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Your data matches 164 different statistics following compositions of up to 3 maps.
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Matching statistic: St000147
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Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
[3,1] => [[3,3],[2]]
=> [2]
=> 2
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 1
[1,3,1] => [[3,3,1],[2]]
=> [2]
=> 2
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1
[2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 1
[2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 2
[3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
[3,2] => [[4,3],[2]]
=> [2]
=> 2
[4,1] => [[4,4],[3]]
=> [3]
=> 3
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 1
[1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> 2
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 1
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> 1
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> 2
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> 2
[1,3,2] => [[4,3,1],[2]]
=> [2]
=> 2
[1,4,1] => [[4,4,1],[3]]
=> [3]
=> 3
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> 1
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> 2
[2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 1
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> 2
[2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 2
[2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> 3
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> 2
[3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 2
[3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> 3
[3,3] => [[5,3],[2]]
=> [2]
=> 2
[4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 3
[4,2] => [[5,4],[3]]
=> [3]
=> 3
[5,1] => [[5,5],[4]]
=> [4]
=> 4
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
=> [1,1]
=> 1
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
=> [2]
=> 2
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> [1,1,1]
=> 1
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> [1,1]
=> 1
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
=> [2,1]
=> 2
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
=> [2,2]
=> 2
[1,1,3,2] => [[4,3,1,1],[2]]
=> [2]
=> 2
[1,1,4,1] => [[4,4,1,1],[3]]
=> [3]
=> 3
[1,2,1,1,1,1] => [[2,2,2,2,2,1],[1,1,1,1]]
=> [1,1,1,1]
=> 1
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 1
[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]]
=> [2,1,1]
=> 2
[1,2,1,3] => [[4,2,2,1],[1,1]]
=> [1,1]
=> 1
[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]]
=> [2,2,1]
=> 2
[1,2,2,2] => [[4,3,2,1],[2,1]]
=> [2,1]
=> 2
[1,2,3,1] => [[4,4,2,1],[3,1]]
=> [3,1]
=> 3
[1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]]
=> [2,2,2]
=> 2
[1,3,1,2] => [[4,3,3,1],[2,2]]
=> [2,2]
=> 2
[1,3,2,1] => [[4,4,3,1],[3,2]]
=> [3,2]
=> 3
Description
The largest part of an integer partition.
Matching statistic: St000676
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: 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
St000676: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[3,1] => [[3,3],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,3,1] => [[3,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[3,2] => [[4,3],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[4,1] => [[4,4],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[1,3,2] => [[4,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,4,1] => [[4,4,1],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 2
[3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
[3,3] => [[5,3],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 3
[4,2] => [[5,4],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[5,1] => [[5,5],[4]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[1,1,3,2] => [[4,3,1,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,4,1] => [[4,4,1,1],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[1,2,1,1,1,1] => [[2,2,2,2,2,1],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,2,1,3] => [[4,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,2,2,2] => [[4,3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,2,3,1] => [[4,4,2,1],[3,1]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 2
[1,3,1,2] => [[4,3,3,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[1,3,2,1] => [[4,4,3,1],[3,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
Description
The number of odd rises of a Dyck path.
This is the number of ones at an odd position, with the initial position equal to 1.
The number of Dyck paths of semilength $n$ with $k$ up steps in odd positions and $k$ returns to the main diagonal are counted by the binomial coefficient $\binom{n-1}{k-1}$ [3,4].
Matching statistic: St001039
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001039: 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
St001039: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[3,1] => [[3,3],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,3,1] => [[3,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[3,2] => [[4,3],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[4,1] => [[4,4],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[1,3,2] => [[4,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,4,1] => [[4,4,1],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 2
[3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
[3,3] => [[5,3],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 3
[4,2] => [[5,4],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[5,1] => [[5,5],[4]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[1,1,3,2] => [[4,3,1,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,4,1] => [[4,4,1,1],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[1,2,1,1,1,1] => [[2,2,2,2,2,1],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,2,1,3] => [[4,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,2,2,2] => [[4,3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,2,3,1] => [[4,4,2,1],[3,1]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 2
[1,3,1,2] => [[4,3,3,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[1,3,2,1] => [[4,4,3,1],[3,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
Description
The maximal height of a column in the parallelogram polyomino associated with a Dyck path.
