Abstract
Let v be a grid path made of north and east steps. The lattice TAM(v), based on all grid paths weakly above the grid path v sharing the same endpoints as v, was introduced by Pre ́ville-Ratelle and Viennot (2014) and corresponds to the usual Tamari lattice in the case v = (NE)n. They showed that TAM(v) is isomorphic to the dual of TAM(←−v ), where ←−v is the reverse of v with N and E exchanged. Our main contribution is a bijection from intervals in TAM(v) to non-separable planar maps. It follows that the number of intervals in TAM(v) over all v of length n is 2(3n+3)! (n+2)!(2n+3)! . This formula was first obtained by Tutte(1963) for non-separable planar maps.
Highlights
Background and main resultsThe well-known Tamari lattice can be defined on Dyck paths or some other combinatorial structures counted by Catalan numbers such as binary trees, and it has many connections with several fields, in1365–8050 c 2016 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, FranceWenjie Fang and Louis-Francois Preville-Ratelle particular in algebraic and enumerative combinatorics
We show that an exploration process gives a bijection between non-separable planar maps and decorated trees, and there is a bijection between decorated trees and synchronized intervals
It is interesting to ask for significant statistics that are transferred by our bijection and other interesting natural involutions
Summary
The well-known Tamari lattice can be defined on Dyck paths or some other combinatorial structures counted by Catalan numbers such as binary trees, and it has many connections with several fields, in. Many results and conjectures about the diagonal coinvariant spaces of the symmetric group (we refer to [Ber, Hag08] for further explanation), called the Garsia-Haiman spaces, led Bergeron to introduce the m-Tamari lattice for any integer m ≥ 1 It was conjectured in [BPR12] and proved in [BMFPR11] and [BMCPR13] that the number of intervals and labeled intervals in the m-Tamari lattice of size n are given respectively by the formulas m + 1 (m + 1)2n + m n(mn + 1). The numbers of usual and labeled intervals in the m-Tamari lattice in [BMFPR11] and [BMCPR13] are given by simple planar-map-like formulas, where a combinatorial explanation is still missing. Theorem 1.2 The isomorphism from TAM(v) to TAM(←−v ) applied on generalized Tamari intervals is equivalent to map duality under our bijection
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