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Generating Trees and the Catalan and SchrÃ¶der Numbers
 DEPARTMENT OF MATHEMATICS, STOCKHOLMS UNIVERSITET, S106 91
, 1995
"... A permutation 2 Sn avoids the subpattern iff has no subsequence having all the same pairwise comparisons as , and we write 2 Sn ( ). We present a new bijective proof of the wellknown result that jS n (123)j = jS n (132)j = c n , the nth Catalan number. A generalization to forbidden pattern ..."
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Cited by 103 (3 self)
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A permutation 2 Sn avoids the subpattern iff has no subsequence having all the same pairwise comparisons as , and we write 2 Sn ( ). We present a new bijective proof of the wellknown result that jS n (123)j = jS n (132)j = c n , the nth Catalan number. A generalization to forbidden patterns of length 4 gives an asymptotic formula for the vexillary permutations. We settle a conjecture of Shapiro and Getu that jS n (3142; 2413)j = s n\Gamma1 , the SchrÃ¶der number, and characterize the dequesortable permutations of Knuth, also counted by s n\Gamma1 .
Exact Enumeration Of 1342Avoiding Permutations A Close Link With Labeled Trees And Planar Maps
, 1997
"... Solving the first nonmonotonic, longerthanthree instance of a classic enumeration problem, we obtain the generating function H(x) of all 1342avoiding permutations of length n as well as an exact formula for their number Sn (1342). While achieving this, we bijectively prove that the number of inde ..."
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Cited by 84 (7 self)
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Solving the first nonmonotonic, longerthanthree instance of a classic enumeration problem, we obtain the generating function H(x) of all 1342avoiding permutations of length n as well as an exact formula for their number Sn (1342). While achieving this, we bijectively prove that the number of indecomposable 1342avoiding permutations of length n equals that of labeled plane trees of a certain type on n vertices recently enumerated by Cori, Jacquard and Schaeffer, 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 first nonmonotonic, longerthanthree instance of a conjecture of Zeilberger and Noonan. We also prove that n p Sn (1342) converges to 8, so in particular, limn!1 (Sn (1342)=Sn (1234)) = 0.