Related papers: Wachs permutations, Bruhat order and weak order

Permutation polynomials are of particular significance in several areas of applied mathematics, such as Coding theory and Cryptography. Many recent constructions are based on the Akbary-Ghioca-Wang (AGW) criterion. Along this line of…

We analyze the problem of determining Waring decompositions of the powers of any quadratic form over the field of complex numbers. Our main goal is to provide information about their rank and also to obtain decompositions whose size is as…

We find a combinatorial setting for the coefficients of the Boros-Moll polynomials $P_m(a)$ in terms of partially 2-colored permutations. Using this model, we give a combinatorial proof of a recurrence relation on the coefficients of…

Partially ordered sets (posets) play a universal role as an abstract structure in many areas of mathematics. For finite posets, an explicit enumeration of distinct partial orders on a set of unlabelled elements is known only up to a…

Clans are combinatorial objects indexing the orbits of $GL(\mathbb{C}^p) \times GL(\mathbb{C}^q)$ on the variety of flags in $\mathbb{C}^{p+q}$. This geometry leads to a partial order on the set of clans analogous to weak Bruhat order on…

We examine ordered graphs, defined as graphs with linearly ordered vertices, from the perspective of homomorphisms (and colorings) and their complexities. We demonstrate the corresponding computational and parameterized complexities, along…

Let $k$ be a field of characteristic $0$, let $\mathsf{C}$ be a finite split category, let $\alpha$ be a 2-cocycle of $\mathsf{C}$ with values in the multiplicative group of $k$, and consider the resulting twisted category algebra…

We introduce a notion of degenerations of graded modules. In relation to it, we also introduce several partial orders as graded analogies of the hom order, the degeneration order and the extension order. We prove that these orders are…

Let $\k$ be a field and let $A$ be a standard $\mathbb{N}$-graded $\k$-algebra. Using numerical information of some invariants in the primary decomposition of $0$ in $A$, namely the so called dimension filtration, we associate a bivariate…

Matrix properties are a type of property of categories which includes the ones of being Mal'tsev, arithmetical, majority, unital, strongly unital and subtractive. Recently, an algorithm has been developed to determine implications…

In this work we discuss a new type of factorisation systems for \textbf{Ord}-enriched categories. We start by defining the new notion of lax weak orthogonality, which involves the existence of lax diagonal morphisms for lax squares. Using…

We provide an algorithm to construct a multicomplex for any lower Bruhat interval of $F_4$, such that its rank--generating function equals that of the Bruhat interval. For Weyl groups, it is always possible to find such a multicomplex…

For an untwisted affine Kac-Moody Lie algebra $\mathfrak{g}$ with Cartan and Borel subalgebras $\mathfrak{h} \subset \mathfrak{b} \subset \mathfrak{g}$, affine Demazure modules are certain $U(\mathfrak{b})$-submodules of the irreducible…

Using the theory of Macdonald, Gordon showed that the graded characters of the simple modules for the restricted rational Cherednik algebra by Etingof and Ginzburg associated to the symmetric group $\mathfrak{S}_n$ are given by…

Partially ordered patterns (POPs) generalize the notion of classical patterns studied in the literature in the context of permutations, words, compositions and partitions. In this paper, we give a number of general, and specific enumerative…

We show that the Zagier-Eisenstein series shares its non-holomorphic part with certain weak Maass forms whose holomorphic parts are generating functions for overpartition rank differences. This has a number of consequences, including exact…

We study permutations on n elements preserving orientation (parity) of every subset of size k. We describe all groups of these permutations. Unexpectedly, these groups (except for some special cases) are either trivial, cyclic or dihedral.…

There is a natural analogue of weak Bruhat order on the involutions in any Coxeter group. The saturated chains of intervals in this order correspond to reduced words for a certain set of group elements called atoms. Brion gives a general…

We investigate permutations in terms of their cycle structure and descent set. To do this, we generalize the classical bijection of Gessel and Reutenauer to deal with permutations that have some ascending and some descending blocks. We then…

This note presents a procedure to determine the reduction of the irreducible and the induced characters of the symmetric group in terms of the irreducible and induced characters of the hyperoctahedral group Key Words: Symmetric Group,…