Identifier
Identifier
Values
[1,0] generating graphics... => 0
[1,0,1,0] generating graphics... => 1
[1,1,0,0] generating graphics... => 0
[1,0,1,0,1,0] generating graphics... => 2
[1,0,1,1,0,0] generating graphics... => 1
[1,1,0,0,1,0] generating graphics... => 1
[1,1,0,1,0,0] generating graphics... => 1
[1,1,1,0,0,0] generating graphics... => 0
[1,0,1,0,1,0,1,0] generating graphics... => 3
[1,0,1,0,1,1,0,0] generating graphics... => 2
[1,0,1,1,0,0,1,0] generating graphics... => 2
[1,0,1,1,0,1,0,0] generating graphics... => 2
[1,0,1,1,1,0,0,0] generating graphics... => 1
[1,1,0,0,1,0,1,0] generating graphics... => 2
[1,1,0,0,1,1,0,0] generating graphics... => 2
[1,1,0,1,0,0,1,0] generating graphics... => 2
[1,1,0,1,0,1,0,0] generating graphics... => 2
[1,1,0,1,1,0,0,0] generating graphics... => 1
[1,1,1,0,0,0,1,0] generating graphics... => 1
[1,1,1,0,0,1,0,0] generating graphics... => 1
[1,1,1,0,1,0,0,0] generating graphics... => 1
[1,1,1,1,0,0,0,0] generating graphics... => 0
[1,0,1,0,1,0,1,0,1,0] generating graphics... => 4
[1,0,1,0,1,0,1,1,0,0] generating graphics... => 3
[1,0,1,0,1,1,0,0,1,0] generating graphics... => 3
[1,0,1,0,1,1,0,1,0,0] generating graphics... => 3
[1,0,1,0,1,1,1,0,0,0] generating graphics... => 2
[1,0,1,1,0,0,1,0,1,0] generating graphics... => 3
[1,0,1,1,0,0,1,1,0,0] generating graphics... => 3
[1,0,1,1,0,1,0,0,1,0] generating graphics... => 3
[1,0,1,1,0,1,0,1,0,0] generating graphics... => 3
[1,0,1,1,0,1,1,0,0,0] generating graphics... => 2
[1,0,1,1,1,0,0,0,1,0] generating graphics... => 2
[1,0,1,1,1,0,0,1,0,0] generating graphics... => 2
[1,0,1,1,1,0,1,0,0,0] generating graphics... => 2
[1,0,1,1,1,1,0,0,0,0] generating graphics... => 1
[1,1,0,0,1,0,1,0,1,0] generating graphics... => 3
[1,1,0,0,1,0,1,1,0,0] generating graphics... => 3
[1,1,0,0,1,1,0,0,1,0] generating graphics... => 3
[1,1,0,0,1,1,0,1,0,0] generating graphics... => 3
[1,1,0,0,1,1,1,0,0,0] generating graphics... => 2
[1,1,0,1,0,0,1,0,1,0] generating graphics... => 3
[1,1,0,1,0,0,1,1,0,0] generating graphics... => 3
[1,1,0,1,0,1,0,0,1,0] generating graphics... => 3
[1,1,0,1,0,1,0,1,0,0] generating graphics... => 3
[1,1,0,1,0,1,1,0,0,0] generating graphics... => 2
[1,1,0,1,1,0,0,0,1,0] generating graphics... => 2
[1,1,0,1,1,0,0,1,0,0] generating graphics... => 2
[1,1,0,1,1,0,1,0,0,0] generating graphics... => 2
[1,1,0,1,1,1,0,0,0,0] generating graphics... => 1
[1,1,1,0,0,0,1,0,1,0] generating graphics... => 2
[1,1,1,0,0,0,1,1,0,0] generating graphics... => 2
[1,1,1,0,0,1,0,0,1,0] generating graphics... => 2
[1,1,1,0,0,1,0,1,0,0] generating graphics... => 2
[1,1,1,0,0,1,1,0,0,0] generating graphics... => 2
[1,1,1,0,1,0,0,0,1,0] generating graphics... => 2
