University of Georgia
Mathematics Department Colloquium
Jean-Louis Nicolas
Universit\'e Claude Bernard (Lyon, France)
March 17, 1999
Some topics in the theory of partitions
Abstract:
A set A of positive
integers will be called stratified if the sum of any t distinct
elements of A is smaller than the sum of any t+1 distinct elements
of A, for 0 <= t < A. The number of stratified sets of largest
element N is equal to p(n), the number of numerical partitions of n.
Let n=n_1+n_2+ ... +n_k be a partition \Pi of n, and a an
integer. The partition \Pi is said to represent a if a can be
written as a subsum n_{i_1}+n_{i_2}+ ... + n_{i_r} of the parts of
\Pi. Let E(\Pi) be the set of integers represented by \Pi, and
\widehat p (n) the number of distinct sets E(\Pi) when \Pi runs
into the p(n) partitions of n. One can prove
p(n)^{0.361} <= \widehat p (n) <= p(n)^{0.768}
for n large enough.
Some other problems will be discussed about the parity of p(n),
and also about the behaviour of the partition function p_A(n)
which counts the partitions of n with parts in A.