On the number of plane partitions and non isomorphic subgroup towers of abelian groups

Details On the number of plane partitions and non isomorphic subgroup towers of abelian groups On the number of plane partitions and non isomorphic subgroup towers of abelian groups

2006-07-27

arxiv.org/abs/math/0607698v1

Mathematics

Number Theory

20 pages, 2 figures

Scientific paper

We study the number of $k \times r$ plane partitions, weighted on the sum of the first row. Using Erhart reciprocity, we prove an identity for the generating function. For the special case $k=1$ this result follows from the classical theory of partitions, and for $k=2$ it was proved in Andersson-Bhowmik with another method. We give an explicit formula in terms of Young tableaux, and study the corresponding zeta-function. We give an application on the average orders of towers of abelian groups. In particular we prove that the number of isomorphism classes of ``subgroups of subgroups of ... ($k-1$ times) ... of abelian groups'' of order at most $N$ is asymptotic to $c_k N (\log N)^{k-1}$. This generalises results from Erd{\H o}s-Szekeres and Andersson-Bhowmik where the corresponding result was proved for $k=1$ and $k=2$.

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