We consider partitions of the positive integernwhose parts satisfy the following condition. For a given sequence of non-negative numbers {bk}k≥1, a part of sizekappears in exactlybkpossible types. Assuming that a weighted partition is selected uniformly at random from the set of all such partitions, we study the asymptotic behaviour of the largest partXn. LetD(s)=∑k=1∞bkk−s,s=σ+iy, be the Dirichlet generating series of the weightsbk. Under certain fairly general assumptions, Meinardus (1954) obtained the asymptotic of the total number of such partitions asn→∞. Using the Meinardus scheme of conditions, we prove thatXn, appropriately normalized, converges weakly to a random variable having Gumbel distribution (i.e., its distribution function equalse−e−t, −∞<t<∞). This limit theorem extends some known results on particular types of partitions and on the Bose–Einstein model of ideal gas.
New Weighted Partition Theorems with the Emphasis on the Smallest Part of Partitions
