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Statistic identifier: St001509
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Collection: Dyck paths
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Description: The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary.
Given two lattice paths $U,L$ from $(0,0)$ to $(d,n-d)$, [1] describes a bijection between lattice paths weakly between $U$ and $L$ and subsets of $\{1,\dots,n\}$ such that the set of all such subsets gives the standard complex of the lattice path matroid $M[U,L]$.
This statistic gives the cardinality of the image of this bijection when a Dyck path is considered as a path weakly below the diagonal and relative to the trivial lower boundary.
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References: [1] Engström, A., Sanyal, R., Stump, C. Standard complexes of matroids and lattice paths [[arXiv:1911.12290]]
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Code:
def standard_monomial(D, low=None, east=1):
n = len(D)
A = [ i+1 for i,s in enumerate(D) if s == 1-east ]
if low is None:
low = [1 .. len(A)]
low = [ i for i in [1 .. n] if i not in low ]
vals = []
j = 0
blocker = low + [Infinity]
for i in [1 .. n]:
if i not in A:
if i < blocker[0]:
j += 1
else:
blocker.pop(0)
else:
if j > 0:
j -= 1
vals.append(i)
blocker.pop(0)
return tuple(vals)
def statistic(D):
return len(standard_monomial(D, east=0))
-----------------------------------------------------------------------------
Statistic values:
[1,0] => 0
[1,0,1,0] => 1
[1,1,0,0] => 0
[1,0,1,0,1,0] => 2
[1,0,1,1,0,0] => 1
[1,1,0,0,1,0] => 1
[1,1,0,1,0,0] => 1
[1,1,1,0,0,0] => 0
[1,0,1,0,1,0,1,0] => 3
[1,0,1,0,1,1,0,0] => 2
[1,0,1,1,0,0,1,0] => 2
[1,0,1,1,0,1,0,0] => 2
[1,0,1,1,1,0,0,0] => 1
[1,1,0,0,1,0,1,0] => 2
[1,1,0,0,1,1,0,0] => 2
[1,1,0,1,0,0,1,0] => 2
[1,1,0,1,0,1,0,0] => 2
[1,1,0,1,1,0,0,0] => 1
[1,1,1,0,0,0,1,0] => 1
[1,1,1,0,0,1,0,0] => 1
[1,1,1,0,1,0,0,0] => 1
[1,1,1,1,0,0,0,0] => 0
[1,0,1,0,1,0,1,0,1,0] => 4
[1,0,1,0,1,0,1,1,0,0] => 3
[1,0,1,0,1,1,0,0,1,0] => 3
[1,0,1,0,1,1,0,1,0,0] => 3
[1,0,1,0,1,1,1,0,0,0] => 2
[1,0,1,1,0,0,1,0,1,0] => 3
[1,0,1,1,0,0,1,1,0,0] => 3
[1,0,1,1,0,1,0,0,1,0] => 3
[1,0,1,1,0,1,0,1,0,0] => 3
[1,0,1,1,0,1,1,0,0,0] => 2
[1,0,1,1,1,0,0,0,1,0] => 2
[1,0,1,1,1,0,0,1,0,0] => 2
[1,0,1,1,1,0,1,0,0,0] => 2
[1,0,1,1,1,1,0,0,0,0] => 1
[1,1,0,0,1,0,1,0,1,0] => 3
[1,1,0,0,1,0,1,1,0,0] => 3
[1,1,0,0,1,1,0,0,1,0] => 3
[1,1,0,0,1,1,0,1,0,0] => 3
[1,1,0,0,1,1,1,0,0,0] => 2
