Log Concavity Involving the Number of Paths from the Origin to Points Along the Line $(a,b,c,d)+t(1,-1,1,-1)$

Abstract

This paper examines the numbers of lattice paths of length $n$ from the origin to integer points along the line $(a,b,c,d)+t(1,-1,1,-1)$. These numbers form a sequence which this paper shows is log concave, and for sufficiently large values of $n$, the location of the maximum of this sequence is shown. This paper also shows unimodality of such sequences for other lines provided that $n$ is sufficiently large.