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arXiv · 1701.02917

A sequent calculus for the Tamari order

Noam Zeilberger
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Abstract

We introduce a sequent calculus with a simple restriction of Lambek's product rules that precisely captures the classical Tamari order, i.e., the partial order on fully-bracketed words (equivalently, binary trees) induced by a semi-associative law (equivalently, tree rotation). We establish a focusing property for this sequent calculus (a strengthening of cut-elimination), which yields the following coherence theorem: every valid entailment in the Tamari order has exactly one focused derivation. One combinatorial application of this coherence theorem is a new proof of the Tutte-Chapoton formula for the number of intervals in the Tamari lattice $Y_n$. We also apply the sequent calculus and the coherence theorem to build a surprising bijection between intervals of the Tamari order and a certain fragment of lambda calculus, consisting of the $β$-normal planar lambda terms with no closed proper subterms.

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Noam Zeilberger. 2017-01-11. A sequent calculus for the Tamari order. https://arxiv.org/abs/1701.02917

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