DC MetaData for: An upper bound for the size of the largest antichain in the poset of partitions of an integer

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