Dendriform dialgebra algorithms to compute using Loday's arithmetic on groves of planar binary trees
Installation of latest release version using Julia:
Provides the types
PBTree for planar binary trees,
Grove for tree collections of constant degree, and
GroveBin to compress grove data. This package defines various essential operations on planar binary trees and groves like
right for branching;
≥ for Tamari's partial ordering;
under); and the
* for dendriform algebra.
The left and right addition are computed on the following recursive principle:
Together these non-commutative binary operations satisfy the properties of an associative dendriform dialgebra. The structures induced by Loday's definition of the sum have the partial ordering of the associahedron known as Tamari lattice.
- Figure: Tamari associahedron, colored to visualize noncommutative sums of [1,2] and [2,1], code: gist
However, in this computational package, a stricter total ordering is constructed using a function that transforms the set-vector isomorphism obtained from the descending greatest integer index position search method:
The structure obtained from this total ordering is used to construct a reliable binary
groveindex representation that encodes the essential data of any grove, using the formula
These algorithms are used in order to facilitate computations that provide insight into the Loday arithmetic.
Basic usage examples:
julia> using Dendriform julia> Grove(3,7) ⊣ [1,2]∪[2,1] [1,2,5,1,2] [1,2,5,2,1] [2,1,5,1,2] [2,1,5,2,1] [1,5,3,1,2] [1,5,2,1,3] [1,5,1,2,3] [1,5,3,2,1] [1,5,1,3,1] Y5 #9/42 julia> Grove(2,3) * [1,2,3]∪[3,2,1] |> GroveBin 2981131286847743360614880957207748817969 Y6 #30/132 [54.75%] julia> [2,1,7,4,1,3,1] < [2,1,7,4,3,2,1] true