Cornerless, peakless, valleyless Motzkin paths (regular and skew) and applications to bargraphs
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912201365782528 |
|---|---|
| author | Prodinger, Helmut |
| author_facet | Prodinger, Helmut |
| contents | Motzkin paths consist of up-steps, down-steps, horizontal steps, never go below the $x$-axis and return to the $x$-axis. Versions where the return to the $x$-axis isn't required are also considered. A path is peakless (valleyless) if $UD$ (if $DU$) never occurs. If it is both peakless and valleyless, it is called cornerless. Deutsch and Elizalde have linked cornerless Motzkin paths and bargraphs bijectly. Thus, instead of prefixes of bargraphs one might consider prefixes of cornerless Motzkin paths. In this paper, this is extended by counting the occurrences of $UD$ resp., $DU$. The concepts are extended to so-called skew Motzkin paths. Methods are generating functions and the kernel method to compute explicit forms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_13645 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cornerless, peakless, valleyless Motzkin paths (regular and skew) and applications to bargraphs Prodinger, Helmut Combinatorics Motzkin paths consist of up-steps, down-steps, horizontal steps, never go below the $x$-axis and return to the $x$-axis. Versions where the return to the $x$-axis isn't required are also considered. A path is peakless (valleyless) if $UD$ (if $DU$) never occurs. If it is both peakless and valleyless, it is called cornerless. Deutsch and Elizalde have linked cornerless Motzkin paths and bargraphs bijectly. Thus, instead of prefixes of bargraphs one might consider prefixes of cornerless Motzkin paths. In this paper, this is extended by counting the occurrences of $UD$ resp., $DU$. The concepts are extended to so-called skew Motzkin paths. Methods are generating functions and the kernel method to compute explicit forms. |
| title | Cornerless, peakless, valleyless Motzkin paths (regular and skew) and applications to bargraphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2501.13645 |