$. In this paper we consider multidimensional permutations (or $d$-permutations), which are identified to…

We study the enumeration of mixed dimer covers on skew Young diagrams of ribbon shape (also called border strips or snake graphs). For the two extreme cases of straight and zigzag shapes, we show that the number of mixed dimer covers are…

The large Schroder numbers are known to count several classes of permutations avoiding two 4-letter patterns. Here we show they count another family of permutations, those whose left to right minima decomposition, when reversed, is…

Let $G$ be the alternating group of degree $n$. Let $\omega(G)$ be the maximal size of a subset $S$ of $G$ such that $\langle x,y \rangle = G$ whenever $x,y \in S$ and $x \neq y$ and let $\sigma(G)$ be the minimal size of a family of proper…

Shallow permutations were defined in 1977 to be those that satisfy the lower bound of the Diaconis-Graham inequality. Recently, there has been renewed interest in these permutations. In particular, Berman and Tenner showed they satisfy…

Let $S_n$ denote the symmetric group of permutations acting on $n$ elements. We investigate the double sequence $\{N_{\ell}(n)\}$ counting the number of $\ell$ tuples of elements of the symmetric group $S_n$, where the components commute,…

In this paper, we count a dual set of Stirling permutations by the number of alternating runs. Properties of the generating functions, including recurrence relations, grammatical interpretations and convolution formulas are studied.

Vertically symmetric alternating sign matrices (VSASMs) of order $2n+1$ are known to be equinumerous with lozenge tilings of a hexagon with side lengths $2n+2$, $2n$, $2n+2$, $2n$, $2n+2$, $2n$ and a central triangular hole of size $2$ that…

The concept of S-permutation matrix is considered in this paper. It defines when two binary matrices are disjoint. For an arbitrary $n^2 \times n^2$ S-permutation matrix, a lower band of the number of all disjoint with it S-permutation…

The reported combined excess of about 4$\sigma$, by BABAR, Belle and LHCb, in the measurements of R($D^{(*)}$) and Br(B $\rightarrow \tau \nu_{\tau}$) from that of the standard model expectation constitute the most significant discrepancy…

An involution is a permutation that is its own inverse. Given a permutation $\sigma$ of $[n],$ let $\mathbf{N}_{n}(\sigma)$ denote the number of ways to write $\sigma$ as a product of two involutions of $[n].$ If we endow the symmetric…