Matching statistic: St000734
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 1
[3,1] => [[3,3],[2]]
=> [2]
=> [[1,2]]
=> 2
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 1
[1,3,1] => [[3,3,1],[2]]
=> [2]
=> [[1,2]]
=> 2
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [[1],[2],[3]]
=> 1
[2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 1
[2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> [[1,2],[3]]
=> 2
[3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> [[1,2],[3,4]]
=> 2
[3,2] => [[4,3],[2]]
=> [2]
=> [[1,2]]
=> 2
[4,1] => [[4,4],[3]]
=> [3]
=> [[1,2,3]]
=> 3
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 1
[1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> [[1,2]]
=> 2
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [[1],[2],[3]]
=> 1
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 1
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> [[1,2],[3]]
=> 2
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> [[1,2],[3,4]]
=> 2
[1,3,2] => [[4,3,1],[2]]
=> [2]
=> [[1,2]]
=> 2
[1,4,1] => [[4,4,1],[3]]
=> [3]
=> [[1,2,3]]
=> 3
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> [[1],[2],[3]]
=> 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 1
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
[2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> [[1,2],[3]]
=> 2
[2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> [[1,2,3],[4]]
=> 3
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> 2
[3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> [[1,2],[3,4]]
=> 2
[3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 3
[3,3] => [[5,3],[2]]
=> [2]
=> [[1,2]]
=> 2
[4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> 3
[4,2] => [[5,4],[3]]
=> [3]
=> [[1,2,3]]
=> 3
[5,1] => [[5,5],[4]]
=> [4]
=> [[1,2,3,4]]
=> 4
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 1
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
=> [2]
=> [[1,2]]
=> 2
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> [1,1,1]
=> [[1],[2],[3]]
=> 1
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 1
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
=> [2,1]
=> [[1,2],[3]]
=> 2
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
=> [2,2]
=> [[1,2],[3,4]]
=> 2
[1,1,3,2] => [[4,3,1,1],[2]]
=> [2]
=> [[1,2]]
=> 2
[1,1,4,1] => [[4,4,1,1],[3]]
=> [3]
=> [[1,2,3]]
=> 3
[1,2,1,1,1,1] => [[2,2,2,2,2,1],[1,1,1,1]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [[1],[2],[3]]
=> 1
[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,2,1,3] => [[4,2,2,1],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 1
[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
[1,2,2,2] => [[4,3,2,1],[2,1]]
=> [2,1]
=> [[1,2],[3]]
=> 2
[1,2,3,1] => [[4,4,2,1],[3,1]]
=> [3,1]
=> [[1,2,3],[4]]
=> 3
[1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> 2
[1,3,1,2] => [[4,3,3,1],[2,2]]
=> [2,2]
=> [[1,2],[3,4]]
=> 2
[1,3,2,1] => [[4,4,3,1],[3,2]]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 3
[6,1,1,1] => [[6,6,6,6],[5,5,5]]
=> [5,5,5]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14,15]]
=> ? = 5
Description
The last entry in the first row of a standard tableau.
