Explicit formulas for a sequence of binary tree polynomials

The W-ary tree corresponding to a word W=w0w1⋯wk is the complete planar tree whose nodes at level i have out-degree wi. Investigating some combinatorial aspects of the Shuffle conjecture (Haglund et al. (2005)  [1], ), Hicks recently discovered (Hicks (2012)  [3], ) a remarkable family of univariate polynomials associated to W-ary trees. She proved that a single functional equation satisfied by her binary tree polynomials reduces all the compositional cases of the Haglund–Morse–Zabrocki conjectures (Haglund et al. (2011)  [2]) to the partition case. This development would entail a substantial advance on the decade-old Shuffle conjecture. We derive here explicit formulas for the Hicks polynomials in the binary tree case, prove her conjecture in this special case and derive some further properties of these binary polynomials that are also conjectured in the thesis work of Hicks.