Related papers: Combinatorics of Multicompositions
Given integers $a_1, a_2, ..., a_n$, with $a_1 + a_2 + ... + a_n \geq 1$, a symmetrically constrained composition $\lambda_1 + lambda_2 + ... + lambda_n = M$ of $M$ into $n$ nonnegative parts is one that satisfies each of the the $n!$…
We study formulas expressing Fibonacci numbers as sums over compositions using free submonoids of the free monoid of compositions with parts 1 and 2.
We study compositions of a positive integer $n$ in which the occurrence of even parts larger than a fixed threshold $k$ is controlled. More precisely, for each composition $m=(m_1,\dots,m_r)$ we consider the number of even parts strictly…
In 2020, Coregliano and Razborov introduced a general framework to study limits of combinatorial objects, using logic and model theory. They introduced the abstract chromatic number and proved/reproved multiple…
This is the first paper that provides a systematic treatment of the $r$-dimensional PTE problem in additive number theory, abbreviated by PTE$_r$, through its connection with combinatorial design theory, the branch of combinatorial…
The non-negative integer cocharge statistic on words was introduced in the 1970's by Lascoux and Sch\"utzenberger to combinatorially characterize the Hall-Littlewood polynomials. Cocharge has since been used to explain phenomena ranging…
In this paper we consider particular generalized compositions of a natural number with a given number of parts. Its number is a weighted polynomial coefficient. The number of all generalized compositions of a natural number is a weighted…
We use the combinatorial properties of central sets to prove a result about the existence of exponential monochromatic patterns, in the style of Hindman's Finite Sums Theorem. More precisely, we prove that for every finite coloring of the…
Utilizing spectral residues of parameterized, recursively defined sequences, we develop a general method for generating identities of composition sums. Specific results are obtained by focusing on coefficient sequences of solutions of first…
This note examines a problem in enumerative and asymptotic combinatorics involving the classical structure of integer compositions. What is sought is an analysis on average and in distribution of the length of the longest run of consecutive…
In this article we are introducing combinatorial spectra of graphs, this is a generalization of $H$-Hamiltonian spectra. The main motivation was to made from $H$-Hamiltonian spectra an operation and develop some algebra in this field. An…
The study of combinatorial properties of mathematical objects is a very important research field and continued fractions have been deeply studied in this sense. However, multidimensional continued fractions, which are a generalization…
In 2019, Andrews investigated integer partitions in which all parts of a given parity are smaller than those of the opposite parity and introduced eight partition functions based on the parity of the smaller parts and parts of a given…
We count the number of countable homogeneous colored linear orderings in $k$ colors. Relatedly, we count the number of countable $C_{n,m}$-homogeneous linear orderings. $C_{n,m}$-homogeneity is a strong homogeneity notion that approximates…
This is an introduction to algebraic combinatorics, written for a quarter-long graduate course. It starts with a rigorous introduction to formal power series with some combinatorial applications, then discusses integer partitions (proving…
We consider colored variants of a class of geometric-combinatorial questions on $k$-gons and empty $k$-gons that have been started around 1935 by Erd\H{o}s and Szekeres. In our setting we have $n$ points in general position in the plane,…
Recently, Andrews and EI Bachraoui discovered several companions for some famous $q$-series formulas, and derived some new identities involving partitions and overpartitions with distinct parts. In this paper, we shall refine their results…
Let c^{k,l}(n) be the number of compositions (ordered partitions) of the integer n whose Ferrers diagram fits inside a k-by-l rectangle. The purpose of this note is to give a simple, algebraic proof of a conjecture of Vatter that the…
We consider a general concept of composition and decomposition of objects, and discuss a few natural properties one may expect from a reasonable choice thereof. It will be demonstrated how this leads to multiplication and co- multiplication…
A paper of the first author and Zilke proposed seven combinatorial problems around formulas for the characteristic polynomial and the exponents of an isolated quasihomogeneous singularity. The most important of them was a conjecture on the…