On the distribution of the summands of unequal partitions in residue classes (Q2495688)

The authors call an integer partition ``unequal'' if its parts are pairwise distinct. Such partitions are more frequently called ``distinct partitions'' or ``strict partitions.'' The question which the authors address in the present paper is how many of the parts of a strict partition belong to a fixed congruence class modulo \(d\), where \(d\) is some fixed given positive integer. Let \(q(n)\) denote the number of strict partitions of \(n\), and let \(w(n)\) be a non-decreasing function with \(\lim_{n\to\infty}w(n)=\infty\). A weak form of the main result of the paper says that, if \(1\leq r\leq d\leq \sqrt{n}\), then for all but \(q(n)/w(n)\) strict partitions of \(n\) the number of parts congruent to \(r\) modulo \(d\) is equal to \[ (1+o(1))\frac {2\sqrt{3n}} {\pi d}\log\left(1+\exp\left(\frac {\pi(d-r)} {2\sqrt{3n}}\right)\right) \] plus an error term which involves the square root of \(w(n)\), which we omit here for the sake of simplicity. This result shows equidistribution of the parts among congruence classes in an approximate sense. It must, however, be compared to the result of \textit{P. Erdős} and \textit{J. Lehner} [Duke Math. J. 8, 335--345 (1941; Zbl 0025.10703)] that says that almost all strict partitions of \(n\) contain \((1+o(1))\frac {2\sqrt{3}\log 2} \pi\sqrt{n}\) parts. The authors prove their result by a probabilistic approach in which they interpret the number of parts of a partition of \(n\) in a given congruence class modulo \(d\) as a random variable on the strict partitions of \(n\). They proceed by determining asymptotically the mean value and the variance of this random variable. The proof is completed by an application of Chebyshev's inequality. This is very much in spirit of the previous paper [Acta Math. Hung. 109, 215--237 (2005; Zbl 1119.11061)] by the same authors, where the same kind of question was answered for unrestricted partitions.