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Your data matches 33 different statistics following compositions of up to 3 maps.
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Matching statistic: St000738
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St000738: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> 1
[[1,2]]
=> 1
[[1],[2]]
=> 2
[[1,2,3]]
=> 1
[[1,3],[2]]
=> 2
[[1,2],[3]]
=> 3
[[1],[2],[3]]
=> 3
[[1,2,3,4]]
=> 1
[[1,3,4],[2]]
=> 2
[[1,2,4],[3]]
=> 3
[[1,2,3],[4]]
=> 4
[[1,3],[2,4]]
=> 2
[[1,2],[3,4]]
=> 3
[[1,4],[2],[3]]
=> 3
[[1,3],[2],[4]]
=> 4
[[1,2],[3],[4]]
=> 4
[[1],[2],[3],[4]]
=> 4
[[1,2,3,4,5]]
=> 1
[[1,3,4,5],[2]]
=> 2
[[1,2,4,5],[3]]
=> 3
[[1,2,3,5],[4]]
=> 4
[[1,2,3,4],[5]]
=> 5
[[1,3,5],[2,4]]
=> 2
[[1,2,5],[3,4]]
=> 3
[[1,3,4],[2,5]]
=> 2
[[1,2,4],[3,5]]
=> 3
[[1,2,3],[4,5]]
=> 4
[[1,4,5],[2],[3]]
=> 3
[[1,3,5],[2],[4]]
=> 4
[[1,2,5],[3],[4]]
=> 4
[[1,3,4],[2],[5]]
=> 5
[[1,2,4],[3],[5]]
=> 5
[[1,2,3],[4],[5]]
=> 5
[[1,4],[2,5],[3]]
=> 3
[[1,3],[2,5],[4]]
=> 4
[[1,2],[3,5],[4]]
=> 4
[[1,3],[2,4],[5]]
=> 5
[[1,2],[3,4],[5]]
=> 5
[[1,5],[2],[3],[4]]
=> 4
[[1,4],[2],[3],[5]]
=> 5
[[1,3],[2],[4],[5]]
=> 5
[[1,2],[3],[4],[5]]
=> 5
[[1],[2],[3],[4],[5]]
=> 5
[[1,2,3,4,5,6]]
=> 1
[[1,3,4,5,6],[2]]
=> 2
[[1,2,4,5,6],[3]]
=> 3
[[1,2,3,5,6],[4]]
=> 4
[[1,2,3,4,6],[5]]
=> 5
[[1,2,3,4,5],[6]]
=> 6
[[1,3,5,6],[2,4]]
=> 2
Description
The first entry in the last row of a standard tableau.
For the last entry in the first row, see [[St000734]].
Matching statistic: St000734
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000734: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [[1]]
=> 1
[[1,2]]
=> [[1],[2]]
=> 1
[[1],[2]]
=> [[1,2]]
=> 2
[[1,2,3]]
=> [[1],[2],[3]]
=> 1
[[1,3],[2]]
=> [[1,2],[3]]
=> 2
[[1,2],[3]]
=> [[1,3],[2]]
=> 3
[[1],[2],[3]]
=> [[1,2,3]]
=> 3
[[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 1
[[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 2
[[1,2,4],[3]]
=> [[1,3],[2],[4]]
=> 3
[[1,2,3],[4]]
=> [[1,4],[2],[3]]
=> 4
[[1,3],[2,4]]
=> [[1,2],[3,4]]
=> 2
[[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 3
[[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 3
[[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> 4
[[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> 4
[[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 4
[[1,2,3,4,5]]
=> [[1],[2],[3],[4],[5]]
=> 1
[[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> 2
[[1,2,4,5],[3]]
=> [[1,3],[2],[4],[5]]
=> 3
[[1,2,3,5],[4]]
=> [[1,4],[2],[3],[5]]
=> 4
[[1,2,3,4],[5]]
=> [[1,5],[2],[3],[4]]
=> 5
[[1,3,5],[2,4]]
=> [[1,2],[3,4],[5]]
=> 2
[[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> 3
[[1,3,4],[2,5]]
=> [[1,2],[3,5],[4]]
=> 2
[[1,2,4],[3,5]]
=> [[1,3],[2,5],[4]]
=> 3
[[1,2,3],[4,5]]
=> [[1,4],[2,5],[3]]
=> 4
[[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 3
[[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> 4
[[1,2,5],[3],[4]]
=> [[1,3,4],[2],[5]]
=> 4
[[1,3,4],[2],[5]]
=> [[1,2,5],[3],[4]]
=> 5
[[1,2,4],[3],[5]]
=> [[1,3,5],[2],[4]]
=> 5
[[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 5
[[1,4],[2,5],[3]]
=> [[1,2,3],[4,5]]
=> 3
[[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 4
[[1,2],[3,5],[4]]
=> [[1,3,4],[2,5]]
=> 4
[[1,3],[2,4],[5]]
=> [[1,2,5],[3,4]]
=> 5
[[1,2],[3,4],[5]]
=> [[1,3,5],[2,4]]
=> 5
[[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 4
[[1,4],[2],[3],[5]]
=> [[1,2,3,5],[4]]
=> 5
