A bijection between nonnegative words and sparse abba-free partitions

  • Authors:

  • Jan Němeček;Martin Klazar

  • Affiliations:

  • Ve Stráni 87, 560 02 Ceská Trebová, Czech Republic;Department of Applied Mathematics and Institute for Theoretical Computer Science, Charles University, Malostranské námestí 25, 118 00 Praha, Czech Republic

  • Venue:
  • Discrete Mathematics
  • Year:
  • 2003

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Abstract

We construct a bijection proving that the following two sets have the same cardinality: (i) the set of words over {-1,0,1} of length m - 2 which have every initial sum nonnegative, and (ii) the set of partitions of {1,2,..., m} such that no two consecutive numbers lie in the same block and for no four numbers the middle two are in one block and the end two are in another block. The words were considered by Gouyou-Beauchamps and Viennot who enumerated by means of them certain animals. The identity connecting (i) and (ii) was observed by Klazar who proved it by generating functions.