Explicit Asymptotics for Signed Binomial Sums and Applications to Carnevale-Voll Conjecture

TL;DR

This paper investigates the asymptotic behavior of signed binomial sums related to the Carnevale-Voll conjecture, providing explicit conditions and ranges where the conjecture holds, especially for large parameters and specific ratios.

Contribution

The paper offers explicit asymptotic analysis and verifies the Carnevale-Voll conjecture for certain parameter ranges and ratios, advancing understanding of the conjecture's validity.

Findings

Conjecture holds for r ≥ 5.8362.

Almost sure truth for fixed r.

Verified for small differences in λ1 and λ2.

Abstract

Carnevale and Voll conjectured that j (--1) j $\lambda$ 1 j $\lambda$ 2 j = 0 when $\lambda$ 1 and $\lambda$ 2 are two distinct integers. We check the conjecture when either $\lambda$ 2 or $\lambda$ 1 -- $\lambda$ 2 is small. We investigate the asymptotic behaviour of their sum when the ratio r := $\lambda$ 1 /$\lambda$ 2 is fixed and $\lambda$ 2 goes to infinity. We find an explicit range r $\ge$ 5.8362 on which the conjecture is true. We show that the conjecture is almost surely true for any fixed r. For r close to 1, we give several explicit intervals on which the conjecture is also true.