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What power of two divides a weighted Catalan number?

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Abstract

Given a sequence of integers b=(b0,b1,b2,) one gives a Dyck path P of length 2n the weightwt(P)=bh1bh2bhn, where hi is the height of the ith ascent of P. The corresponding weighted Catalan number isCnb=Pwt(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 i0. 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(Cnb)=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).

Keywords

Difference operator
Divisibility
Group actions
Morse links
Orbits
Power of two
Shift operator
Weighted Catalan numbers

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