On the number of distinct block sizes in partitions of a set

AbstractThe average number of distinct block sizes in a partition of a set of n elements is asymptotic to e log n as n → ∞. In addition, almost all partitions have approximately e log n distinct block sizes. This is in striking contrast to the fact that the average total number of blocks in a partition is ∼n(log n)−1 as n → ∞

Асимптотични резултати за случайни целочислени разлагания на големи числа

ИМИ-БАН, 30.05.2012 г., присъждане на образователна и научна степен "доктор" на Емил Петков Каменов по научна специалност 01.01.10. теория на вероятностите и математическа статистика. [Kamenov Emil Petkov; Каменов Емил Петков

Covering Irrep(Sn)Irrep(S_n) With Tensor Products and Powers

We study when a tensor product of irreducible representations of the symmetric group $S_n$ contains all irreducibles as subrepresentations - we say such a tensor product covers $Irrep(S_n)$. Our results show that this behavior is typical. We first give a general criterion for such a tensor product to have this property. Using this criterion we show that the tensor product of a constant number of random irreducibles covers $Irrep(S_n)$ asymptotically almost surely. We also consider, for a fixed irreducible representation, the degree of tensor power needed to cover $Irrep(S_n)$. We show that the simple lower bound based on dimension is tight up to a universal constant factor for every irreducible representation, as was recently conjectured by Liebeck, Shalev, and Tiep

The Automorphism Group of a Finite p-Group is Almost Always a p-Group

Many common finite p-groups admit automorphisms of order coprime to p, and when p is odd, it is reasonably difficult to find finite p-groups whose automorphism group is a p-group. Yet the goal of this paper is to prove that the automorphism group of a finite p-group is almost always a p-group. The asymptotics in our theorem involve fixing any two of the following parameters and letting the third go to infinity: the lower p-length, the number of generators, and p. The proof of this theorem depends on a variety of topics: counting subgroups of a p-group; analyzing the lower p-series of a free group via its connection with the free Lie algebra; counting submodules of a module via Hall polynomials; and using numerical estimates on Gaussian coefficients.Comment: 38 pages, to appear in the Journal of Algebra; improved references, changes in terminolog