[1,1,1,0,1,0,0,1,0,0] generating graphics... => 2
[1,1,1,0,1,0,1,0,0,0] generating graphics... => 2
[1,1,1,0,1,1,0,0,0,0] generating graphics... => 1
[1,1,1,1,0,0,0,0,1,0] generating graphics... => 1
[1,1,1,1,0,0,0,1,0,0] generating graphics... => 1
[1,1,1,1,0,0,1,0,0,0] generating graphics... => 1
[1,1,1,1,0,1,0,0,0,0] generating graphics... => 1
[1,1,1,1,1,0,0,0,0,0] generating graphics... => 0
[1,0,1,0,1,0,1,0,1,0,1,0] generating graphics... => 5
[1,0,1,0,1,0,1,0,1,1,0,0] generating graphics... => 4
[1,0,1,0,1,0,1,1,0,0,1,0] generating graphics... => 4
[1,0,1,0,1,0,1,1,0,1,0,0] generating graphics... => 4
[1,0,1,0,1,0,1,1,1,0,0,0] generating graphics... => 3
[1,0,1,0,1,1,0,0,1,0,1,0] generating graphics... => 4
[1,0,1,0,1,1,0,0,1,1,0,0] generating graphics... => 4
[1,0,1,0,1,1,0,1,0,0,1,0] generating graphics... => 4
[1,0,1,0,1,1,0,1,0,1,0,0] generating graphics... => 4
[1,0,1,0,1,1,0,1,1,0,0,0] generating graphics... => 3
[1,0,1,0,1,1,1,0,0,0,1,0] generating graphics... => 3
[1,0,1,0,1,1,1,0,0,1,0,0] generating graphics... => 3
[1,0,1,0,1,1,1,0,1,0,0,0] generating graphics... => 3
[1,0,1,0,1,1,1,1,0,0,0,0] generating graphics... => 2
[1,0,1,1,0,0,1,0,1,0,1,0] generating graphics... => 4
[1,0,1,1,0,0,1,0,1,1,0,0] generating graphics... => 4
[1,0,1,1,0,0,1,1,0,0,1,0] generating graphics... => 4
[1,0,1,1,0,0,1,1,0,1,0,0] generating graphics... => 4
[1,0,1,1,0,0,1,1,1,0,0,0] generating graphics... => 3
[1,0,1,1,0,1,0,0,1,0,1,0] generating graphics... => 4
[1,0,1,1,0,1,0,0,1,1,0,0] generating graphics... => 4
[1,0,1,1,0,1,0,1,0,0,1,0] generating graphics... => 4
[1,0,1,1,0,1,0,1,0,1,0,0] generating graphics... => 4
[1,0,1,1,0,1,0,1,1,0,0,0] generating graphics... => 3
[1,0,1,1,0,1,1,0,0,0,1,0] generating graphics... => 3
[1,0,1,1,0,1,1,0,0,1,0,0] generating graphics... => 3
[1,0,1,1,0,1,1,0,1,0,0,0] generating graphics... => 3
[1,0,1,1,0,1,1,1,0,0,0,0] generating graphics... => 2
[1,0,1,1,1,0,0,0,1,0,1,0] generating graphics... => 3
[1,0,1,1,1,0,0,0,1,1,0,0] generating graphics... => 3
[1,0,1,1,1,0,0,1,0,0,1,0] generating graphics... => 3
[1,0,1,1,1,0,0,1,0,1,0,0] generating graphics... => 3
[1,0,1,1,1,0,0,1,1,0,0,0] generating graphics... => 3
[1,0,1,1,1,0,1,0,0,0,1,0] generating graphics... => 3
[1,0,1,1,1,0,1,0,0,1,0,0] generating graphics... => 3
[1,0,1,1,1,0,1,0,1,0,0,0] generating graphics... => 3
[1,0,1,1,1,0,1,1,0,0,0,0] generating graphics... => 2
[1,0,1,1,1,1,0,0,0,0,1,0] generating graphics... => 2
[1,0,1,1,1,1,0,0,0,1,0,0] generating graphics... => 2
[1,0,1,1,1,1,0,0,1,0,0,0] generating graphics... => 2
[1,0,1,1,1,1,0,1,0,0,0,0] generating graphics... => 2
[1,0,1,1,1,1,1,0,0,0,0,0] generating graphics... => 1
