What power of two divides a weighted Catalan number?

Given a sequence of integers b=(b0,b1,b2,…)b=(b0,b1,b2,…) one gives a Dyck path P of length 2n the weightwt(P)=bh1bh2⋯bhn,wt(P)=bh1bh2⋯bhn, where hihi is the height of the ith ascent of P. The corresponding weighted Catalan number isCnb=∑Pwt(P), where the sum is over all Dyck paths of length 2n  . So, in particular, the ordinary Catalan numbers CnCn correspond to bi=1bi=1 for all i⩾0i⩾0. Let ξ(n)ξ(n) stand for the base two exponent of n, i.e., the largest power of 2 dividing n. We give a condition on b   which implies that ξ(Cnb)=ξ(Cn). In the special case bi=(2i+1)2bi=(2i+1)2, this settles a conjecture of Postnikov about the number of plane Morse links. Our proof generalizes the recent combinatorial proof of Deutsch and Sagan of the classical formula for ξ(Cn)ξ(Cn).