[1,1,0,1,0,0,1,0,1,0] => 3
[1,1,0,1,0,0,1,1,0,0] => 3
[1,1,0,1,0,1,0,0,1,0] => 3
[1,1,0,1,0,1,0,1,0,0] => 3
[1,1,0,1,0,1,1,0,0,0] => 2
[1,1,0,1,1,0,0,0,1,0] => 2
[1,1,0,1,1,0,0,1,0,0] => 2
[1,1,0,1,1,0,1,0,0,0] => 2
[1,1,0,1,1,1,0,0,0,0] => 1
[1,1,1,0,0,0,1,0,1,0] => 2
[1,1,1,0,0,0,1,1,0,0] => 2
[1,1,1,0,0,1,0,0,1,0] => 2
[1,1,1,0,0,1,0,1,0,0] => 2
[1,1,1,0,0,1,1,0,0,0] => 2
[1,1,1,0,1,0,0,0,1,0] => 2
[1,1,1,0,1,0,0,1,0,0] => 2
[1,1,1,0,1,0,1,0,0,0] => 2
[1,1,1,0,1,1,0,0,0,0] => 1
[1,1,1,1,0,0,0,0,1,0] => 1
[1,1,1,1,0,0,0,1,0,0] => 1
[1,1,1,1,0,0,1,0,0,0] => 1
[1,1,1,1,0,1,0,0,0,0] => 1
[1,1,1,1,1,0,0,0,0,0] => 0
[1,0,1,0,1,0,1,0,1,0,1,0] => 5
[1,0,1,0,1,0,1,0,1,1,0,0] => 4
[1,0,1,0,1,0,1,1,0,0,1,0] => 4
[1,0,1,0,1,0,1,1,0,1,0,0] => 4
[1,0,1,0,1,0,1,1,1,0,0,0] => 3
[1,0,1,0,1,1,0,0,1,0,1,0] => 4
[1,0,1,0,1,1,0,0,1,1,0,0] => 4
[1,0,1,0,1,1,0,1,0,0,1,0] => 4
[1,0,1,0,1,1,0,1,0,1,0,0] => 4
[1,0,1,0,1,1,0,1,1,0,0,0] => 3
[1,0,1,0,1,1,1,0,0,0,1,0] => 3
[1,0,1,0,1,1,1,0,0,1,0,0] => 3
[1,0,1,0,1,1,1,0,1,0,0,0] => 3
[1,0,1,0,1,1,1,1,0,0,0,0] => 2
[1,0,1,1,0,0,1,0,1,0,1,0] => 4
[1,0,1,1,0,0,1,0,1,1,0,0] => 4
[1,0,1,1,0,0,1,1,0,0,1,0] => 4
[1,0,1,1,0,0,1,1,0,1,0,0] => 4
[1,0,1,1,0,0,1,1,1,0,0,0] => 3
[1,0,1,1,0,1,0,0,1,0,1,0] => 4
[1,0,1,1,0,1,0,0,1,1,0,0] => 4
[1,0,1,1,0,1,0,1,0,0,1,0] => 4
[1,0,1,1,0,1,0,1,0,1,0,0] => 4
[1,0,1,1,0,1,0,1,1,0,0,0] => 3
[1,0,1,1,0,1,1,0,0,0,1,0] => 3
[1,0,1,1,0,1,1,0,0,1,0,0] => 3
[1,0,1,1,0,1,1,0,1,0,0,0] => 3
[1,0,1,1,0,1,1,1,0,0,0,0] => 2
[1,0,1,1,1,0,0,0,1,0,1,0] => 3
[1,0,1,1,1,0,0,0,1,1,0,0] => 3
[1,0,1,1,1,0,0,1,0,0,1,0] => 3
[1,0,1,1,1,0,0,1,0,1,0,0] => 3
[1,0,1,1,1,0,0,1,1,0,0,0] => 3
[1,0,1,1,1,0,1,0,0,0,1,0] => 3
[1,0,1,1,1,0,1,0,0,1,0,0] => 3
[1,0,1,1,1,0,1,0,1,0,0,0] => 3
[1,0,1,1,1,0,1,1,0,0,0,0] => 2
[1,0,1,1,1,1,0,0,0,0,1,0] => 2
[1,0,1,1,1,1,0,0,0,1,0,0] => 2
[1,0,1,1,1,1,0,0,1,0,0,0] => 2
[1,0,1,1,1,1,0,1,0,0,0,0] => 2
[1,0,1,1,1,1,1,0,0,0,0,0] => 1
[1,1,0,0,1,0,1,0,1,0,1,0] => 4
[1,1,0,0,1,0,1,0,1,1,0,0] => 4
[1,1,0,0,1,0,1,1,0,0,1,0] => 4
[1,1,0,0,1,0,1,1,0,1,0,0] => 4
[1,1,0,0,1,0,1,1,1,0,0,0] => 3
[1,1,0,0,1,1,0,0,1,0,1,0] => 4
[1,1,0,0,1,1,0,0,1,1,0,0] => 4
[1,1,0,0,1,1,0,1,0,0,1,0] => 4
[1,1,0,0,1,1,0,1,0,1,0,0] => 4
[1,1,0,0,1,1,0,1,1,0,0,0] => 3
[1,1,0,0,1,1,1,0,0,0,1,0] => 3
[1,1,0,0,1,1,1,0,0,1,0,0] => 3
[1,1,0,0,1,1,1,0,1,0,0,0] => 3
[1,1,0,0,1,1,1,1,0,0,0,0] => 2
[1,1,0,1,0,0,1,0,1,0,1,0] => 4
[1,1,0,1,0,0,1,0,1,1,0,0] => 4
[1,1,0,1,0,0,1,1,0,0,1,0] => 4
