Polynomials Arising from Sorted Binomial Coefficients
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866915598774042624 |
|---|---|
| author | Levens, Owen John |
| author_facet | Levens, Owen John |
| contents | The triangle of sorted binomial coefficients $\left\langle {n \atop k} \right\rangle = \binom{n}{\lfloor \frac{n - k}{2} \rfloor}$ for $0 \leq k \leq n$ has appeared several times in recent combinatorial works but has evaded dedicated study. Here we refer to $\left\langle {n \atop k} \right\rangle$ as the Pascalian numbers and unify the various perspectives of $\left\langle {n \atop k} \right\rangle$. We then view each row of the $\left\langle {n \atop k} \right\rangle$ triangle as the coefficients of the $n$th Pascalian polynomial, which we denote $P_n(z)$. We derive recursions, formulae, and bounds on $P_n(z)$'s roots in $\mathbb{C}$, and characterize the asymptotics of these roots. We show the roots of $P_n(z)$ converge uniformly to a curve $\partial Γ\subset \mathbb{C}$ and asymptotically fill the curve densely. We conclude with a discussion of the reducibility and Galois groups of $P_n(z)$. Our work has natural connections to the truncated binomial polynomials, asymptotic analysis, and well-known integer families. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_03082 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Polynomials Arising from Sorted Binomial Coefficients Levens, Owen John Combinatorics Complex Variables 12D10 (Primary), 11B65 (Secondary) The triangle of sorted binomial coefficients $\left\langle {n \atop k} \right\rangle = \binom{n}{\lfloor \frac{n - k}{2} \rfloor}$ for $0 \leq k \leq n$ has appeared several times in recent combinatorial works but has evaded dedicated study. Here we refer to $\left\langle {n \atop k} \right\rangle$ as the Pascalian numbers and unify the various perspectives of $\left\langle {n \atop k} \right\rangle$. We then view each row of the $\left\langle {n \atop k} \right\rangle$ triangle as the coefficients of the $n$th Pascalian polynomial, which we denote $P_n(z)$. We derive recursions, formulae, and bounds on $P_n(z)$'s roots in $\mathbb{C}$, and characterize the asymptotics of these roots. We show the roots of $P_n(z)$ converge uniformly to a curve $\partial Γ\subset \mathbb{C}$ and asymptotically fill the curve densely. We conclude with a discussion of the reducibility and Galois groups of $P_n(z)$. Our work has natural connections to the truncated binomial polynomials, asymptotic analysis, and well-known integer families. |
| title | Polynomials Arising from Sorted Binomial Coefficients |
| topic | Combinatorics Complex Variables 12D10 (Primary), 11B65 (Secondary) |
| url | https://arxiv.org/abs/2511.03082 |