The number of permutations of length 2n-1 avoiding 312 in the classical sense which can be realized as labels on an increasing binary tree read in the order they appear in a breadth-first search. (Note that breadth-first search reading word is equivalent to reading the tree left to right by levels, starting with the root.)
In some cases, more than one tree results in the same breadth-first search reading word, but here we count the permutations, not the trees.
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