2006/01/13 by Alexander Postnikov, Postnikov, Alexander, Bruce E. Sagan +1 · 1 citation
Mathematics · #11B75 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #Primary 05A10 #Secondary 11A55
paper · pdf · doi:10.48550/arxiv.math/0601339
openalex publication_date 2006/01/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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).