Note: What power of two divides a weighted Catalan number?

  • Authors:

  • Alexander Postnikov;Bruce E. Sagan

  • Affiliations:

  • Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA;Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA

  • Venue:
  • Journal of Combinatorial Theory Series A
  • Year:
  • 2007

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Abstract

Given a sequence of integers b=(b"0,b"1,b"2,...) one gives a Dyck path P of length 2n the weightwt(P)=b"h"""1b"h"""2...b"h"""n, where h"i is the height of the ith ascent of P. The corresponding weighted Catalan number isC"n^b=@?Pwt(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 @x(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 @x(C"n^b)=@x(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 @x(C"n).