Related papers: Restricted Color $n$-color Compositions

We consider colored compositions where only some parts are allowed different colors, depending on their locations in the composition. The counting sequences are obtained through generating functions. Connections to many other combinatorial…

We study compositions whose parts are colored by subsequences of the Fibonacci numbers. We give explicit bijections between Fibonacci colored compositions and several combinatorial objects, including certain restricted ternary and…

In this paper, we settle the open complexity status of interval constrained coloring with a fixed number of colors. We prove that the problem is already NP-complete if the number of different colors is 3. Previously, it has only been known…

In this paper we introduce mixed coloured permutation, permutations with certain coloured cycles, and study the enumerative properties of these combinatorial objects. We derive the generating function, closed forms, recursions and…

Integer compositions restricted by inequalities on certain pairs of parts were first considered by J\"{o}rg Arndt in 2013 and several variations have been studied recently. Here we consider a broad two-parameter generalization that scales…

Carlitz considered integer compositions in which adjacent parts must be unequal. Arndt recently initiated the study of restricted compositions based on conditions applied to certain pairs of parts rather than to individual parts. Here, we…

A model named `Colored Percolation' has been introduced with its infinite number of versions in two dimensions. The sites of a regular lattice are randomly occupied with probability $p$ and are then colored by one of the $n$ distinct colors…

Colored interlacing triangles, introduced by Aggarwal-Borodin-Wheeler (2024), provide the combinatorial framework for the Central Limit Theorem for probability measures arising from the Lascoux-Leclerc-Thibon (LLT) polynomials. Colored…

Let $k$ and $r$ be two integers with $k \ge 2$ and $k\ge r \ge 1$. In this paper we show that (1) if a strongly connected digraph $D$ contains no directed cycle of length $1$ modulo $k$, then $D$ is $k$-colorable; and (2) if a digraph $D$…

MacMahon's definition of self-inverse composition is extended to $n$-colour self-inverse composition. This introduces four new sequences which satisfy the same recurrence relation with different initial conditions like the famous Fibonacci…

Bright, saturated structural colors in birds have inspired synthesis of self-assembled, disordered arrays of assembled nanoparticles with varied particle spacings and refractive indices. However, predicting colors of assembled…

We study the number of factorizations of a positive integer, where the parts of the factorization are of l different colors (or kinds). Recursive or explicit formulas are derived for the case of unordered and ordered, distinct and…

By the Road Coloring Theorem (Trahtman, 2008), the edges of any aperiodic directed multigraph with a constant out-degree can be colored such that the resulting automaton admits a reset word. There may also be a need for a particular reset…

This paper shows how the study of colored compositions of integers reveals some unexpected and original connection with the Invert operator. The Invert operator becomes an important tool to solve the problem of directly counting the number…

In his recent work, Andrews revisited two-color partitions with certain restrictions on the differences between consecutive parts, and he established three theorems linking these two-color partitions with more familiar kinds of partitions.…

In responding to a question on Math Stackexchange, the author formulated the problem of determining the number of strings of balls colored in most $n$ colors with a number $k$ of repeated colors. In this paper, we formulate the problem more…

Let $p_k(n)$ denote the number of $2$-color partitions of $n$ where one of the colors appears only in parts that are multiples of $k$. We will prove a conjecture of Ahmed, Baruah, and Dastidar on congruences modulo $5$ for $p_k(n)$.…

Integer compositions with certain colored parts were introduced by Andrews in 2007 to address a number-theoretic problem. Integer compositions allowing zero as some parts were introduced by Ouvry and Polychronakos in 2019. We give a…

Let $a_k(n)$ denote the number of partitions of $n$ wherein even parts come in only one color, while the odd parts may be ``colored" with one of $k$ colors, for fixed $k$. In this note, we find some congruences for $a_k(n)$ in the spirit of…

The coloring problem is studied in the paper for graph classes defined by two small forbidden induced subgraphs. We prove some sufficient conditions for effective solvability of the problem in such classes. As their corollary we determine…