Matching statistic: St000288
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00280: Binary words —path rowmotion⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 88% ●values known / values provided: 99%●distinct values known / distinct values provided: 88%
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00280: Binary words —path rowmotion⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 88% ●values known / values provided: 99%●distinct values known / distinct values provided: 88%
Values
[2,1,1] => [1,3] => 1100 => 0111 => 3 = 1 + 2
[3,1] => [1,1,2] => 1110 => 1111 => 4 = 2 + 2
[1,2,1,1] => [2,3] => 10100 => 11001 => 3 = 1 + 2
[1,3,1] => [2,1,2] => 10110 => 11011 => 4 = 2 + 2
[2,1,1,1] => [1,4] => 11000 => 00111 => 3 = 1 + 2
[2,1,2] => [1,3,1] => 11001 => 01110 => 3 = 1 + 2
[2,2,1] => [1,2,2] => 11010 => 11101 => 4 = 2 + 2
[3,1,1] => [1,1,3] => 11100 => 01111 => 4 = 2 + 2
[3,2] => [1,1,2,1] => 11101 => 11110 => 4 = 2 + 2
[4,1] => [1,1,1,2] => 11110 => 11111 => 5 = 3 + 2
[1,1,2,1,1] => [3,3] => 100100 => 011001 => 3 = 1 + 2
[1,1,3,1] => [3,1,2] => 100110 => 011011 => 4 = 2 + 2
[1,2,1,1,1] => [2,4] => 101000 => 110001 => 3 = 1 + 2
[1,2,1,2] => [2,3,1] => 101001 => 110010 => 3 = 1 + 2
[1,2,2,1] => [2,2,2] => 101010 => 110101 => 4 = 2 + 2
[1,3,1,1] => [2,1,3] => 101100 => 110011 => 4 = 2 + 2
[1,3,2] => [2,1,2,1] => 101101 => 110110 => 4 = 2 + 2
[1,4,1] => [2,1,1,2] => 101110 => 110111 => 5 = 3 + 2
[2,1,1,1,1] => [1,5] => 110000 => 000111 => 3 = 1 + 2
[2,1,1,2] => [1,4,1] => 110001 => 001110 => 3 = 1 + 2
[2,1,2,1] => [1,3,2] => 110010 => 011101 => 4 = 2 + 2
[2,1,3] => [1,3,1,1] => 110011 => 011100 => 3 = 1 + 2
[2,2,1,1] => [1,2,3] => 110100 => 111001 => 4 = 2 + 2
[2,2,2] => [1,2,2,1] => 110101 => 111010 => 4 = 2 + 2
[2,3,1] => [1,2,1,2] => 110110 => 111011 => 5 = 3 + 2
[3,1,1,1] => [1,1,4] => 111000 => 001111 => 4 = 2 + 2
[3,1,2] => [1,1,3,1] => 111001 => 011110 => 4 = 2 + 2
[3,2,1] => [1,1,2,2] => 111010 => 111101 => 5 = 3 + 2
[3,3] => [1,1,2,1,1] => 111011 => 111100 => 4 = 2 + 2
[4,1,1] => [1,1,1,3] => 111100 => 011111 => 5 = 3 + 2
[4,2] => [1,1,1,2,1] => 111101 => 111110 => 5 = 3 + 2
[5,1] => [1,1,1,1,2] => 111110 => 111111 => 6 = 4 + 2
[1,1,1,2,1,1] => [4,3] => 1000100 => 0011001 => 3 = 1 + 2
[1,1,1,3,1] => [4,1,2] => 1000110 => 0011011 => 4 = 2 + 2
[1,1,2,1,1,1] => [3,4] => 1001000 => 0110001 => 3 = 1 + 2
[1,1,2,1,2] => [3,3,1] => 1001001 => 0110010 => 3 = 1 + 2
[1,1,2,2,1] => [3,2,2] => 1001010 => 0110101 => 4 = 2 + 2
[1,1,3,1,1] => [3,1,3] => 1001100 => 0110011 => 4 = 2 + 2
[1,1,3,2] => [3,1,2,1] => 1001101 => 0110110 => 4 = 2 + 2
[1,1,4,1] => [3,1,1,2] => 1001110 => 0110111 => 5 = 3 + 2
[1,2,1,1,1,1] => [2,5] => 1010000 => 1100001 => 3 = 1 + 2
[1,2,1,1,2] => [2,4,1] => 1010001 => 1100010 => 3 = 1 + 2
[1,2,1,2,1] => [2,3,2] => 1010010 => 1100101 => 4 = 2 + 2
[1,2,1,3] => [2,3,1,1] => 1010011 => 1100100 => 3 = 1 + 2
[1,2,2,1,1] => [2,2,3] => 1010100 => 1101001 => 4 = 2 + 2
[1,2,2,2] => [2,2,2,1] => 1010101 => 1101010 => 4 = 2 + 2
[1,2,3,1] => [2,2,1,2] => 1010110 => 1101011 => 5 = 3 + 2
[1,3,1,1,1] => [2,1,4] => 1011000 => 1100011 => 4 = 2 + 2
[1,3,1,2] => [2,1,3,1] => 1011001 => 1100110 => 4 = 2 + 2
[1,3,2,1] => [2,1,2,2] => 1011010 => 1101101 => 5 = 3 + 2
[9,1] => [1,1,1,1,1,1,1,1,2] => 1111111110 => 1111111111 => ? = 8 + 2
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000394
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4 = 3 + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> 4 = 3 + 1
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> 4 = 3 + 1
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 4 = 3 + 1
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 5 = 4 + 1
