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Polynomials Arising from Sorted Binomial Coefficients

Owen John Levens

TL;DR

This work introduces the Pascalian numbers generated by sorting binomial coefficients and studies the associated Pascalian polynomials P_n. It develops recursions, generating functions, and root bounds, establishing a characterization of root locations including a limit curve ∂Γ to which roots converge densely. A geometric-analytic framework shows nontrivial roots lie in a tight annulus and accumulate on ∂Γ, with a detailed treatment of real and purely imaginary roots and a demonstration that P_n shares no nontrivial roots with P_{n-2}. The paper also investigates irreducibility and Galois groups, proving a factorization pattern for odd n and embedding the Galois group inside the hyperoctahedral group, while outlining conjectures for even n and connections to truncated binomial polynomials. Overall, it links combinatorial interpretations to analytic and algebraic properties, producing a rich structure around polynomials whose coefficients are Pascalian numbers and offering groundwork for broader explorations in root asymptotics and algebraic symmetries.

Abstract

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.

Polynomials Arising from Sorted Binomial Coefficients

TL;DR

This work introduces the Pascalian numbers generated by sorting binomial coefficients and studies the associated Pascalian polynomials P_n. It develops recursions, generating functions, and root bounds, establishing a characterization of root locations including a limit curve ∂Γ to which roots converge densely. A geometric-analytic framework shows nontrivial roots lie in a tight annulus and accumulate on ∂Γ, with a detailed treatment of real and purely imaginary roots and a demonstration that P_n shares no nontrivial roots with P_{n-2}. The paper also investigates irreducibility and Galois groups, proving a factorization pattern for odd n and embedding the Galois group inside the hyperoctahedral group, while outlining conjectures for even n and connections to truncated binomial polynomials. Overall, it links combinatorial interpretations to analytic and algebraic properties, producing a rich structure around polynomials whose coefficients are Pascalian numbers and offering groundwork for broader explorations in root asymptotics and algebraic symmetries.

Abstract

The triangle of sorted binomial coefficients for has appeared several times in recent combinatorial works but has evaded dedicated study. Here we refer to as the Pascalian numbers and unify the various perspectives of . We then view each row of the triangle as the coefficients of the th Pascalian polynomial, which we denote . We derive recursions, formulae, and bounds on 's roots in , and characterize the asymptotics of these roots. We show the roots of converge uniformly to a curve and asymptotically fill the curve densely. We conclude with a discussion of the reducibility and Galois groups of . Our work has natural connections to the truncated binomial polynomials, asymptotic analysis, and well-known integer families.

Paper Structure

This paper contains 19 sections, 22 theorems, 66 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Combinatorial Foundations
  3. Standard Domino Tableaux
  4. Rightward diagonal lattice walks
  5. $B_n$ to $D_n$
  6. Pascalian Polynomials
  7. Recursions on the Pascalian Polynomials
  8. Generating Functionology
  9. Roots of Pascalian Polynomials
  10. Bounds on the roots of $P_n(z)$ within the complex plane
  11. Real and imaginary roots
  12. Convergence of Roots to $\Gamma_n$
  13. Properties of Key Curves
  14. Uniform convergence to $\Gamma$
  15. Irreducibility and Galois Groups
  16. ...and 4 more sections

Key Result

Theorem 2.2

There are $\binom{n}{\lfloor\frac{n-k}{2}\rfloor}$ SDT of shape $(n+k,n-k)$, so $\genfrac{\langle}{\rangle}{0pt}{}{n}{k}=\binom{n}{\lfloor\frac{n-k}{2}\rfloor}$. Particularly, the Pascalian numbers satisfy the recursion with $\genfrac{\langle}{\rangle}{0pt}{}{n}{n}=\genfrac{\langle}{\rangle}{0pt}{}{n}{n-1}=1$ for each $n$.

Figures (7)

  • Figure 1: The $6$ standard domino tableaux with $2$ dominos. From left to right, they have shape $(1,1,1,1), (2,1,1), (2,2), (2,2),( 3,1),$ and $(4)$.
  • Figure 2: $-3,1,2\in S_3^B$ represented as a pair of SDT of shape $(4,2)$
  • Figure 3: Pascalian numbers represented in a triangular array for $0\leq n\leq5$.
  • Figure 4: The elements of $D_2$ with heights $0,0,1,2$, respectively
  • Figure 5: The roots of $P_n(z)$ plotted for $n\leq50$ with the annulus of Theorem \ref{['thm: PP roots annuli']} in gray. The color of $P_n(z)$'s roots flows from blue to red as $n$ grows.
  • ...and 2 more figures

Theorems & Definitions (42)

  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • proof
  • Proposition 2.4
  • Theorem 2.5
  • Theorem 2.6
  • proof
  • Corollary 2.7
  • Proposition 3.1
  • ...and 32 more