Compositions with distinct parts
Let \(q(n,k)\) denote the number of partitions of \(n\) into \(k\) distinct parts. The function \(q(n,k)\) has been studied in detail by \textit{G. Szekeres} [Q. J. Math., Oxf. II. Ser. 2, 85-108 (1951; Zbl 0042.041); ibid. 4, 96-111 (1953; Zbl 0050.041)]. In the paper under review the authors prove some analogous results for \(C(n,k)= k! q(n,k)\), the number of compositions of \(n\) with \(k\) distinct parts. Let \(n= m(m+1)/2+ j\), \(0\leq j\leq m\), \(n\geq n_ 0\). Theorem 1 asserts the unimodality of the sequence \(\{C (n, k)\}\): there is an integer \(k_ 1\) such that \[ C(n,1)< C(n,2)< \dots< C(n, k_ 1)\geq C(n, k_ 1+ 1)> \dots > C(n,m). \] Theorem 2 yields an asymptotic estimate for \(C(n)= \sum_{k=1}^ m C(n,k)\). The proof of Theorem 2 shows that the \(C(n,k)/ C(n)\) tend to a normal distribution with mean \(k_ 1\) and standard deviation \(\sim (2m/ \log m)^{1/2}\).
scientific article; zbMATH DE number 729147