What power of two divides a weighted Catalan number?

Given a sequence of integers b = ( b 0 , b 1 , b 2 , … ) one gives a Dyck path P of length 2n the weight wt ( P ) = b h 1 b h 2 ⋯ b h n , where h i is the height of the ith ascent of P. The corresponding weighted Catalan number is C n b = ∑ P wt ( P ) , where the sum is over all Dyck paths of length 2n. So, in particular, the ordinary Catalan numbers C n correspond to b i = 1 for all i ⩾ 0 . Let 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 ( C n b ) = ξ ( C n ) . In the special case b i = ( 2 i + 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 ( C n ) .