[[1,3],[2],[4],[5]]
=> [[1,2,4,5],[3]]
=> 5
[[1,2],[3],[4],[5]]
=> [[1,3,4,5],[2]]
=> 5
[[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> 5
[[1,2,3,4,5,6]]
=> [[1],[2],[3],[4],[5],[6]]
=> 1
[[1,3,4,5,6],[2]]
=> [[1,2],[3],[4],[5],[6]]
=> 2
[[1,2,4,5,6],[3]]
=> [[1,3],[2],[4],[5],[6]]
=> 3
[[1,2,3,5,6],[4]]
=> [[1,4],[2],[3],[5],[6]]
=> 4
[[1,2,3,4,6],[5]]
=> [[1,5],[2],[3],[4],[6]]
=> 5
[[1,2,3,4,5],[6]]
=> [[1,6],[2],[3],[4],[5]]
=> 6
[[1,3,5,6],[2,4]]
=> [[1,2],[3,4],[5],[6]]
=> 2
Description
The last entry in the first row of a standard tableau.
Matching statistic: St000505
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Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000505: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00284: Standard tableaux —rows⟶ Set partitions
St000505: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [[1]]
=> {{1}}
=> 1
[[1,2]]
=> [[1],[2]]
=> {{1},{2}}
=> 1
[[1],[2]]
=> [[1,2]]
=> {{1,2}}
=> 2
[[1,2,3]]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 1
[[1,3],[2]]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 2
[[1,2],[3]]
=> [[1,3],[2]]
=> {{1,3},{2}}
=> 3
[[1],[2],[3]]
=> [[1,2,3]]
=> {{1,2,3}}
=> 3
[[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 1
[[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 2
[[1,2,4],[3]]
=> [[1,3],[2],[4]]
=> {{1,3},{2},{4}}
=> 3
[[1,2,3],[4]]
=> [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> 4
[[1,3],[2,4]]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 2
[[1,2],[3,4]]
=> [[1,3],[2,4]]
=> {{1,3},{2,4}}
=> 3
[[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 3
[[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> {{1,2,4},{3}}
=> 4
[[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 4
[[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> {{1,2,3,4}}
=> 4
[[1,2,3,4,5]]
=> [[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> 1
[[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 2
[[1,2,4,5],[3]]
=> [[1,3],[2],[4],[5]]
=> {{1,3},{2},{4},{5}}
=> 3
[[1,2,3,5],[4]]
=> [[1,4],[2],[3],[5]]
=> {{1,4},{2},{3},{5}}
=> 4
[[1,2,3,4],[5]]
=> [[1,5],[2],[3],[4]]
=> {{1,5},{2},{3},{4}}
=> 5
[[1,3,5],[2,4]]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 2
[[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> {{1,3},{2,4},{5}}
=> 3
[[1,3,4],[2,5]]
=> [[1,2],[3,5],[4]]
=> {{1,2},{3,5},{4}}
=> 2
[[1,2,4],[3,5]]
=> [[1,3],[2,5],[4]]
=> {{1,3},{2,5},{4}}
=> 3
[[1,2,3],[4,5]]
=> [[1,4],[2,5],[3]]
=> {{1,4},{2,5},{3}}
=> 4
[[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 3
[[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> {{1,2,4},{3},{5}}
=> 4
[[1,2,5],[3],[4]]
=> [[1,3,4],[2],[5]]
=> {{1,3,4},{2},{5}}
=> 4
[[1,3,4],[2],[5]]
=> [[1,2,5],[3],[4]]
=> {{1,2,5},{3},{4}}
=> 5
[[1,2,4],[3],[5]]
=> [[1,3,5],[2],[4]]
=> {{1,3,5},{2},{4}}
=> 5
[[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> 5
[[1,4],[2,5],[3]]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 3
[[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> {{1,2,4},{3,5}}
=> 4
[[1,2],[3,5],[4]]
=> [[1,3,4],[2,5]]
=> {{1,3,4},{2,5}}
=> 4
[[1,3],[2,4],[5]]
=> [[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> 5
[[1,2],[3,4],[5]]
=> [[1,3,5],[2,4]]
=> {{1,3,5},{2,4}}
=> 5
[[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 4
[[1,4],[2],[3],[5]]
=> [[1,2,3,5],[4]]
=> {{1,2,3,5},{4}}
=> 5
[[1,3],[2],[4],[5]]
=> [[1,2,4,5],[3]]
=> {{1,2,4,5},{3}}
=> 5
[[1,2],[3],[4],[5]]
