@article {
author = {Bandpey, Zeinab and Farley, Jonathan},
title = {The number of graph homomorphisms between paths and cycles with loops, a problem from Stanleyâ€™s enumerative combinatorics},
journal = {Transactions on Combinatorics},
volume = {12},
number = {3},
pages = {115-130},
year = {2023},
publisher = {University of Isfahan},
issn = {2251-8657},
eissn = {2251-8665},
doi = {10.22108/toc.2022.131646.1949},
abstract = {Let $g_{k}(n)$ denote the number of sequences $t_{1},ldots,t_{n}$ in $\{0, 1,\ldots,k-1\}$ such that $t_{j+1}\equiv t_{j}-1, t_{j}$ or $t_{j}+1$ (mod $k$), $1\leq j\leq n$, (where $t_{n+1}$ is identified with $t_{1}$). It is proved combinatorially that $g_{4}(n)= 3^{n}+2+(-1)^{n}$ and $g_{6}(n)= 3^{n}+2^{n+1}+(-1)^{n}$. This solves a problem from Richard P. Stanley's 1986 text, $Enumerative$ $Combinatorics$.},
keywords = {Trinomial coefficient,Path,cycle,(graph) homomorphism,transfer matrix method},
url = {https://toc.ui.ac.ir/article_26788.html},
eprint = {https://toc.ui.ac.ir/article_26788_8767f1e10b0ecd627858fb572c491887.pdf}
}