An Upper Bound for the Size of the 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 re-nement. 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 d(P i n) b(P i n) e + o(1) as n ! 1: 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 .