=> [[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> 5
[[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> 5
[[1,2,3,4,5,6]]
=> [[1],[2],[3],[4],[5],[6]]
=> {{1},{2},{3},{4},{5},{6}}
=> 1
[[1,3,4,5,6],[2]]
=> [[1,2],[3],[4],[5],[6]]
=> {{1,2},{3},{4},{5},{6}}
=> 2
[[1,2,4,5,6],[3]]
=> [[1,3],[2],[4],[5],[6]]
=> {{1,3},{2},{4},{5},{6}}
=> 3
[[1,2,3,5,6],[4]]
=> [[1,4],[2],[3],[5],[6]]
=> {{1,4},{2},{3},{5},{6}}
=> 4
[[1,2,3,4,6],[5]]
=> [[1,5],[2],[3],[4],[6]]
=> {{1,5},{2},{3},{4},{6}}
=> 5
[[1,2,3,4,5],[6]]
=> [[1,6],[2],[3],[4],[5]]
=> {{1,6},{2},{3},{4},{5}}
=> 6
[[1,3,5,6],[2,4]]
=> [[1,2],[3,4],[5],[6]]
=> {{1,2},{3,4},{5},{6}}
=> 2
Description
The biggest entry in the block containing the 1.
Matching statistic: St000382
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => [1,0]
=> [1] => 1
[[1,2]]
=> [1,2] => [1,0,1,0]
=> [1,1] => 1
[[1],[2]]
=> [2,1] => [1,1,0,0]
=> [2] => 2
[[1,2,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1] => 1
[[1,3],[2]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [2,1] => 2
[[1,2],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [3] => 3
[[1],[2],[3]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [3] => 3
[[1,2,3,4]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1
[[1,3,4],[2]]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [2,1,1] => 2
[[1,2,4],[3]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3,1] => 3
[[1,2,3],[4]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [4] => 4
[[1,3],[2,4]]
=> [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,2] => 2
[[1,2],[3,4]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [3,1] => 3
[[1,4],[2],[3]]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3,1] => 3
[[1,3],[2],[4]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [4] => 4
[[1,2],[3],[4]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [4] => 4
[[1],[2],[3],[4]]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [4] => 4
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 1
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 2
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => 3
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => 4
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => 2
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> [3,1,1] => 3
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> [2,3] => 2
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> [3,2] => 3
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [4,1] => 4
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => 3
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => 4
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => 4
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
=> [3,2] => 3
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> [4,1] => 4
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [4,1] => 4
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => 4
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1] => 1
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1] => 2
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [3,1,1,1] => 3
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [4,1,1] => 4
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,1] => 5
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6] => 6
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [1,1,0,1,1,0,0,0,1,0,1,0]
=> [2,2,1,1] => 2
[[1,5,8],[2,6],[3],[4],[7]]
=> [7,4,3,2,6,1,5,8] => ?
