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Partitions without small parts. (English) Zbl 0707.11072

Number theory. Vol. I. Elementary and analytic, Proc. Conf., Budapest/Hung. 1987, Colloq. Math. Soc. János Bolyai 51, 9-33 (1990).
Let \(r(n,m)\) denote the number of partitions of \(n\) into parts \(\ge m\). The present authors show that the asymptotic formula \[ r(n,m)=p(n)\left(\frac{C}{2\sqrt{n}}\right)^{m-1}(m-1)!\{1+O(m^2/\sqrt{n})\} \tag{*} \] holds uniformly for \(1\le m\le n^{1/4}\); here \(C=\pi \sqrt{2/3}\) and \(p(n)=r(n,1)\) is the number of (unrestricted) partitions of \(n\). Their proof is based on the representation \(r(n,m)=D^{(m-1)}(p(n))\) where \(D^{(m-1)}\) is a difference operator reflecting the relation \[ \sum_{n\geq 1}r(n,m)x^ n=\prod^{m-1}_{k-1}(1-x^ k)\sum_{n\geq 1}p(n)x^ n. \] Here \(p(n)\) may be replaced by the Hardy-Ramanujan formula \[ p(n)=\frac{C^3}{2\pi \sqrt{2}}F'(C^2(n-1/24)) + \text{error} \] where \(F(x)=\exp (\sqrt{x})/\sqrt{x}\), hence a simple extension of the mean value theorem gives for \(m\le \sqrt{n}\) \[ r(n,m)=(m-1)!\frac{C^{2m+1}}{2\pi \sqrt{2}}F^{(m)}(t) + \text{error} \] with \(C^2(n- m^2/2)\le t\le C^2n\). Evaluation of \(F^{(m)}(t)\) in terms of Bessel polynomials and some straightforward estimates yield (*) uniformly in \(m\le n^{1/4}\).
As an application of (*) an asymptotic formula of P. Erdős and M. Szalay [Studies in Pure Math., Memory of P. Turán, 187–212 (1983; Zbl 0523.10029)] for the number of nonpractical partitions of \(n\) is considerably sharpened. (A partition \(n=n_1+\cdots+n_k\) of \(n\) is said to be practical if each \(b\in \{1,\ldots,n\}\) is representable as a subsum \(b=\sum^k_{i=1}a_i n_i\) with \(a_i\in \{0,1\}\).)
[For the entire collection see Zbl 0694.00005.]

MSC:

11P82 Analytic theory of partitions