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 thatd(Pi n )b(Pi n )=<e+o(1)asn->~.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.