Dyck Word in 3d¶
3d DyckWords
Generalisation of Dyck Word to Surface of cubes in the n x n x n cube above the plane x+y+z=2n.
EXAMPLES:
sage: from slabbe.dyck_3d import DyckBlocks3d
sage: L = [len(DyckBlocks3d(i)) for i in range(1, 7)] # not tested
[1, 2, 9, 96, 2498, 161422]
sage: L = [1, 2, 9, 96, 2498, 161422]
sage: oeis.find_by_subsequence(L) # not tested internet
0: A115965: Number of planar subpartitions of size n pyramidal planar partition.
AUTHOR:
Sébastien Labbé, 31 october 2014
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slabbe.dyck_3d.
DyckBlocks3d
(n)¶ EXAMPLES:
sage: from slabbe.dyck_3d import DyckBlocks3d sage: L = [len(DyckBlocks3d(i)) for i in range(1, 6)] sage: L [1, 2, 9, 96, 2498]
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slabbe.dyck_3d.
Possible
(n)¶ Possible stack of DyckWords inside a n x n cube.
EXAMPLES:
sage: from slabbe.dyck_3d import Possible sage: Possible(1) The Cartesian product of ({[1, 0]},) sage: Possible(2) The Cartesian product of ({[1, 1, 0, 0]}, {[1, 0, 1, 0], [1, 1, 0, 0]}) sage: Possible(3).list() [([1, 1, 1, 0, 0, 0], [1, 1, 0, 1, 0, 0], [1, 0, 1, 0, 1, 0]), ([1, 1, 1, 0, 0, 0], [1, 1, 0, 1, 0, 0], [1, 0, 1, 1, 0, 0]), ([1, 1, 1, 0, 0, 0], [1, 1, 0, 1, 0, 0], [1, 1, 0, 0, 1, 0]), ([1, 1, 1, 0, 0, 0], [1, 1, 0, 1, 0, 0], [1, 1, 0, 1, 0, 0]), ([1, 1, 1, 0, 0, 0], [1, 1, 0, 1, 0, 0], [1, 1, 1, 0, 0, 0]), ([1, 1, 1, 0, 0, 0], [1, 1, 1, 0, 0, 0], [1, 0, 1, 0, 1, 0]), ([1, 1, 1, 0, 0, 0], [1, 1, 1, 0, 0, 0], [1, 0, 1, 1, 0, 0]), ([1, 1, 1, 0, 0, 0], [1, 1, 1, 0, 0, 0], [1, 1, 0, 0, 1, 0]), ([1, 1, 1, 0, 0, 0], [1, 1, 1, 0, 0, 0], [1, 1, 0, 1, 0, 0]), ([1, 1, 1, 0, 0, 0], [1, 1, 1, 0, 0, 0], [1, 1, 1, 0, 0, 0])]
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slabbe.dyck_3d.
is_larger_than
(x, y)¶ EXAMPLES:
sage: from slabbe.dyck_3d import is_larger_than sage: w = DyckWord([1,1,1,0,0,0]) sage: w.heights() (0, 1, 2, 3, 2, 1, 0) sage: z = DyckWord([1,0,1,0,1,0]) sage: is_larger_than(w,z) True sage: is_larger_than(z,w) False sage: is_larger_than(w,w) True