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3 + 1
[1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> ? = 4 + 1
[2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 4 + 1
[3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 4 + 1
[4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 5 + 1
Description
The sum of the heights of the peaks of a Dyck path minus the number of peaks.
Matching statistic: St001291
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 62% ●values known / values provided: 92%●distinct values known / distinct values provided: 62%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 62% ●values known / values provided: 92%●distinct values known / distinct values provided: 62%
Values
[2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[3,1] => [[3,3],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,1] => [[3,3,1],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[3,2] => [[4,3],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[4,1] => [[4,4],[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,3,2] => [[4,3,1],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[1,4,1] => [[4,4,1],[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[3,3] => [[5,3],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 4 = 3 + 1
[4,2] => [[5,4],[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[5,1] => [[5,5],[4]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,3,2] => [[4,3,1,1],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,4,1] => [[4,4,1,1],[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,2,1,1,1,1] => [[2,2,2,2,2,1],[1,1,1,1]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,2,1,3] => [[4,2,2,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,2,2,2] => [[4,3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,2,3,1] => [[4,4,2,1],[3,1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,3,1,2] => [[4,3,3,1],[2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,3,2,1] => [[4,4,3,1],[3,2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[2,1,1,1,1,1,1] => [[2,2,2,2,2,2,2],[1,1,1,1,1,1]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[3,1,1,1,1,1] => [[3,3,3,3,3,3],[2,2,2,2,2]]
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[4,1,1,1,1] => [[4,4,4,4,4],[3,3,3,3]]
=> [3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 3 + 1
[5,1,1,1] => [[5,5,5,5],[4,4,4]]
=> [4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 4 + 1
[6,1,1] => [[6,6,6],[5,5]]
=> [5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 5 + 1
[7,1] => [[7,7],[6]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 6 + 1
[2,1,1,1,1,1,1,1] => [[2,2,2,2,2,2,2,2],[1,1,1,1,1,1,1]]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[3,1,1,1,1,1,1] => [[3,3,3,3,3,3,3],[2,2,2,2,2,2]]
=> [2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 + 1
[4,1,1,1,1,1] => [[4,4,4,4,4,4],[3,3,3,3,3]]
=> [3,3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[5,1,1,1,1] => [[5,5,5,5,5],[4,4,4,4]]
=> [4,4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4 + 1
[6,1,1,1] => [[6,6,6,6],[5,5,5]]
=> [5,5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 5 + 1
[7,1,1] => [[7,7,7],[6,6]]
=> [6,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 6 + 1
[8,1] => [[8,8],[7]]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7 + 1
[9,1] => [[9,9],[8]]
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 8 + 1
Description
The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path.
Let $A$ be the Nakayama algebra associated to a Dyck path as given in [[DyckPaths/NakayamaAlgebras]]. This statistics is the number of indecomposable summands of $D(A) \otimes D(A)$, where $D(A)$ is the natural dual of $A$.
