Related papers: Descent sets of cyclic permutations in types B and…

It is well known that descents and excedances are equidistributed in the symmetric group. We show that the descent and excedance enumerators, summed over permutations with a fixed first letter are identical when we perform a simple change…

We explicitly describe the derived Picard groups of symmetric representation-finite algebras of type $D$. In particular, we prove that these groups are generated by spherical twists along collections of $0$-spherical objects, the shift and…

For a positive integer $n$, a collection $S$ of subsets of $[n]=\{1,\ldots,n\}$ is called symmetric if $X\in S$ implies $X^\ast\in S$, where $X^\ast:=\{i\in [n]\colon n-i+1\notin X\}$ (the involution $\ast$ was introduced by Karpman).…

Let $R(\phi)$ be the number of $\phi$-conjugacy (or Reidemeister) classes of an endomorphism $\phi$ of a group $G$. We prove for several classes of groups (including polycyclic) that the number $R(\phi)$ is equal to the number of fixed…

Let $\pi \in \mathfrak{S}_m$ and $\sigma \in \mathfrak{S}_n$ be permutations. An occurrence of $\pi$ in $\sigma$ as a consecutive pattern is a subsequence $\sigma_i \sigma_{i+1} \cdots \sigma_{i+m-1}$ of $\sigma$ with the same order…

Pairwise non-isomorphic semigroups obtained from the finite inverse symmetric semigroup $\mathcal{IS}_n ,$ finite symmetric semigroup $\mathcal{T}_n$ and bicyclic semigroup by the deformed multiplication proposed by Ljapin are classified.

Let Sym_n denote the symmetric group of all permutations pi = a_1...a_n of {1,...,n}. An index i is a peak of pi if a_{i-1} < a_i > a_{i+1} and we let P(pi) be the set of peaks of pi. Given any set S of positive integers we define P(S;n) to…

We derive a generating function for the number of integer compositions of $n$ into $k$ parts (i.e., $k$-compositions of $n$) with a given number of inversions, and obtain similar results for $k$-compositions of $n$ with a given number of…

For a fixed prime $p$, we consider a filtration of the commuting complex of elements of order $p$ in the symmetric group $\mathfrak{S}_n$. The filtration is obtained by imposing successively relaxed bounds on the number of disjoint…

A general setting to study a certain type of formulas, expressing characters of the symmetric group $\mathfrak{S}_n$ explicitly in terms of descent sets of combinatorial objects, has been developed by two of the authors. This theory is…

Given a set $I \subseteq \mathbb{N}$, consider the sequences $\{d_n(I)\},\{p_n(I)\}$ where for any $n$, $d_n(I)$ and $p_n(I)$ respectively count the number of permutations in the symmetric group $\mathfrak{S}_n$ whose descent set…

A permutation $\pi$ is said to be {\em Dumont permutations of the first kind} if each even integer in $\pi$ must be followed by a smaller integer, and each odd integer is either followed by a larger integer or is the last element of $\pi$…

Reifegerste and independently, Petersen and Tenner studied a statistic $\mathrm{drops}()$ on permutations in $\mathfrak{S}_n$. Two other studied statistics on $\mathfrak{S}_n$ are $\mathrm{depth}$ and $\mathrm{exc}$. Using descents in ${\it…

Let $N_\sigma(\pi)$ denote the number of occurrences of a permutation pattern $\sigma\in S_k$ in a permutation $\pi\in S_n$. Gaetz and Ryba (2021) showed using partition algebras that the $d$-th moment $M_{\sigma,d,n}(\pi)$ of $N_\sigma$ on…

Let $\phi$ be a birational map of the complex projective plane. We know that $\phi$ can be written as a composition of automorphisms of $\mathbb{P}^2_\mathbb{C}$ and the standard quadratic birational map $\sigma$. This writing, that is…

Computing the reversal distances of signed permutations is an important topic in Bioinformatics. Recently, a new lower bound for the reversal distance was obtained via the plane permutation framework. This lower bound appears different from…

Ascent sequences were introduced by Bousquet-M\'elou, Claesson, Dukes and Kitaev, and are in bijection with unlabeled $(2+2)$-free posets, Fishburn matrices, permutations avoiding a bivincular pattern of length $3$, and Stoimenow matchings.…

Let $G$ be a finite group having a factorisation $G=AB$ into subgroups $A$ and $B$ with $B$ cyclic and $A\cap B=1,$ and let $b$ be a generator of $B$. The associated skew-morphism is the bijective mapping $f:A \to A$ well defined by the…

The periodic (ordinal) patterns of a map are the permutations realized by the relative order of the points in its periodic orbits. We give a combinatorial characterization of the periodic patterns of an arbitrary signed shift, in terms of…

In this article we give an order-dividing bijective function between cyclic and non cyclic groups of finite order. In particular, we prove that there exists a bijective function from D_{2n} to Z_{2n} for any natural integer n; and from Z_p…