[1,1,0,0,1,0,1,0,1,0,1,0] generating graphics... => 4
[1,1,0,0,1,0,1,0,1,1,0,0] generating graphics... => 4
[1,1,0,0,1,0,1,1,0,0,1,0] generating graphics... => 4
[1,1,0,0,1,0,1,1,0,1,0,0] generating graphics... => 4
[1,1,0,0,1,0,1,1,1,0,0,0] generating graphics... => 3
[1,1,0,0,1,1,0,0,1,0,1,0] generating graphics... => 4
[1,1,0,0,1,1,0,0,1,1,0,0] generating graphics... => 4
[1,1,0,0,1,1,0,1,0,0,1,0] generating graphics... => 4
[1,1,0,0,1,1,0,1,0,1,0,0] generating graphics... => 4
[1,1,0,0,1,1,0,1,1,0,0,0] generating graphics... => 3
[1,1,0,0,1,1,1,0,0,0,1,0] generating graphics... => 3
[1,1,0,0,1,1,1,0,0,1,0,0] generating graphics... => 3
[1,1,0,0,1,1,1,0,1,0,0,0] generating graphics... => 3
[1,1,0,0,1,1,1,1,0,0,0,0] generating graphics... => 2
[1,1,0,1,0,0,1,0,1,0,1,0] generating graphics... => 4
[1,1,0,1,0,0,1,0,1,1,0,0] generating graphics... => 4
[1,1,0,1,0,0,1,1,0,0,1,0] generating graphics... => 4
[1,1,0,1,0,0,1,1,0,1,0,0] generating graphics... => 4
[1,1,0,1,0,0,1,1,1,0,0,0] generating graphics... => 3
[1,1,0,1,0,1,0,0,1,0,1,0] generating graphics... => 4
[1,1,0,1,0,1,0,0,1,1,0,0] generating graphics... => 4
[1,1,0,1,0,1,0,1,0,0,1,0] generating graphics... => 4
[1,1,0,1,0,1,0,1,0,1,0,0] generating graphics... => 4
[1,1,0,1,0,1,0,1,1,0,0,0] generating graphics... => 3
[1,1,0,1,0,1,1,0,0,0,1,0] generating graphics... => 3
[1,1,0,1,0,1,1,0,0,1,0,0] generating graphics... => 3
[1,1,0,1,0,1,1,0,1,0,0,0] generating graphics... => 3
[1,1,0,1,0,1,1,1,0,0,0,0] generating graphics... => 2
[1,1,0,1,1,0,0,0,1,0,1,0] generating graphics... => 3
[1,1,0,1,1,0,0,0,1,1,0,0] generating graphics... => 3
[1,1,0,1,1,0,0,1,0,0,1,0] generating graphics... => 3
[1,1,0,1,1,0,0,1,0,1,0,0] generating graphics... => 3
[1,1,0,1,1,0,0,1,1,0,0,0] generating graphics... => 3
[1,1,0,1,1,0,1,0,0,0,1,0] generating graphics... => 3
[1,1,0,1,1,0,1,0,0,1,0,0] generating graphics... => 3
[1,1,0,1,1,0,1,0,1,0,0,0] generating graphics... => 3
[1,1,0,1,1,0,1,1,0,0,0,0] generating graphics... => 2
[1,1,0,1,1,1,0,0,0,0,1,0] generating graphics... => 2
[1,1,0,1,1,1,0,0,0,1,0,0] generating graphics... => 2
[1,1,0,1,1,1,0,0,1,0,0,0] generating graphics... => 2
[1,1,0,1,1,1,0,1,0,0,0,0] generating graphics... => 2
[1,1,0,1,1,1,1,0,0,0,0,0] generating graphics... => 1
[1,1,1,0,0,0,1,0,1,0,1,0] generating graphics... => 3
[1,1,1,0,0,0,1,0,1,1,0,0] generating graphics... => 3
[1,1,1,0,0,0,1,1,0,0,1,0] generating graphics... => 3
[1,1,1,0,0,0,1,1,0,1,0,0] generating graphics... => 3
[1,1,1,0,0,0,1,1,1,0,0,0] generating graphics... => 3
[1,1,1,0,0,1,0,0,1,0,1,0] generating graphics... => 3
[1,1,1,0,0,1,0,0,1,1,0,0] generating graphics... => 3
