What power of two divides a weighted Catalan number?

13 years 4 months ago

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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 Cb n = P 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 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(Cb n) = ξ(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 ξ(Cn). © 2006 Elsevier Inc. All rights reserved.

Alexander Postnikov, Bruce E. Sagan

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