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Your data matches 1 statistic following compositions of up to 3 maps.
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Matching statistic: St001956
Mapping
St001956: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => 0
[1,0,1,0]
 => 0
[1,1,0,0]
 => 1
[1,0,1,0,1,0]
 => 1
[1,0,1,1,0,0]
 => 2
[1,1,0,0,1,0]
 => 2
[1,1,0,1,0,0]
 => 3
[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]
 => 3
[1,0,1,1,0,1,0,0]
 => 4
[1,0,1,1,1,0,0,0]
 => 4
[1,1,0,0,1,0,1,0]
 => 4
[1,1,0,0,1,1,0,0]
 => 5
[1,1,0,1,0,0,1,0]
 => 5
[1,1,0,1,0,1,0,0]
 => 6
[1,1,0,1,1,0,0,0]
 => 3
[1,1,1,0,0,0,1,0]
 => 0
[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]
 => 2
[1,0,1,0,1,0,1,0,1,0]
 => 6
[1,0,1,0,1,0,1,1,0,0]
 => 5
[1,0,1,0,1,1,0,0,1,0]
 => 3
[1,0,1,0,1,1,0,1,0,0]
 => 4
[1,0,1,0,1,1,1,0,0,0]
 => 4
[1,0,1,1,0,0,1,0,1,0]
 => 5
[1,0,1,1,0,0,1,1,0,0]
 => 6
[1,0,1,1,0,1,0,0,1,0]
 => 6
[1,0,1,1,0,1,0,1,0,0]
 => 7
[1,0,1,1,0,1,1,0,0,0]
 => 7
[1,0,1,1,1,0,0,0,1,0]
 => 6
[1,0,1,1,1,0,0,1,0,0]
 => 7
[1,0,1,1,1,0,1,0,0,0]
 => 7
[1,0,1,1,1,1,0,0,0,0]
 => 4
[1,1,0,0,1,0,1,0,1,0]
 => 7
[1,1,0,0,1,0,1,1,0,0]
 => 6
[1,1,0,0,1,1,0,0,1,0]
 => 7
[1,1,0,0,1,1,0,1,0,0]
 => 8
[1,1,0,0,1,1,1,0,0,0]
 => 8
[1,1,0,1,0,0,1,0,1,0]
 => 8
[1,1,0,1,0,0,1,1,0,0]
 => 9
[1,1,0,1,0,1,0,0,1,0]
 => 9
[1,1,0,1,0,1,0,1,0,0]
 => 10
[1,1,0,1,0,1,1,0,0,0]
 => 7
[1,1,0,1,1,0,0,0,1,0]
 => 4
[1,1,0,1,1,0,0,1,0,0]
 => 5
[1,1,0,1,1,0,1,0,0,0]
 => 5
[1,1,0,1,1,1,0,0,0,0]
 => 6
Description
The comajor index for set-valued two-row standard Young tableaux. The comajorindex is the sum ∑k(n+1−k) over all natural descents k. Bijections via bicolored Motzkin paths (with two restrictions, see [1]) give the following for Dyck paths. Let j be smallest integer such that 2j is a down step. Then k is a natural descent if * k−2≥j and positions 2(k−1)−1,2(k−1) are a valley i.e. [0,1], or * k−2≥j and positions 2(k−1)−1,2(k−1) are a peak i.e. [1,0], or * k−1≥j and positions 2(k−1),2k−1,2k form [0,1,1], or * k=j and positions 2k−1,2k are double down i.e. [0,0], or * k<j and positions 2k−1,2k are a valley i.e. [0,1].