[1,1,0,1,0,0,1,1,0,1,0,0] => 4
[1,1,0,1,0,0,1,1,1,0,0,0] => 3
[1,1,0,1,0,1,0,0,1,0,1,0] => 4
[1,1,0,1,0,1,0,0,1,1,0,0] => 4
[1,1,0,1,0,1,0,1,0,0,1,0] => 4
[1,1,0,1,0,1,0,1,0,1,0,0] => 4
[1,1,0,1,0,1,0,1,1,0,0,0] => 3
[1,1,0,1,0,1,1,0,0,0,1,0] => 3
[1,1,0,1,0,1,1,0,0,1,0,0] => 3
[1,1,0,1,0,1,1,0,1,0,0,0] => 3
[1,1,0,1,0,1,1,1,0,0,0,0] => 2
[1,1,0,1,1,0,0,0,1,0,1,0] => 3
[1,1,0,1,1,0,0,0,1,1,0,0] => 3
[1,1,0,1,1,0,0,1,0,0,1,0] => 3
[1,1,0,1,1,0,0,1,0,1,0,0] => 3
[1,1,0,1,1,0,0,1,1,0,0,0] => 3
[1,1,0,1,1,0,1,0,0,0,1,0] => 3
[1,1,0,1,1,0,1,0,0,1,0,0] => 3
[1,1,0,1,1,0,1,0,1,0,0,0] => 3
[1,1,0,1,1,0,1,1,0,0,0,0] => 2
[1,1,0,1,1,1,0,0,0,0,1,0] => 2
[1,1,0,1,1,1,0,0,0,1,0,0] => 2
[1,1,0,1,1,1,0,0,1,0,0,0] => 2
[1,1,0,1,1,1,0,1,0,0,0,0] => 2
[1,1,0,1,1,1,1,0,0,0,0,0] => 1
[1,1,1,0,0,0,1,0,1,0,1,0] => 3
[1,1,1,0,0,0,1,0,1,1,0,0] => 3
[1,1,1,0,0,0,1,1,0,0,1,0] => 3
[1,1,1,0,0,0,1,1,0,1,0,0] => 3
[1,1,1,0,0,0,1,1,1,0,0,0] => 3
[1,1,1,0,0,1,0,0,1,0,1,0] => 3
[1,1,1,0,0,1,0,0,1,1,0,0] => 3
[1,1,1,0,0,1,0,1,0,0,1,0] => 3
[1,1,1,0,0,1,0,1,0,1,0,0] => 3
[1,1,1,0,0,1,0,1,1,0,0,0] => 3
[1,1,1,0,0,1,1,0,0,0,1,0] => 3
[1,1,1,0,0,1,1,0,0,1,0,0] => 3
[1,1,1,0,0,1,1,0,1,0,0,0] => 3
[1,1,1,0,0,1,1,1,0,0,0,0] => 2
[1,1,1,0,1,0,0,0,1,0,1,0] => 3
[1,1,1,0,1,0,0,0,1,1,0,0] => 3
[1,1,1,0,1,0,0,1,0,0,1,0] => 3
[1,1,1,0,1,0,0,1,0,1,0,0] => 3
[1,1,1,0,1,0,0,1,1,0,0,0] => 3
[1,1,1,0,1,0,1,0,0,0,1,0] => 3
[1,1,1,0,1,0,1,0,0,1,0,0] => 3
[1,1,1,0,1,0,1,0,1,0,0,0] => 3
[1,1,1,0,1,0,1,1,0,0,0,0] => 2
[1,1,1,0,1,1,0,0,0,0,1,0] => 2
[1,1,1,0,1,1,0,0,0,1,0,0] => 2
[1,1,1,0,1,1,0,0,1,0,0,0] => 2
[1,1,1,0,1,1,0,1,0,0,0,0] => 2
[1,1,1,0,1,1,1,0,0,0,0,0] => 1
[1,1,1,1,0,0,0,0,1,0,1,0] => 2
[1,1,1,1,0,0,0,0,1,1,0,0] => 2
[1,1,1,1,0,0,0,1,0,0,1,0] => 2
[1,1,1,1,0,0,0,1,0,1,0,0] => 2
[1,1,1,1,0,0,0,1,1,0,0,0] => 2
[1,1,1,1,0,0,1,0,0,0,1,0] => 2
[1,1,1,1,0,0,1,0,0,1,0,0] => 2
[1,1,1,1,0,0,1,0,1,0,0,0] => 2
[1,1,1,1,0,0,1,1,0,0,0,0] => 2
[1,1,1,1,0,1,0,0,0,0,1,0] => 2
[1,1,1,1,0,1,0,0,0,1,0,0] => 2
[1,1,1,1,0,1,0,0,1,0,0,0] => 2
[1,1,1,1,0,1,0,1,0,0,0,0] => 2
[1,1,1,1,0,1,1,0,0,0,0,0] => 1
[1,1,1,1,1,0,0,0,0,0,1,0] => 1
[1,1,1,1,1,0,0,0,0,1,0,0] => 1
[1,1,1,1,1,0,0,0,1,0,0,0] => 1
[1,1,1,1,1,0,0,1,0,0,0,0] => 1
[1,1,1,1,1,0,1,0,0,0,0,0] => 1
[1,1,1,1,1,1,0,0,0,0,0,0] => 0
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Created: Nov 28, 2019 at 17:31 by Christian Stump
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Last Updated: Dec 29, 2019 at 22:07 by Martin Rubey