=> ? => ? = 7
[[1,4],[2,8],[3],[5],[6],[7]]
=> [7,6,5,3,2,8,1,4] => ?
=> ? => ? = 7
[[1,5],[2,7],[3],[4],[6],[8]]
=> [8,6,4,3,2,7,1,5] => ?
=> ? => ? = 8
[[1,3],[2,7],[4],[5],[6],[8]]
=> [8,6,5,4,2,7,1,3] => ?
=> ? => ? = 8
[[1,2],[3,6],[4],[5],[7],[8]]
=> [8,7,5,4,3,6,1,2] => ?
=> ? => ? = 8
[[1,2],[3,5],[4],[6],[7],[8]]
=> [8,7,6,4,3,5,1,2] => ?
=> ? => ? = 8
Description
The first part of an integer composition.
Matching statistic: St000439
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => [1] => [1,0]
=> 2 = 1 + 1
[[1,2]]
=> [1,2] => [1,2] => [1,0,1,0]
=> 2 = 1 + 1
[[1],[2]]
=> [2,1] => [2,1] => [1,1,0,0]
=> 3 = 2 + 1
[[1,2,3]]
=> [1,2,3] => [1,3,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 3 = 2 + 1
[[1,2],[3]]
=> [3,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> 4 = 3 + 1
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 4 = 3 + 1
[[1,2,3,4]]
=> [1,2,3,4] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[[1,3,4],[2]]
=> [2,1,3,4] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[[1,2,4],[3]]
=> [3,1,2,4] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 4 = 3 + 1
[[1,2,3],[4]]
=> [4,1,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[[1,3],[2,4]]
=> [2,4,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[[1,2],[3,4]]
=> [3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0]
=> 4 = 3 + 1
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0]
=> 5 = 4 + 1
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> 4 = 3 + 1
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> 4 = 3 + 1
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> 5 = 4 + 1
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 4 = 3 + 1
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [4,2,1,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> 5 = 4 + 1
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> 5 = 4 + 1
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
=> 4 = 3 + 1
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 5 = 4 + 1
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> 5 = 4 + 1
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 1 + 1
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,1,6,5,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [3,1,6,5,4,2] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [4,1,6,5,3,2] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 5 = 4 + 1
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [5,1,6,4,3,2] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 6 = 5 + 1
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 7 = 6 + 1
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [2,6,1,5,4,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> 3 = 2 + 1
[[1,5,8],[2,6],[3],[4],[7]]
=> [7,4,3,2,6,1,5,8] => ? => ?
=> ? = 7 + 1
[[1,7],[2,8],[3],[4],[5],[6]]
=> [6,5,4,3,2,8,1,7] => [6,5,4,3,2,8,1,7] => [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> ? = 6 + 1
[[1,4],[2,8],[3],[5],[6],[7]]
=> [7,6,5,3,2,8,1,4] => ? => ?
=> ? = 7 + 1
[[1,5],[2,7],[3],[4],[6],[8]]
=> [8,6,4,3,2,7,1,5] => [8,6,4,3,2,7,1,5] => ?
=> ? = 8 + 1
[[1,3],[2,7],[4],[5],[6],[8]]
=> [8,6,5,4,2,7,1,3] => [8,6,5,4,2,7,1,3] => ?
=> ? = 8 + 1
[[1,2],[3,6],[4],[5],[7],[8]]
=> [8,7,5,4,3,6,1,2] => ? => ?
=> ? = 8 + 1
[[1,2],[3,5],[4],[6],[7],[8]]
=> [8,7,6,4,3,5,1,2] => ? => ?
=> ? = 8 + 1
[[1,3,5,7,9,11],[2,4,6,8,10,12]]
=> [2,4,6,8,10,12,1,3,5,7,9,11] => [2,12,11,10,9,8,1,7,6,5,4,3] => ?
=> ? = 2 + 1
Description
The position of the first down step of a Dyck path.
