{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/lattice-paths-and-submonoids-of-mathbb-z-2","title":"Lattice paths and submonoids of $\\mathbb Z^2$","arxiv_id":"1811.05735","date":"2018-11-14","proceeding":null,"authors":["James East","Nicholas Ham"],"abstract":"We study a number of combinatorial and algebraic structures arising from walks on the two-dimensional integer lattice. To a given step set $X\\subseteq\\mathbb Z^2$, there are two naturally associated monoids: $\\mathscr F_X$, the monoid of all $X$-walks/paths; and $\\mathscr A_X$, the monoid of all endpoints of $X$-walks starting from the origin $O$. For each $A\\in\\mathscr A_X$, write $\\pi_X(A)$ for the number of $X$-walks from $O$ to $A$. Calculating the numbers $\\pi_X(A)$ is a classical problem, leading to Fibonacci, Catalan, Motzkin, Delannoy and Schroder numbers, among many other well-studied sequences and arrays. Our main results give relationships between finiteness properties of the numbers $\\pi_X(A)$, geometrical properties of the step set $X$, algebraic properties of the monoid $\\mathscr A_X$, and combinatorial properties of a certain bi-labelled digraph naturally associated to $X$. There is an intriguing divergence between the cases of finite and infinite step sets, and some constructions rely on highly non-trivial properties of real numbers. We also consider the case of walks constrained to stay within a given region of the plane. Several examples are considered throughout to highlight the sometimes-subtle nature of the theoretical results.","url_abs":"https://arxiv.org/abs/1811.05735v3","url_pdf":"https://arxiv.org/pdf/1811.05735v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"lattice-paths-and-submonoids-of-mathbb-z-2","repo_url":"https://gitlab.com/n-ham-paper-files/lattice-path-algorithms","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}