Counting Nondecreasing Integer Sequences that Lie Below a Barrier

Given a barrier 0 ≤ b0 ≤ b1 ≤ · · · , let f(n) be the number of nondecreasing integer sequences 0 ≤ a0 ≤ a1 ≤ · · · ≤ an for which aj ≤ bj for all 0 ≤ j ≤ n. Known formulæ for f(n) include an n × n determinant whose entries are binomial coefficients (Kreweras, 1965) and, in the special case of bj = rj + s, a short explicit formula (Proctor, 1988, p.320). A relatively easy bivariate recursion, decomposing all sequences according to n and an, 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.