A Statistic on 𝑛-Color Compositions and Related Sequences

Proceedings Mathematical Sciences

Mark Shattuck¹  Toufik Mansour² 

[1] $$;Department of Mathematics, University of Haifa, Haifa, Israel$$

关键词: Compositions;    𝑛-color compositions;    𝑞-generalization.;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

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 ≥ 1, where $F_m$ denotes the Fibonacci number defined by $F_m = F_{m-1}+F_{m-2}$ if 𝑚 ≥ 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.

【 授权许可】

Unknown