[1,1,1,0,0,1,0,1,0,0,1,0] generating graphics... => 3
[1,1,1,0,0,1,0,1,0,1,0,0] generating graphics... => 3
[1,1,1,0,0,1,0,1,1,0,0,0] generating graphics... => 3
[1,1,1,0,0,1,1,0,0,0,1,0] generating graphics... => 3
[1,1,1,0,0,1,1,0,0,1,0,0] generating graphics... => 3
[1,1,1,0,0,1,1,0,1,0,0,0] generating graphics... => 3
[1,1,1,0,0,1,1,1,0,0,0,0] generating graphics... => 2
[1,1,1,0,1,0,0,0,1,0,1,0] generating graphics... => 3
[1,1,1,0,1,0,0,0,1,1,0,0] generating graphics... => 3
[1,1,1,0,1,0,0,1,0,0,1,0] generating graphics... => 3
[1,1,1,0,1,0,0,1,0,1,0,0] generating graphics... => 3
[1,1,1,0,1,0,0,1,1,0,0,0] generating graphics... => 3
[1,1,1,0,1,0,1,0,0,0,1,0] generating graphics... => 3
[1,1,1,0,1,0,1,0,0,1,0,0] generating graphics... => 3
[1,1,1,0,1,0,1,0,1,0,0,0] generating graphics... => 3
[1,1,1,0,1,0,1,1,0,0,0,0] generating graphics... => 2
[1,1,1,0,1,1,0,0,0,0,1,0] generating graphics... => 2
[1,1,1,0,1,1,0,0,0,1,0,0] generating graphics... => 2
[1,1,1,0,1,1,0,0,1,0,0,0] generating graphics... => 2
[1,1,1,0,1,1,0,1,0,0,0,0] generating graphics... => 2
[1,1,1,0,1,1,1,0,0,0,0,0] generating graphics... => 1
[1,1,1,1,0,0,0,0,1,0,1,0] generating graphics... => 2
[1,1,1,1,0,0,0,0,1,1,0,0] generating graphics... => 2
[1,1,1,1,0,0,0,1,0,0,1,0] generating graphics... => 2
[1,1,1,1,0,0,0,1,0,1,0,0] generating graphics... => 2
[1,1,1,1,0,0,0,1,1,0,0,0] generating graphics... => 2
[1,1,1,1,0,0,1,0,0,0,1,0] generating graphics... => 2
[1,1,1,1,0,0,1,0,0,1,0,0] generating graphics... => 2
[1,1,1,1,0,0,1,0,1,0,0,0] generating graphics... => 2
[1,1,1,1,0,0,1,1,0,0,0,0] generating graphics... => 2
[1,1,1,1,0,1,0,0,0,0,1,0] generating graphics... => 2
[1,1,1,1,0,1,0,0,0,1,0,0] generating graphics... => 2
[1,1,1,1,0,1,0,0,1,0,0,0] generating graphics... => 2
[1,1,1,1,0,1,0,1,0,0,0,0] generating graphics... => 2
[1,1,1,1,0,1,1,0,0,0,0,0] generating graphics... => 1
[1,1,1,1,1,0,0,0,0,0,1,0] generating graphics... => 1
[1,1,1,1,1,0,0,0,0,1,0,0] generating graphics... => 1
[1,1,1,1,1,0,0,0,1,0,0,0] generating graphics... => 1
[1,1,1,1,1,0,0,1,0,0,0,0] generating graphics... => 1
[1,1,1,1,1,0,1,0,0,0,0,0] generating graphics... => 1
[1,1,1,1,1,1,0,0,0,0,0,0] generating graphics... => 0
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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.
References
[1] Engström, A., Sanyal, R., Stump, C. Standard complexes of matroids and lattice paths arXiv:1911.12290
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 ]
    low = [ i+1 for i,s in enumerate(low) 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))
Created
Nov 28, 2019 at 17:31 by Christian Stump
Updated
Dec 29, 2019 at 22:07 by Martin Rubey