Counting nondecreasing integer sequences that lie below a barrier

TL;DR

This paper studies the enumeration of nondecreasing integer sequences constrained by a barrier, deriving formulas, recursions, and generating functions, and explores their probabilistic interpretations and related inequalities.

Contribution

It introduces a new recursive approach to count constrained sequences, deriving generating functions and identities, and connects combinatorial enumeration with probabilistic analysis.

Findings

Derived a bivariate recursion for counting sequences.

Established generating functions and linear recursions for the sequence counts.

Linked combinatorial counts to probabilistic interpretations and inequalities.

Abstract

Given a barrier $0 \leq b_0 \leq b_1 \leq ...$, let $f(n)$ be the number of nondecreasing integer sequences $0 \leq a_0 \leq a_1 \leq ... \leq a_n$ for which $a_j \leq b_j$ for all $0 \leq j \leq n$. Known formul\ae for $f(n)$ 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.