计数位于障碍之下的非减整数序列
Counting nondecreasing integer sequences that lie below a barrier
摘要：
给定一个障碍$0 \leq b_0 \leq b_1 \leq ...$，令$f(n)$为非减整数序列$0 \leq a_0 \leq a_1 \leq ... \leq a_n$的个数，其中$a_j \leq b_j$对所有$0 \leq j \leq n$。已知的$f(n)$公式包括一个$n \times n$行列式，其元素是二项式系数（Kreweras，1965），在$b_j = rj+s$的特殊情况下，还有一个简短的显式公式（Proctor，1988，p.320）。一个相对简单的二元递归，根据$n$和$a_n$分解所有序列，导致一个二元生成函数，然后是一个单变量生成函数，然后是$\{f(n) \}$的线性递归。此外，二元生成函数的系数具有概率解释，导致分析不等式是其自变量的某些值的恒等式。
英文摘要：
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 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.
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