• A Statistic on 𝑛-Color Compositions and Related Sequences

• # Fulltext

https://www.ias.ac.in/article/fulltext/pmsc/124/02/0127-0140

• # Keywords

Compositions; 𝑛-color compositions; 𝑞-generalization.

• # Abstract

A composition of a positive integer in which a part of size 𝑛 may be assigned one of 𝑛 colors is called an 𝑛-color composition. Let $a_m$ denote the number of 𝑛-color compositions of the integer 𝑚. It is known that $a_m = F_{2m}$ for all $m \geq 1$, where $F_m$ denotes the Fibonacci number defined by $F_m = F_{m-1}+F_{m-2}$ if $m\geq 2$, with $F_0=0$ and $F_1=1$. A statistic is studied on the set of 𝑛-color compositions of 𝑚 thus providing a polynomial generalization of the sequence $F_{2m}$. The statistic may be described, equivalently, in terms of statistics on linear tilings and lattice paths. The restriction to the set of 𝑛-color compositions having a prescribed number of parts is considered and an explicit formula for the distribution is derived. We also provide 𝑞-generalizations of relations between $a_m$ and the number of self-inverse 𝑛-compositions of $2m+1$ or $2m$. Finally, we consider a more general recurrence than that satisfied by the numbers $a_m$ and note some particular cases.

• # Author Affiliations

1. Department of Mathematics, University of Haifa, 3498838 Haifa, Israel

• # Proceedings – Mathematical Sciences

Current Issue
Volume 129 | Issue 5
November 2019

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019