Matching statistic: St000010
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Values
[2,1,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[3,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
[1,2,1,1] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[1,3,1] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 2
[2,1,1,1] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[2,1,2] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
[2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 2
[3,1,1] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
[3,2] => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 2
[4,1] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 3
[1,1,2,1,1] => [3,3] => [[5,3],[2]]
=> [2]
=> 1
[1,1,3,1] => [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 2
[1,2,1,1,1] => [4,2] => [[5,4],[3]]
=> [3]
=> 1
[1,2,1,2] => [1,3,2] => [[4,3,1],[2]]
=> [2]
=> 1
[1,2,2,1] => [2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 2
[1,3,1,1] => [3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 2
[1,3,2] => [1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> 2
[1,4,1] => [2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> 3
[2,1,1,1,1] => [5,1] => [[5,5],[4]]
=> [4]
=> 1
[2,1,1,2] => [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 1
[2,1,2,1] => [2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> 2
[2,1,3] => [1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> 1
[2,2,1,1] => [3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> 2
[2,2,2] => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> 2
[2,3,1] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> 3
[3,1,1,1] => [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 2
[3,1,2] => [1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> 2
[3,2,1] => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> 3
[3,3] => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 2
[4,1,1] => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> 3
[4,2] => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 3
[5,1] => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> 4
[1,1,1,2,1,1] => [3,4] => [[6,3],[2]]
=> [2]
=> 1
[1,1,1,3,1] => [2,1,4] => [[5,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,2,1,1,1] => [4,3] => [[6,4],[3]]
=> [3]
=> 1
[1,1,2,1,2] => [1,3,3] => [[5,3,1],[2]]
=> [2]
=> 1
[1,1,2,2,1] => [2,2,3] => [[5,3,2],[2,1]]
=> [2,1]
=> 2
[1,1,3,1,1] => [3,1,3] => [[5,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,3,2] => [1,2,1,3] => [[4,2,2,1],[1,1]]
=> [1,1]
=> 2
[1,1,4,1] => [2,1,1,3] => [[4,2,2,2],[1,1,1]]
=> [1,1,1]
=> 3
[1,2,1,1,1,1] => [5,2] => [[6,5],[4]]
=> [4]
=> 1
[1,2,1,1,2] => [1,4,2] => [[5,4,1],[3]]
=> [3]
=> 1
[1,2,1,2,1] => [2,3,2] => [[5,4,2],[3,1]]
=> [3,1]
=> 2
[1,2,1,3] => [1,1,3,2] => [[4,3,1,1],[2]]
=> [2]
=> 1
[1,2,2,1,1] => [3,2,2] => [[5,4,3],[3,2]]
=> [3,2]
=> 2
[1,2,2,2] => [1,2,2,2] => [[4,3,2,1],[2,1]]
=> [2,1]
=> 2
[1,2,3,1] => [2,1,2,2] => [[4,3,2,2],[2,1,1]]
=> [2,1,1]
=> 3
[1,3,1,1,1] => [4,1,2] => [[5,4,4],[3,3]]
=> [3,3]
=> 2
[1,3,1,2] => [1,3,1,2] => [[4,3,3,1],[2,2]]
=> [2,2]
=> 2
[1,3,2,1] => [2,2,1,2] => [[4,3,3,2],[2,2,1]]
=> [2,2,1]
=> 3
[1,1,1,1,2,1,1] => [3,5] => [[7,3],[2]]
=> ?
=> ? = 1
[1,1,1,2,1,2] => [1,3,4] => [[6,3,1],[2]]
=> ?
=> ? = 1
[1,1,1,2,2,1] => [2,2,4] => [[6,3,2],[2,1]]
=> ?
=> ? = 2
[1,1,1,3,2] => [1,2,1,4] => [[5,2,2,1],[1,1]]
=> ?
