ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

13. Combinatorics

1991 MS Classification: 05, 06 Engel, Konrad, Universität Rostock, Germany, and Canfield, E. Rodney, University of Georgia, Athens, USA An upper bound for the size of a largest antichain in the poset of partitions of an integer Let P i_{n} be the poset of partitions of an integer n, ordered by refinement. Let b(P i_{n}) be the largest size of a level and d(P i_{n}) be the largest size of an antichain of P i_{n}. We prove that

\frac{d(P i_{n})}{b(P i_{n})} ≤ e + o(1) as n → ∞.

The denominator is determined asymptotically. In addition, we show that the incidence matrices in the lower half of P i_{n} have full rank, and we prove a tight upper bound for the ratio from above if P i_{n} is replaced by any graded poset P.