Matching statistic: St000011
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => [1,0]
=> [1,0]
=> 1
[[1,2]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[[1],[2]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 2
[[1,2,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[[1,3],[2]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[[1,2],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[[1],[2],[3]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[[1,2,3,4]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[[1,3,4],[2]]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[[1,2,4],[3]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[[1,2,3],[4]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[[1,3],[2,4]]
=> [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[[1,2],[3,4]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[[1,4],[2],[3]]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[[1,3],[2],[4]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[[1,2],[3],[4]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[[1],[2],[3],[4]]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 3
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 4
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 5
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> 2
[[1,3,4,7],[2,5,6,8]]
=> [2,5,6,8,1,3,4,7] => [1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 2
[[1,2,4,7],[3,5,6,8]]
=> [3,5,6,8,1,2,4,7] => [1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 3
[[1,3,5,6],[2,4,7,8]]
=> [2,4,7,8,1,3,5,6] => [1,1,0,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 2
[[1,3,4,6],[2,5,7,8]]
=> [2,5,7,8,1,3,4,6] => [1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 2
[[1,2,4,6],[3,5,7,8]]
=> [3,5,7,8,1,2,4,6] => [1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 3
[[1,2,3,6],[4,5,7,8]]
=> [4,5,7,8,1,2,3,6] => [1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> ? = 4
[[1,3,4,5],[2,6,7,8]]
=> [2,6,7,8,1,3,4,5] => [1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 2
[[1,2,4,5],[3,6,7,8]]
=> [3,6,7,8,1,2,4,5] => [1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 3
[[1,2,3,5],[4,6,7,8]]
=> [4,6,7,8,1,2,3,5] => [1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> ? = 4
[[1,5,8],[2,6],[3],[4],[7]]
=> [7,4,3,2,6,1,5,8] => ?
=> ?
=> ? = 7
[[1,4],[2,8],[3],[5],[6],[7]]
=> [7,6,5,3,2,8,1,4] => ?
=> ?
=> ? = 7
[[1,5],[2,7],[3],[4],[6],[8]]
=> [8,6,4,3,2,7,1,5] => ?
=> ?
=> ? = 8
[[1,3],[2,7],[4],[5],[6],[8]]
=> [8,6,5,4,2,7,1,3] => ?
=> ?
=> ? = 8
[[1,2],[3,6],[4],[5],[7],[8]]
=> [8,7,5,4,3,6,1,2] => ?
=> ?
=> ? = 8
[[1,2],[3,5],[4],[6],[7],[8]]
=> [8,7,6,4,3,5,1,2] => ?
=> ?
=> ? = 8
Description
The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St000971
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000971: Set partitions ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000971: Set partitions ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => [1,0]
=> {{1}}
=> 1
[[1,2]]
=> [1,2] => [1,0,1,0]
=> {{1},{2}}
=> 1
[[1],[2]]
=> [2,1] => [1,1,0,0]
=> {{1,2}}
=> 2
[[1,2,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 1
[[1,3],[2]]
=> [2,1,3] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 2
[[1,2],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 3
[[1],[2],[3]]
=> [3,2,1] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 3
[[1,2,3,4]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 1
[[1,3,4],[2]]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 2
[[1,2,4],[3]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 3
[[1,2,3],[4]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4
[[1,3],[2,4]]
=> [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 2
[[1,2],[3,4]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 3
[[1,4],[2],[3]]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 3
[[1,3],[2],[4]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4
[[1,2],[3],[4]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4
[[1],[2],[3],[4]]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 1
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> 2
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 3
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 4
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 5
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> 2
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> {{1,2,4},{3},{5}}
