What power of two divides a weighted Catalan number?
Alexander Postnikov, Bruce Sagan
Abstract
Given a sequence of integers b = (b0,b1,b2,...) one gives a Dyck path P of length 2n the weight wt(P) = bh1 bh2 ... bhn, where hi is the height of the ith ascent of P. The corresponding weighted Catalan number is Cnb = sumP wt(P), where the sum is over all Dyck paths of length 2n. So, in particular, the ordinary Catalan numbers Cn correspond to bi = 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(Cnb) = xi(Cn). In the special case bi=(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(Cn).
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