| Title:
|
Enumerating 2D and 3D lattice paths with arbitrary steps (English) |
| Author:
|
Karaçam, Cemil |
| Author:
|
Vural, Alper |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0011-4642 |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
151 |
| Issue:
|
3 |
| Year:
|
2026 |
| Pages:
|
481-493 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $S$ be a finite set of integer vectors. We consider lattice paths that use only the vectors in $S$. We focus on paths that use a fixed number of vectors. We generally assume vectors in $S$ have a fixed coordinate sum, which allows us to determine the number of vectors in a path, which we call its length. We count the number of paths with fixed length for various sets of vectors $S$. We then use our enumeration results to determine the minimal length path given a terminal point. First, we explore this problem in $S \subseteq \mathbb {N}^3$. After solving the problem of enumeration and determining the minimal length for various sets $S\subseteq \mathbb {N}^3$, we solve these problems for a general case $S=\{(1,0),(0,1),(u,v),(v,u)\}$. We conclude with an enumeration problem of paths that stay weakly below the line $y=x$. (English) |
| Keyword:
|
lattice path |
| Keyword:
|
enumeration |
| Keyword:
|
shortest path |
| Keyword:
|
generating function |
| MSC:
|
05A15 |
| MSC:
|
05A19 |
| DOI:
|
10.21136/MB.2025.0084-24 |
| . |
| Date available:
|
2026-08-24T07:53:14Z |
| Last updated:
|
2026-08-24 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/153718 |
| . |
| Reference:
|
[1] Evoniuk, J., Klee, S., Magnan, V.: Enumerating minimal length lattice paths.J. Integer Seq. 21 (2018), Article ID 18.3.6, 12 pages. Zbl 1384.05020, MR 3805751 |
| Reference:
|
[2] Firoozi, F.: Enumeration of Lattice Paths with Respect to a Linear Boundary.Simon Fraser University, Burnaby (2023), Available at https://summit.sfu.ca/item/36159\kern0pt. |
| Reference:
|
[3] Humphreys, K.: A history and a survey of lattice path enumeration.J. Stat. Plann. Inference 140 (2010), 2237-2254. Zbl 1204.05015, MR 2609483, 10.1016/j.jspi.2010.01.020 |
| Reference:
|
[4] Iwanojko, N., Klee, S., Lasher, B., Volpi, E.: Enumerating lattice walks with prescribed steps.J. Integer Seq. 23 (2020), Article ID 20.4.3, 15 pages. Zbl 1439.05017, MR 4105870 |
| Reference:
|
[5] Krattenthaler, C.: Lattice path enumeration.Handbook of Enumerative Combinatorics Discrete Mathematics and its Applications. CRC Press, Boca Raton (2015), 589-678. Zbl 1332.05009, MR 3409351, 10.1201/b18255-16 |
| Reference:
|
[6] Sloane, N. J. A.: The On-Line Encyclopedia of Integer Sequences.Available at\ https://oeis.org/. Zbl 1159.11327 |
| Reference:
|
[7] White, V.: Enumeration of Lattice Paths with Restrictions.Georgia Southern University, Statesboro (2024), Available at\ https://digitalcommons.georgiasouthern.edu/etd/2799/\kern0pt. |
| . |
