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Your data matches 6 different statistics following compositions of up to 3 maps.
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Matching statistic: St000015
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000015: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000015: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> [1]
=> [1,0]
=> 1
Description
The number of peaks of a Dyck path.
Matching statistic: St001068
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001068: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001068: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> [1]
=> [1,0]
=> 1
Description
Number of torsionless simple modules in the corresponding Nakayama algebra.
Matching statistic: St000053
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
Description
The number of valleys of the Dyck path.
Matching statistic: St000331
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000331: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000331: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
Description
The number of upper interactions of a Dyck path.
An ''upper interaction'' in a Dyck path is defined as the occurrence of a factor '''$A^{k}$$B^{k}$''' for any '''${k ≥ 1}$''', where '''${A}$''' is a down-step and '''${B}$''' is a up-step.
Matching statistic: St001169
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001169: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001169: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
Description
Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra.
Matching statistic: St001509
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001509: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001509: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
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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