=> ? = 2
[1,1,2,1,1,1,1] => [5,3] => [[7,5],[4]]
=> ?
=> ? = 1
[1,1,3,3] => [1,1,2,1,3] => [[4,2,2,1,1],[1,1]]
=> ?
=> ? = 2
[1,2,1,1,1,1,1] => [6,2] => [[7,6],[5]]
=> ?
=> ? = 1
[1,2,1,1,1,2] => [1,5,2] => [[6,5,1],[4]]
=> ?
=> ? = 1
[2,1,1,1,1,2] => [1,6,1] => [[6,6,1],[5]]
=> ?
=> ? = 1
[2,1,1,1,2,1] => [2,5,1] => [[6,6,2],[5,1]]
=> ?
=> ? = 2
[2,1,1,2,2] => [1,2,4,1] => [[5,5,2,1],[4,1]]
=> ?
=> ? = 2
[2,1,1,3,1] => [2,1,4,1] => [[5,5,2,2],[4,1,1]]
=> ?
=> ? = 3
[2,2,1,1,1,1] => [5,2,1] => [[6,6,5],[5,4]]
=> ?
=> ? = 2
[3,1,1,3] => [1,1,4,1,1] => [[4,4,4,1,1],[3,3]]
=> ?
=> ? = 2
[4,1,3] => [1,1,3,1,1,1] => [[3,3,3,3,1,1],[2,2,2]]
=> ?
=> ? = 3
Description
The length of the partition.
Matching statistic: St000012
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St000012: Dyck paths ⟶ ℤResult quality: 62% ●values known / values provided: 89%●distinct values known / distinct values provided: 62%
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St000012: Dyck paths ⟶ ℤResult quality: 62% ●values known / values provided: 89%●distinct values known / distinct values provided: 62%
Values
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4 = 3 + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> 4 = 3 + 1
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> 4 = 3 + 1
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 4 = 3 + 1
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 5 = 4 + 1
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3 + 1
[1,2,3,2] => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 + 1
[1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> ? = 4 + 1
[1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 3 + 1
[2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 4 + 1
[2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 3 + 1
[2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 4 + 1
[3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 4 + 1
[3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 5 + 1
[4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 5 + 1
[4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6 + 1
[2,1,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1 + 1
[3,1,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2 + 1
[4,1,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[5,1,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4 + 1
[6,1,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 5 + 1
[7,1,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6 + 1
[8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 7 + 1
[9,1] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 8 + 1
Description
The area of a Dyck path.
This is the number of complete squares in the integer lattice which are below the path and above the x-axis. The 'half-squares' directly above the axis do not contribute to this statistic.
1. Dyck paths are bijection with '''area sequences''' $(a_1,\ldots,a_n)$ such that $a_1 = 0, a_{k+1} \leq a_k + 1$.
2. The generating function $\mathbf{D}_n(q) = \sum_{D \in \mathfrak{D}_n} q^{\operatorname{area}(D)}$ satisfy the recurrence $$\mathbf{D}_{n+1}(q) = \sum q^k \mathbf{D}_k(q) \mathbf{D}_{n-k}(q).$$
3. The area is equidistributed with [[St000005]] and [[St000006]]. Pairs of these statistics play an important role in the theory of $q,t$-Catalan numbers.
Matching statistic: St000984
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St000984: Dyck paths ⟶ ℤResult quality: 62% ●values known / values provided: 89%●distinct values known / distinct values provided: 62%
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St000984: Dyck paths ⟶ ℤResult quality: 62% ●values known / values provided: 89%●distinct values known / distinct values provided: 62%
Values
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4 = 3 + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> 4 = 3 + 1
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> 4 = 3 + 1
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 4 = 3 + 1
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 5 = 4 + 1
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3 + 1
[1,2,3,2] => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 + 1
[1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> ? = 4 + 1
[1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 3 + 1
[2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 4 + 1
[2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 3 + 1
[2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 4 + 1
[3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 4 + 1
[3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 5 + 1
[4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 5 + 1
[4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6 + 1
[2,1,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1 + 1
[3,1,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2 + 1
[4,1,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[5,1,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4 + 1
[6,1,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 5 + 1
[7,1,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6 + 1
[8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 7 + 1
[9,1] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 8 + 1
Description
The number of boxes below precisely one peak.
