Counting nondecreasing integer sequences that lie below a barrier

Abstract

Given a barrier $0\le b_0\le b_1\le ...$, let $f(n)$ be the number of nondecreasing integer sequences $0\le a_0\le a_1\le ...\le a_n$ for which $a_j\le b_j$ for all $0\le j\le n$. Known formulæfor $f(n)$ include an $n\times n$ determinant whose entries are binomial coefficients (Kreweras, 1965) and, in the special case of $b_j=rj+s$, a short explicit formula (Proctor, 1988, p.320). A relatively easy bivariate recursion, decomposing all sequences according to $n$ and $a_n$, leads to a bivariate generating function, then a univariate generating function, then a linear recursion for $\{f(n)\}$. Moreover, the coefficients of the bivariate generating function have a probabilistic interpretation, leading to an analytic inequality which is an identity for certain values of its argument.