An upper bound for the size of the largest antichain in the poset of partitions of an integer Original Research Article

Let Pin be the poset of partitions of an integer n, ordered by refinement. Let b(Pin) be the largest size of a level and d(Pin) be the largest size of an antichain of Pin. We prove thatd(Pin)b(Pin)⩽e+o(1) as n→∞.The denominator is determined asymptotically. In addition, we show that the incidence matrices in the lower half of Pin have full rank, and we prove a tight upper bound for the ratio from above if Pin is replaced by any graded poset P.