Reifegerste and independently, Petersen and Tenner studied a statistic $\text{drops}$ on permutations in ${\mathfrak{S}}_n$. Two other studied statistics on ${\mathfrak{S}}_n$ are $\text{depth}$ and $\text{exc}$. Using descents in canonical reduced words of elements in ${\mathfrak{S}}_n$, we give an involution $f_A:{\mathfrak{S}}_n\mapsto {\mathfrak{S}}_n$ ...$$${\mathfrak{}}_{}$${\mathfrak{}}_{}$${\mathfrak{}}_{}$${}_{\mathit{}}$$$${}_{}$$$$$$$$$${\mathfrak{}}_{}$$$$$${\mathcal{}}_{}$

Let q = 2 m. In this paper, we investigate permutation pentanomials over F q 2 of the form f ( x ) = x t + x r 1 ( q − 1 ) + t + x r 2 ( q − 1 ) + t + x r 3 ( q − 1 ) + t + x r 4 ( q − 1 ) + t with gcd ( x r 4 + x r 3 + x r 2 + x r 1 + 1 , x t + ...

We introduce a kind of ( p , q , t )-Catalan numbers of Type A by generalizing the J-type continued fraction formula, we prove that the corresponding expansions could be expressed by the polynomials counting permutations on S n ( 321 ) by various ...

Two totally order relations are defined on the hyperoctahedral group ${\mathfrak{B}}_n$. Regarding ${\mathfrak{B}}_n$ as a Coxeter group of type B, we always use the natural order. By taking ${\mathfrak{B}}_n$ as a color permutation group, it is convenient to use the r-order. Considering ${\mathfrak{B}}_n$ as ...${\mathfrak{}}_{}$$$${\mathfrak{}}_{}$$$${\mathfrak{}}_{}$$$${\mathfrak{}}_{}$

We introduce a class of permutation polynomial over F q n that can be written in the form L ( x ) x q + 1 or L ( x q + 1 ) x for some q-linear polynomial L over F q n. Specifically, we present those permutation polynomials explicitly as well as ...

This paper investigates two involutions on binary trees. One is the mirror symmetry of binary trees which combined with the classical bijection φ between binary trees and plane trees answers an open problem posed by Bai and Chen. This involution ...

Building off recent work of Garg and Peng, we continue the investigation into classical and consecutive pattern avoidance in rooted forests, resolving some of their conjectures and questions and proving generalizations whenever possible. Through ...

In this paper, we generalize a theorem of R. P. Stanley regarding the enumeration of paths in acyclic digraphs.

As a natural generalization of permutations, transversals of Young diagrams play an important role in the study of pattern avoiding permutations. Let T λ ( τ ) and ST λ ( τ ) denote the set of τ-avoiding transversals and τ-avoiding symmetric ...

We show that if a permutation statistic can be written as a linear combination of bivincular patterns, then its moments can be expressed as a linear combination of factorials with constant coefficients. This generalizes a result of Zeilberger. We ...

• General approach to find formulas for moments of permutation statistics.
• Intuitive proof of the Central Limit Theorem for occurrences of classical patterns.
• Computationally useful summation formula for arbitrary descent statistic ...

A permutation $\pi$ is a merge of a permutation $\sigma$ and a permutation $\tau$, if we can color the elements of $\pi$ red and blue so that the red elements have the same relative order as $\sigma$ and the blue ones as $\tau$. We consider, for fixed hereditary permutation ...$\mathcal{}$$\mathcal{}$$$$\mathcal{}$$\mathcal{}$

Over the past several years, there are numerous papers about permutation polynomials of the form x r h ( x q − 1 ) over F q 2. A bijection between the multiplicative subgroup μ q + 1 of F q 2 and the projective line PG ( 1 , q ) = F q ∪ { ∞ } ...

In this paper, we construct a mixed-base number system over the generalized symmetric group G ( m , 1 , n ), which is a complex reflection group with a root system of type B n ( m ). We also establish one-to-one correspondence between all ...

Given integers $k,l\ge 2$, where either l is odd or k is even, let n(k, l) denote the largest integer n such that each element of $A_n$ is a product of k many l-cycles. M. Herzog, G. Kaplan and A. Lev conjectured that $\lfloor \frac{2kl}{3}\rfloor \le n(k,l)\le \lfloor \frac{2kl}{3}\rfloor +1$ [Herzog et al. ...$\mathrm{}\mathrm{}\mathrm{}$$\mathrm{}$$\mathrm{}$$\mathrm{}$$\frac{\mathrm{}}{\mathrm{}}$$\mathrm{}$$\mathrm{}$

Let L be a finite lattice. Inspired by Ungar’s solution to the famous slopes problem, we define an Ungar move to be an operation that sends an element $x\in L$ to the meet of $\{x\}\cup T$, where T is a subset of the set of elements covered by x. We introduce ...${}_{}$$\mathrm{}{\mathrm{}}^{}$

We consider multipermutations and a certain partial order, the weak Bruhat order, on this set. This generalizes the Bruhat order for permutations, and is defined in terms of containment of inversions. Different characterizations of this order are ...

Let q = 2 m and F q 3 be the finite field with q 3 elements. In this paper, based on the multivariate method, resultant elimination, and transforming into dealing with some equations over finite fields, we propose two classes of permutation ...

Motivated by the work of Visontai and Dey–Sivasubramanian on the gamma-positivity of some polynomials, we discover the commutative property of a pair of Eulerian operators. As an application, we show the bi-gamma-positivity of the descent ...

We prove two conjectures involving permutation polynomials in a paper of Dmytrenko, Lazebnik, Williford, in a low degree regime, using the theory of algebraic curves over finite fields. More precisely, we prove that Conjecture A holds whenever q ≥...

We introduce an algorithmic approach based on a generating tree method for enumerating the inversion sequences with various pattern-avoidance restrictions. For a given set of patterns, we propose an algorithm that outputs either an ...