E. Rodney Canfield, Konrad Engel
An upper bound for the size of the largest antichain in the poset of partitions of an integer

The paper is published: Discrete Applied Mathematics

MSC:
06A07 Combinatorics of partially ordered sets
11P82 Analytic theory of partitions
Abstract: Let $Pi_n$ be the poset of partitions of an integer $n$, ordered by
refinement. Let $b(Pi_n)$ be the largest size of a level and $d(Pi_n)$ be
the largest size of an antichain of $Pi_n$. We prove that
$\frac{d(Pi_n)}{b(Pi_n)} \le e + o(1) \mbox{ as } n \rightarrow \infty.$
The denominator is determined asymptotically. In addition, we show that the
incidence matrices in the lower half of $Pi_n$ have full rank, and we prove
a tight upper bound for the ratio from above if $Pi_n$ is replaced by any
graded poset $P$.

Keywords: integer partitions, antichain, Sperner theory, Hardy-Ramanujan formula, incidence matrix