Gilbert Labelle ; Annie Lacasse
-
Closed paths whose steps are roots of unity
dmtcs:2937 -
Discrete Mathematics & Theoretical Computer Science,
January 1, 2011,
DMTCS Proceedings vol. AO, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011)
-
https://doi.org/10.46298/dmtcs.2937
Closed paths whose steps are roots of unityArticle
Authors: Gilbert Labelle 1; Annie Lacasse 2
NULL##NULL
Gilbert Labelle;Annie Lacasse
1 Laboratoire de combinatoire et d'informatique mathématique [Montréal]
We give explicit formulas for the number $U_n(N)$ of closed polygonal paths of length $N$ (starting from the origin) whose steps are $n^{\textrm{th}}$ roots of unity, as well as asymptotic expressions for these numbers when $N \rightarrow \infty$. We also prove that the sequences $(U_n(N))_{N \geq 0}$ are $P$-recursive for each fixed $n \geq 1$ and leave open the problem of determining the values of $N$ for which the $\textit{dual}$ sequences $(U_n(N))_{n \geq 1}$ are $P$-recursive.