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, d...

We show that if P (x) is a polynomial with nondecreasing, nonnegative coefficients, then the coefficient sequence of P (x+ 1) is unimodal. Applications are given.

The paper solves the enumeration of set partitions in which nondecreasing sequence parts , is contained a degree-2 polynomial . This generalization number into arithmetic progressions. We also study problem dividing whose differences between consecutive are integers. In particular, we focus on sum triangular numbers.

Two finite real sequences (aI , . . . . ak) and (b, , . . . . bk) are cross-monotone if each is nondecreasing and ai + 1 -ci > bi + 1 bt for all i < k. A sequence ((x, , . . . , on) of nondecreasing reals is in class CM(k) if it has disjoint k-term subsequences that are cross-monotone. The paper shows that f(k), the smallest n such that every nondecreasing (o, , . . . , on) is in CM(k), is boun...