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Exceptional and modern intervals of the Tamari lattice
- Mathematics
- 2018
In this article we use the theory of interval-posets recently introduced by Ch{\^a}tel and Pons in order to describe some interesting families of intervals in the Tamari lattices. These families are…
Toward the Enumeration of Maximal Chains in the Tamari Lattices
- Mathematics
- 2016
The Tamari lattices have been intensely studied since they first appeared in Dov Tamari’s thesis around 1952. He defined the n-th Tamari lattice Tn on bracketings of a set of n + 1 objects, with a…
Meeting Covered Elements in $\nu$-Tamari Lattices
- Mathematics
- 2021
For each meet-semilattice M , we define an operator PopM : M →M by PopM (x) = ∧ ({y ∈M : y l x} ∪ {x}). When M is the right weak order on a symmetric group, PopM is the pop-stack-sorting map. We…
Two bijections on Tamari Intervals
- Mathematics
- 2013
We use a recently introduced combinatorial object, the $\textit{interval-poset}$, to describe two bijections on intervals of the Tamari lattice. Both bijections give a combinatorial proof of some…
m-dendriform algebras
- Mathematics
- 2014
The Fuss-Catalan numbers are a generalization of the Catalan numbers. They enumerate a large class of objects and in particular m-Dyck paths and m+1-ary trees. Recently, F. Bergeron defined an…
Geometry of $\nu $-Tamari lattices in types $A$ and $B$
- MathematicsTransactions of the American Mathematical Society
- 2018
In this paper, we exploit the combinatorics and geometry of triangulations of products of simplices to derive new results in the context of Catalan combinatorics of $\nu$-Tamari lattices. In our…
Cubic realizations of Tamari interval lattices
- Mathematics
- 2019
We introduce cubic coordinates, which are integer words encoding intervals in the Tamari lattices. Cubic coordinates are in bijection with interval-posets, themselves known to be in bijection with…
GEOMETRIC REALIZATIONS OF TAMARI INTERVAL LATTICES VIA CUBIC COORDINATES
- Mathematics
- 2020
We introduce cubic coordinates, which are integer words encoding intervals in the Tamari lattices. Cubic coordinates are in bijection with interval-posets, themselves known to be in bijection with…
An extension of Tamari lattices
- Mathematics
- 2014
For any finite path $v$ on the square grid consisting of north and east unit steps, starting at (0,0), we construct a poset Tam$(v)$ that consists of all the paths weakly above $v$ with the same…
References
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The number of intervals in this lattice is proved to be $$ m+1}{n(mn+1)} {(m+1)^2 n+m\choose n-1}.
The algebra of binary search trees
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is a sub-Hopf algebra. There is a basis Qn of Soln such that the composite map k[Qn]&Soln k[Sn] has the following property: its linear dual k[Sn] k[Qn] is induced by a set-theoretic map Sn Qn . In…
Problems of Associativity: A Simple Proof for the Lattice Property of Systems Ordered by a Semi-associative Law
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