Imagine that each peak of the Dyck path, drawn with north and east steps, casts a shadow onto the triangular region between it and the diagonal. This statistic is the number of cells which are in the shade of precisely one peak.
The following 154 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001280The number of parts of an integer partition that are at least two. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000052The number of valleys of a Dyck path not on the x-axis. St000225Difference between largest and smallest parts in a partition. St000356The number of occurrences of the pattern 13-2. St000463The number of admissible inversions of a permutation. St000497The lcb statistic of a set partition. St000572The dimension exponent of a set partition. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000996The number of exclusive left-to-right maxima of a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St000451The length of the longest pattern of the form k 1 2. St000617The number of global maxima of a Dyck path. St000171The degree of the graph. St000204The number of internal nodes of a binary tree. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000454The largest eigenvalue of a graph if it is integral. St000536The pathwidth of a graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001120The length of a longest path in a graph. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001644The dimension of a graph. St001962The proper pathwidth of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000172The Grundy number of a graph. St000363The number of minimal vertex covers of a graph. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000822The Hadwiger number of the graph. St001029The size of the core of a graph. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001330The hat guessing number of a graph. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001368The number of vertices of maximal degree in a graph. St001458The rank of the adjacency matrix of a graph. St001459The number of zero columns in the nullspace of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001725The harmonious chromatic number of a graph. St001883The mutual visibility number of a graph. St001963The tree-depth of a graph. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St000053The number of valleys of the Dyck path. St000093The cardinality of a maximal independent set of vertices of a graph. St000636The hull number of a graph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001286The annihilation number of a graph. St001315The dissociation number of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001654The monophonic hull number of a graph. St000686The finitistic dominant dimension of a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St001674The number of vertices of the largest induced star graph in the graph. St001812The biclique partition number of a graph. St000141The maximum drop size of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000065The number of entries equal to -1 in an alternating sign matrix. St001323The independence gap of a graph. St000028The number of stack-sorts needed to sort a permutation. St000741The Colin de Verdière graph invariant. St001642The Prague dimension of a graph. St000145The Dyson rank of a partition. St000803The number of occurrences of the vincular pattern |132 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001964The interval resolution global dimension of a poset. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001645The pebbling number of a connected graph. St000358The number of occurrences of the pattern 31-2. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000223The number of nestings in the permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000956The maximal displacement of a permutation. St001091The number of parts in an integer partition whose next smaller part has the same size. St000233The number of nestings of a set partition. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St001082The number of boxed occurrences of 123 in a permutation. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St000299The number of nonisomorphic vertex-induced subtrees. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000039The number of crossings of a permutation. St000317The cycle descent number of a permutation. St000711The number of big exceedences of a permutation. St000732The number of double deficiencies of a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001411The number of patterns 321 or 3412 in a permutation. St001552The number of inversions between excedances and fixed points of a permutation. St001727The number of invisible inversions of a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St000021The number of descents of a permutation. St000024The number of double up and double down steps of a Dyck path. St000155The number of exceedances (also excedences) of a permutation. St000238The number of indices that are not small weak excedances. St000316The number of non-left-to-right-maxima of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000339The maf index of a permutation. St000354The number of recoils of a permutation. St000798The makl of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001480The number of simple summands of the module J^2/J^3. St001489The maximum of the number of descents and the number of inverse descents. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St000015The number of peaks of a Dyck path. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001692The number of vertices with higher degree than the average degree in a graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001691The number of kings in a graph. St001305The number of induced cycles on four vertices in a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St000455The second largest eigenvalue of a graph if it is integral.
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