Exact Enumeration of 1342-avoiding Permutations a Close Link with Labeled Trees and Planar Maps

Solving the rst nonmonotonic, longer-than-three instance of a classic enumeration problem, we obtain the generating function H(x) of all 1342-avoiding permutations of length n as well as an exact formula for their number S n (1342). While achieving this, we bijectively prove that the number of indecomposable 1342-avoiding permutations of length n equals that of labeled plane trees of a certain type on n vertices recently enumerated by Cori, Jacquard and Schaeeer, which is in turn known to be equal to the number of rooted bicubic maps enumerated by Tutte in 1963. Moreover, H(x) turns out to be algebraic, proving the rst nonmonotonic, longer-than-three instance of a conjecture of Zeilberger and Noonan. We also prove that n p S n (1342) converges to 8, so in particular, lim n!1 (S n (1342)=S n (1234)) = 0.