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Counting smaller trees in the Tamari order

Abstract : We introduce new combinatorial objects, the interval-posets, that encode intervals of the Tamari lattice. We then find a combinatorial interpretation of the bilinear form that appears in the functional equation of Tamari intervals described by Chapoton. Thus, we retrieve this functional equation and prove that the polynomial recursively computed from the bilinear form on each tree $T$ counts the number of trees smaller than $T$ in the Tamari order.
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https://hal.inria.fr/hal-00824154
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Grégory Chatel, Viviane Pons. Counting smaller trees in the Tamari order. 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), 2013, Paris, France. pp.433-444. ⟨hal-00824154v2⟩

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