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Some general problems on the number of parts in partitions. (English) Zbl 0823.11059

Let \(p_ A (m,n)\) (resp. \(q_ A (m,n)\)) denote the number of partitions (resp. unequal partitions) of \(n\) into exactly \(m\) parts chosen from a general sequence \(A\) of positive integers. Let \(\overline {p}_ A (m,n)= p_ A (m,1)+ p_ A (m,2)+ \ldots+ p_ A (m,n)\), \(\overline {q}_ A (m,n)= q_ A (m,1)+ q_ A (m,2)+ \ldots+ q_ A (m,n)\). For each of these functions P. Turán [Colloq. Int. Teor. Comb. (Roma, 1973), Tomo II, 181-200 (1976; Zbl 0359.10041)] proposed the problem of finding a general class of sequences \(A\) and a suitable \(f(n, A)= f(n)\) so that, for almost all such partitions, \(m\sim f(n)\) as \(n\to \infty\). P. Erdős and P. Turán [Acta Arith. 18, 53-62 (1971; Zbl 0217.322)] have found that for \(\overline {q}_ A (m,n)\) alone a density requirement on \(A\) implies the existence of an \(f(n)\) and that for \(q_ A (m,n)\) somewhat stronger conditions are sufficient.
In the paper under review the author determines the asymptotic behaviour of \(p_ A (m,n)\) and \(\overline {p}_ A (m,n)\) for those \(m\) near the values of \(m\) which maximize each function. The proof follows the approach of C. B. Haselgrove and H. N. V. Temperley [Proc. Camb. Philos. Soc. 50, 225-241 (1954; Zbl 0055.274)] but their conditions are replaced by more useful ones. The author’s density requirements are weaker than those of Erdős and Turán.
A problematic condition of Haselgrove and Temperley is replaced by the following arithmetical condition: We say \(A\) has property \(Q_ k\) if it has more than \(k\) elements and if we remove an arbitrary subset of \(k\) elements from \(A\) the remaining elements are not in the same residue class modulo \(m\) for any \(m>1\). Besides the density requirements in his main theorem the author supposes \(Q_ k\) for \(A\) and \(Q_ \ell\) for certain “quotient” sequence derived from \(A\).

MSC:

11P82 Analytic theory of partitions
05A16 Asymptotic enumeration
05A17 Combinatorial aspects of partitions of integers
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