Related papers: Signed words and permutations, IV; Fixed and pixed…

Let G(q) be the group of permutations of the finite field F_q which is generated by those permutations that can be written as c-->ac^m+bc^n with 0<m<n<q and a,b in F_q with ab nonzero. We show that there are infinitely many q for which G(q)…

Starting from the moment sequences of classical orthogonal polynomials we derive the orthogonality purely algebraically. We consider also the moments of ($q=1$) classical orthogonal polynomials, and study those cases in which the…

A classical result states that the parity balance of the number of excedances of all permutations (derangements, respectively) of length $n$ is the Euler number. In 2010, Josuat-Verg\`{e}s gives a $q$-analogue with $q$ representing the…

To a univariate monic polynomial is attached a special planar forest that is called the picture of the polynomial. Isotopy classes of pictures are called signatures. All combinatorially possible signatures are realized and spaces of…

We show the classical $q$-Stirling numbers of the second kind can be expressed compactly as a pair of statistics on a subset of restricted growth words. The resulting expressions are polynomials in $q$ and $1+q$. We extend this enumerative…

Operators acting on the discrete random chaos yield signed multiplicative systems, extending the notion of spin matrices and quaternions. We investigate signed groups through the associated sign matrices, focusing on generators and their…

For a prime number $p$ and a free pro-$p$ group $G$ on a totally ordered basis $X$, we consider closed normal subgroups $G^\Phi$ of $G$ which are generated by $p$-powers of iterated commutators associated with Lyndon words in the alphabet…

In this paper, we start by giving the definitions and basic facts about hyperoctahedral number system. There is a natural correspondence between the integers expressed in the latter and the elements of the hyperoctahedral group when we use…

A signed labeled forest is defined as a (plane) forest labeled by {1,2,..., n} along with minus signs associated to some vertices. Signed labeled forests can be viewed as an extension of signed permutations. We define the inversion number,…

We prove a new signed elementary symmetric function expansion of the chromatic quasisymmetric function of any natural unit interval graph. We then use a sign-reversing involution to prove a new combinatorial formula for K-chains, which are…

For all finite fields of $q$ elements where $q\equiv1\pmod4$ we have constructed permutation polynomials which have order 2 as permutations, and have 3 terms, or 4 terms as polynomials. Explicit formulas for their coefficients are given in…

Smith normal form evaluations found by Bessenrodt and Stanley for some Hankel matrices of q-Catalan numbers are proven in two ways. One argument generalizes the Bessenrodt-Stanley results for the Smith normal form of a certain multivariate…

The use of permutation polynomials has appeared, along to their compositional inverses, as a good choice in the implementation of cryptographic systems. Hence, there has been a demand for constructions of these polynomials which…

For a fixed positive integer n, let S_n denote the symmetric group of n! permutations on n symbols, and let maj(sigma) denote the major index of a permutation sigma. For positive integers k<m not greater than n and non-negative integers i…

We work an analogue of a classical arithmetic problem over polynomials. More precisely, we study the fixed points $F$ of the sum of divisors function $\sigma : F_2[x] \mapsto F_2[x]$ (defined \emph{mutatis mutandi} like the usual sum of…

The paper designs five graph operations, and proves that every signed graph with chromatic number $q$ can be obtained from all-positive complete graphs $(K_q,+)$ by repeatedly applying these operations. This result gives a signed version of…

First introduced in the study of the Sturmian words, the iterated palindromic closure was recently generalized to pseudopalindromes. This operator allows one to construct words with an infinity of pseudopalindromic prefixes, called…

A signed graph has edge signs. A gain graph has oriented edge gains drawn from a group. We define the combination of the two for the abelian case, in which each oriented edge of a signed graph has a gain from an abelian group, concentrating…

We study biinvariant word metrics on groups. We provide an efficient algorithm for computing the biinvariant word norm on a finitely generated free group and we construct an isometric embedding of a locally compact tree into the biinvariant…

We exhibit a recurrence on the number of discrete line segments joining two integer points in the plane using an encoding of such segments as balanced words of given length and height over the two-letter alphabet $\{0,1\}$. We give…