On the Number of Containments in P-free Families

A subfamily {F-1, F-2, ..., F-vertical bar P vertical bar} subset of F is a copy of the poset P if there exists a bijection i : P -> {F-1, F-2, ..., F-vertical bar P vertical bar}, such that p k, then this number is of the order Theta(Pi(k+1)(i=1) ((li-1)(li))), where l(0) = n and l(1) >= l(2) >= ... >= l(k+1) are such that n - l(1), l(1) - l(2), ..., l(k) - l(k+1), l(k+1) differ by at most one. On the other hand if h(P)