A regular determinant of binomial coefficients

Abstract:

Let $n$ be a positive integer and suppose that each of $\{ {a_p}\} _1^n$ and $\{ {c_p}\} _1^n$ is an increasing sequence of nonnegative integers. Let $M$ be the $n \times n$ matrix such that ${M_{ij}} = C({a_i},{c_j})$, where $C(m,n)$ is the number of combinations of $m$ objects taken $n$ at a time. We give an explicit formula for the determinant of $M$ as a sum of nonnegative quantities. Further, if ${a_i} \geqq {c_i},i = 1,2, \cdots ,n$, we show that the determinant of $M$ is positive.