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 = sum_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 xi(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   xi(C_n^b) = xi(C_n).   In the special case b_i=(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 xi(C_n). 
Alexander Postnikov (MIT) and Bruce Sagan (Michigan State U)