Related papers: On the cycle index and the weight enumerator
We introduce the notion of a weighted $\delta$-vector of a lattice polytope. Although the definition is motivated by motivic integration, we study weighted $\delta$-vectors from a combinatorial perspective. We present a version of Ehrhart…
The notion of the weighted core inverse in a ring with involution was introduced, recently [Mosic et al. Comm. Algebra, 2018; 46(6); 2332-2345]. In this paper, we explore new representation and characterization of the weighted core inverse…
This paper is concentrated on the classification of permutation matrix with the permutation similarity relation, mainly about the canonical form of a permutational similar equivalence class, the cycle matrix decomposition of a permutation…
Studying the generalized Hamming weights of linear codes is a significant research area within coding theory, as it provides valuable structural information about the codes and plays a crucial role in determining their performance in…
The weight distribution and weight hierarchy of linear codes are two important research topics in coding theory. In this paper, by choosing proper defining sets from inhomogeneous quadratic functions over $\mathbb{F}_{q}^{2},$ we construct…
In this paper we introduce a binomial ideal derived from a binary linear code. We present some applications of a Gr\"obner basis of this ideal with respect to a total degree ordering. In the first application we give a decoding method for…
We describe a new notation for finite transformations. This attractor-cycle notation extends the orbit-cycle notation for permutations and builds upon existing transformation notations. How the basins of attraction of a finite…
We consider the complexity of two questions on polynomials given by arithmetic circuits: testing whether a monomial is present and counting the number of monomials. We show that these problems are complete for subclasses of the counting…
We introduce a new series $R_k$, $k=2,3,4,\dots$, of integer valued weight systems. The value of the weight system $R_k$ on a chord diagram is a signed number of cycles of even length $2k$ in the intersection graph of the diagram. We show…
Cylindric plane partitions may be thought of as a natural generalization of reverse plane partitions. A generating series for the enumeration of cylindric plane partitions was recently given by Borodin. The first result of this paper is a…
We consider the task of weighted first-order model counting (WFOMC) used for probabilistic inference in the area of statistical relational learning. Given a formula $\phi$, domain size $n$ and a pair of weight functions, what is the…
We apply the theory of weighted bicategorical colimits to study the problem of existence and computation of such colimits of birepresentations of finitary bicategories. The main application of our results is the complete classification of…
Using the notion of generalized weight we improve estimates on the parameters of quantum codes obtained by Steane's construction from binary codes. This yields several new families of quantum codes.
The Hamiltonian cycle polynomial can be evaluated to count the number of Hamiltonian cycles in a graph. It can also be viewed as a list of all spanning cycles of length $n$. We adopt the latter perspective and present a pair of original…
We determine a condition on the minimum Hamming weight of some special abelian group codes and, as a consequence of this result, we establish that any such code is, up to permutational equivalence, a subspace of the direct sum of $s$ copies…
The cyclic codes with parity check polynomial the reciprocal of the characteristic polynomial of the Fibonacci recurrence over a prime finite field are shown to have either one weight or two weights. When these codes are irreducible cyclic…
The study of pinnacle sets has been a recent area of interest in combinatorics. Given a permutation, its pinnacle set is the set of all values larger than the values on either side of it. Largely inspired by conjectures posed by Davis,…
Cyclic codes over finite fields are widely implemented in data storage systems, communication systems, and consumer electronics, as they have very efficient encoding and decoding algorithms. They are also important in theory, as they are…
Fully commutative elements in types $B$ and $D$ are completely characterized and counted by Stembridge. Recently, Feinberg-Kim-Lee-Oh have extended the study of fully commutative elements from Coxeter groups to the complex setting, giving…
This paper has two objectives. The first is to develop the theory of bicategories enriched in a monoidal bicategory -- categorifying the classical theory of categories enriched in a monoidal category -- up to a description of the free…