%I
%S 1,2,6,22,84,330,1308,5210,20796,83100,332232,1328598,5313732,
%T 21253620,85011864,340042246,1360158564
%N a(n) is the number of permutations avoiding 231 and 312 realizable on increasing strict binary trees with 2n1 nodes.
%C The number of permutations avoiding 231 and 312 in the classical sense which can be realized as labels on an increasing strict binary tree with 2n1 nodes. A strict binary tree is a tree graph where each node has 0 or 2 children. The permutation is found by reading the labels in the order they appear in a breadthfirst search. (Note that breadthfirst search reading word is equivalent to reading the tree left to right by levels, starting with the root.)
%C In some cases, more than one tree results in the same breadthfirst search reading word, but here we count the permutations, not the trees.
%e For example, when n=3, the permutations 12543, 12435, 13245, 13254, 12345,and 12354. all avoid 231 and 312 in the classical sense and occur as breadthfirst search reading words on an increasing strict binary tree with 5 nodes.
%e . 1 1 1 1 1 1
%e . / \ / \ / \ / \ / \ / \
%e . 2 5 2 4 3 2 3 2 2 3 2 3
%e . / \ / \ / \ / \ / \ / \
%e . 4 3 3 5 4 5 5 4 4 5 5 4
%Y A bisection of A002083.
%K nonn,more
%O 1,2
%A _Manda Riehl_, Aug 05 2014
%E More terms from _N. J. A. Sloane_, Jul 07 2015
