The number of intervals in the m-Tamari lattices

Mireille Bousquet-Mélou (1) , Eric Fusy (2) , Louis-François Préville Ratelle (3)

1 LaBRI - Laboratoire Bordelais de Recherche en Informatique
2 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]
3 LaCIM - Laboratoire de combinatoire et d'informatique mathématique [Montréal]

An m-ballot path of size n is a path on the square grid consisting of north and east steps, starting at (0,0), ending at (mn,n), and never going below the line {x=my}. The set of these paths can be equipped with a lattice structure, called the m-Tamari lattice, which generalizes the usual Tamari lattice obtained when m=1. We prove that the number of intervals in this lattice is $$ \frac {m+1}{n(mn+1)} {(m+1)^2 n+m\choose n-1}. $$ This formula was recently conjectured by Bergeron in connection with the study of coinvariant spaces. The case m=1 was proved a few years ago by Chapoton. Our proof is based on a recursive description of intervals, which translates into a functional equation satisfied by the associated generating function. The solution of this equation is an algebraic series, obtained by a guess-and-check approach. Finding a bijective proof remains an open problem.

Keywords: Enumeration; Lattice paths; Tamari lattices

Domains: Combinatorics [math.CO]

Mireille Bousquet-Mélou, Eric Fusy, Louis-François Préville Ratelle. The number of intervals in the m-Tamari lattices. 2011. ⟨hal-00598840v1⟩