On representations by determinants of P(n) and Pm(n)

AbstractP(n) and Pm(n) denote the number of (unordered) partitions of n and the number of partitions of n into m parts, respectively. For P(n), there exists a recursion formula which is shown in Eq. (3) and a complicated formula indicated in J. L. Doob et al. (“Hans Rademacher: Topic Analytic Number Theory,” Springer-Verlag, Berlin/New York, 1973, p. 275, which is accompanied with the error term. For Pm(n), there is no general rule known covering all m (Doob et al., p. 222). In this article, P(n) and Pm(n) are represented by determinants. Note that the determinant of the former agrees with the above recursion formula and the finite product of binomials analogous to Euler identity, which is indicated in (5), leads to the representation of the latter. The computation of determinant is a little troublesome, but it is very important that the representations themselves of the number of partitions are simple, if we make use of the determinant