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%I #12 May 08 2019 15:54:21
%S 1,1,2,5,14,44,148,528,1972,7647,30605,125801,529131,2270481,9914870,
%T 43973755,197744417,900327160,4145285618,19280282194,90507546094,
%U 428476211848,2044274855774,9823314566417,47516954475991,231260870664189
%N Number of valid hook configurations of 312-avoiding permutations of [n].
%C The class of a Motzkin path is the set of indices i such that the i-th non-down step is an east step. For n > 0, a(n) is the number of pairs (P,Q) of Motzkin paths such that P and Q have the same class and P lies below or is equal to Q.
%C Conjecture: This sequence is the binomial transform of A151347.
%C The Defant article gives a functional equation that defines a generating function Q(x,y,z) such that Q(x,0,0) is the ordinary generating function of this sequence.
%H Colin Defant, <a href="http://arxiv.org/abs/1904.10451">Motzkin intervals and valid hook configurations</a>, arXiv preprint arXiv:1904.10451 [math.CO], 2019.
%H Wenjie Fang, <a href="http://arxiv.org/abs/1801.04809">A partial order on Motzkin paths</a>, arXiv preprint arXiv:1801.04809 [math.CO], 2018.
%Y Appears to be the binomial transform of A151347.
%K nonn
%O 0,3
%A _Colin Defant_, Apr 28 2019
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