Density and Symmetry in the Generalized Motzkin Numbers mod $p$
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910787538255872 |
|---|---|
| author | Kohen, Nadav |
| author_facet | Kohen, Nadav |
| contents | We give a formula for the density of $0$ in the sequence of generalized Motzkin numbers, $M^{a, b}_n$, modulo a prime, $p$, in terms of the first $p$ generalized central trinomial coefficients $T^{a, b}_n\bmod p$ (with $n<p$). We apply our method to various other sequences to obtain similar formulas. We also prove that $T^{a, b}_{p-1-n}\equiv (b^2-4a^2)^{\frac{p-1}{2}-n}T^{a, b}_n\pmod p$ to obtain tight lower bounds for the density of $0$ in our sequences. This symmetry of the first $p$ central trinomial coefficients mod $p$ also appears in a couple of other applications, including the proof of a novel symmetry of the first $p-2$ Motzkin numbers that is of independent interest: $M^{a, b}_{p-3-n}\equiv (b^2-4a^2)^{\frac{p-3}{2}-n}M^{a, b}_n\pmod p$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_03681 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Density and Symmetry in the Generalized Motzkin Numbers mod $p$ Kohen, Nadav Combinatorics Number Theory 11B05, 11B50 (Primary) 68R15, 05A15, 11B85 (Secondary) We give a formula for the density of $0$ in the sequence of generalized Motzkin numbers, $M^{a, b}_n$, modulo a prime, $p$, in terms of the first $p$ generalized central trinomial coefficients $T^{a, b}_n\bmod p$ (with $n<p$). We apply our method to various other sequences to obtain similar formulas. We also prove that $T^{a, b}_{p-1-n}\equiv (b^2-4a^2)^{\frac{p-1}{2}-n}T^{a, b}_n\pmod p$ to obtain tight lower bounds for the density of $0$ in our sequences. This symmetry of the first $p$ central trinomial coefficients mod $p$ also appears in a couple of other applications, including the proof of a novel symmetry of the first $p-2$ Motzkin numbers that is of independent interest: $M^{a, b}_{p-3-n}\equiv (b^2-4a^2)^{\frac{p-3}{2}-n}M^{a, b}_n\pmod p$. |
| title | Density and Symmetry in the Generalized Motzkin Numbers mod $p$ |
| topic | Combinatorics Number Theory 11B05, 11B50 (Primary) 68R15, 05A15, 11B85 (Secondary) |
| url | https://arxiv.org/abs/2411.03681 |