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

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.