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Valid hook configurations are combinatorial objects used to understand West'sstack sorting map as well as cumulants in noncommutative probability theory. Weshow a bijection between reduced valid hook configurations on 312-avoidingpermutations with the maximal allowed number of points and 3D-Dyck words,proving a conjecture of Sankar's. We extend to a bijection between all312-avoiding reduced valid hook configurations and 3D-Dyck words with specifiedmodifications. We show how these can be counted in terms of the number of3D-Dyck words of length 3k in which exactly i Y's do not have an X immediatelybefore them, the (k,i)-Duck words, and use this relationship to prove severalproperties about sums of 312-avoiding reduced valid hook configurations,including two more of Sankar's conjectures. We also show that the number of(k,1)-Duck words is given by a variant of the tennis ball numbers.

312-Avoiding Reduced Valid Hook Configurations and Duck Words