On the convex partitions

Abstract

In the present paper, first, we translate the generating function for convex partitions of a strictly increasing positive integer sequence into the accumulated weight generating function for lecture hall sequences of a positive integer sequence which corresponds to the strictly increasing positive integer sequence. And then, we compute the accumulated weight generating function for the lecture hall sequences of special sequences: trivial sequences and (s, t)-alternating sequences, by the MacMahon``s operator and semi-combinatorial reasoning. Furthermore, by decomposing a lecture hall sequence into standard lecture hall sequences and reduced lecture hall sequences, we conjecture what sequences derive that its accumulated weight generating function equals to the generating function for the integer partitions with all parts in some finite subset of set of all positive integer.