F Uredi{hajnal Conjecture Implies Stanley{wilf Conjecture Two Nite Sequences U = a 1 a 2 : : : a K 2 N

We show that Stanley{Wilf enumerative conjecture on permutations follows easily from F uredi{Hajnal extremal conjecture on 0-1 matrices. We apply the method of deriving an (almost) exponential bound on number of objects from a (almost) linear bound on their sizes, which was discovered by Alon and Friedgut. They proved by it a weaker form of Stanley{Wilf conjecture. Using bipartite graphs, we give a simpler proof of their result. Stanley{Wilf conjecture asserts that the number of n-permutations not containing a given permutation is exponential in n. Alon and Friedgut 1] proved that it is true provided we have a linear upper bound on lengths of certain words over an ordered alphabet. They also proved a weaker version of it with an almost exponential upper bound. In the present note we want to inform the reader about this interesting development by reproving the latter result in a simpler way. We use bipartite graphs instead of words. We point out that in 1992 F uredi and Hajnal almost made an extremal conjecture on 0-1 matrices that now can easily be seen to imply Stanley{Wilf conjecture. We prove that both extremal conjectures are logically equivalent. of positive integers are isomorphic if k = l and, for every i and j, a i < a j is equivalent to b i < b j. We say that v contains u if v has a subsequence that is isomorphic to u; we write v < u. Replacing < by =, we deene in the same way the corresponding isomorphism and containment relation =. For example, 31225345 = 2121 but 31225345 6 6 < 2121. The length of a sequence u is juj and n] is the set f1; 2; : : : ; ng.