2025/11/05 by Levens, Owen John
Mathematics · Computer Science · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Polynomial and algebraic computation
paper · doi:10.48550/arxiv.2511.03082
The triangle of sorted binomial coefficients ⟨ n \atop k ⟩ = \binomn\lfloor (n - k)/(2) \rfloor for 0 ≤ k ≤ n has appeared several times in recent combinatorial works but has evaded dedicated study. Here we refer to ⟨ n \atop k ⟩ as the Pascalian numbers and unify the various perspectives of ⟨ n \atop k ⟩. We then view each row of the ⟨ n \atop k ⟩ triangle as the coefficients of the nth Pascalian polynomial, which we denote Pn(z). We derive recursions, formulae, and bounds on Pn(z)'s roots in ℂ, and characterize the asymptotics of these roots. We show the roots of Pn(z) converge uniformly to a curve ∂ Γ⊂ ℂ and asymptotically fill the curve densely. We conclude with a discussion of the reducibility and Galois groups of Pn(z). Our work has natural connections to the truncated binomial polynomials, asymptotic analysis, and well-known integer families.