Carnevale and Voll conjectured that
![$\sum_j (-1)^j{\lambda_1\choose
j}{\lambda_2\choose j}\neq0$](abs/img1.gif)
when
![$\lambda_1$](abs/img2.gif)
and
![$\lambda_2$](abs/img3.gif)
are two
distinct integers. We check the conjecture when either
![$\lambda_2$](abs/img3.gif)
or
![$\lambda_1-\lambda_2$](abs/img4.gif)
is small. We investigate the asymptotic
behaviour of the sum when the ratio
![$r:=\lambda_1/\lambda_2$](abs/img5.gif)
is fixed
and
![$\lambda_2$](abs/img3.gif)
goes to infinity. We find an explicit range
![$r\ge 5.8362$](abs/img6.gif)
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.
Received July 3 2019; revised versions received February 5 2020; February 9 2020; May 27 2020.
Published in Journal of Integer Sequences,
June 10 2020.