=> 3
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> 2
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> {{1,2,4,5},{3}}
=> 3
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> {{1,2,3,5},{4}}
=> 4
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 3
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 4
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 4
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 5
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 5
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 5
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
=> {{1,4,5},{2,3}}
=> 3
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> {{1,2,5},{3,4}}
=> 4
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> {{1,2,5},{3,4}}
=> 4
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 5
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 5
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 4
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 5
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 5
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 5
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 5
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5},{6}}
=> 1
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5},{6}}
=> 2
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> {{1,2,3},{4},{5},{6}}
=> 3
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> {{1,2,3,4},{5},{6}}
=> 4
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> {{1,2,3,4,5},{6}}
=> 5
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> {{1,2,3,4,5,6}}
=> 6
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [1,1,0,1,1,0,0,0,1,0,1,0]
=> {{1,3,4},{2},{5},{6}}
=> 2
[[1,3,5,7],[2,4,6,8]]
=> [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> {{1,3,5,7,8},{2},{4},{6}}
=> ? = 2
[[1,2,5,7],[3,4,6,8]]
=> [3,4,6,8,1,2,5,7] => [1,1,1,0,1,0,1,1,0,1,1,0,0,0,0,0]
=> {{1,2,5,7,8},{3},{4},{6}}
=> ? = 3
[[1,3,4,7],[2,5,6,8]]
=> [2,5,6,8,1,3,4,7] => [1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> {{1,3,4,7,8},{2},{5},{6}}
=> ? = 2
[[1,2,4,7],[3,5,6,8]]
=> [3,5,6,8,1,2,4,7] => [1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0]
=> {{1,2,4,7,8},{3},{5},{6}}
=> ? = 3
[[1,2,3,7],[4,5,6,8]]
=> [4,5,6,8,1,2,3,7] => [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> {{1,2,3,7,8},{4},{5},{6}}
=> ? = 4
[[1,3,5,6],[2,4,7,8]]
=> [2,4,7,8,1,3,5,6] => [1,1,0,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> {{1,3,5,6,8},{2},{4},{7}}
=> ? = 2
[[1,2,5,6],[3,4,7,8]]
=> [3,4,7,8,1,2,5,6] => [1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0]
=> {{1,2,5,6,8},{3},{4},{7}}
=> ? = 3
[[1,3,4,6],[2,5,7,8]]
=> [2,5,7,8,1,3,4,6] => [1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> {{1,3,4,6,8},{2},{5},{7}}
=> ? = 2
[[1,2,4,6],[3,5,7,8]]
=> [3,5,7,8,1,2,4,6] => [1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> {{1,2,4,6,8},{3},{5},{7}}
=> ? = 3
[[1,2,3,6],[4,5,7,8]]
=> [4,5,7,8,1,2,3,6] => [1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0]
=> {{1,2,3,6,8},{4},{5},{7}}
=> ? = 4
[[1,3,4,5],[2,6,7,8]]
=> [2,6,7,8,1,3,4,5] => [1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> {{1,3,4,5,8},{2},{6},{7}}
=> ? = 2
[[1,2,4,5],[3,6,7,8]]
=> [3,6,7,8,1,2,4,5] => [1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0]
=> {{1,2,4,5,8},{3},{6},{7}}
=> ? = 3
[[1,2,3,5],[4,6,7,8]]
=> [4,6,7,8,1,2,3,5] => [1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0]
=> {{1,2,3,5,8},{4},{6},{7}}
=> ? = 4
[[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5},{6},{7}}
=> ? = 5
[[1,5,8],[2,6],[3],[4],[7]]
=> [7,4,3,2,6,1,5,8] => ?
=> ?
=> ? = 7
[[1,7],[2,8],[3],[4],[5],[6]]
=> [6,5,4,3,2,8,1,7] => [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> {{1,7,8},{2,3,4,5,6}}
=> ? = 6
[[1,6],[2,8],[3],[4],[5],[7]]
=> [7,5,4,3,2,8,1,6] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> {{1,2,8},{3,4,5,6,7}}
=> ? = 7
[[1,5],[2,8],[3],[4],[6],[7]]
=> [7,6,4,3,2,8,1,5] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> {{1,2,8},{3,4,5,6,7}}
=> ? = 7
[[1,4],[2,8],[3],[5],[6],[7]]
=> [7,6,5,3,2,8,1,4] => ?
=> ?
=> ? = 7
[[1,3],[2,8],[4],[5],[6],[7]]
=> [7,6,5,4,2,8,1,3] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> {{1,2,8},{3,4,5,6,7}}
=> ? = 7
[[1,2],[3,8],[4],[5],[6],[7]]
=> [7,6,5,4,3,8,1,2] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> {{1,2,8},{3,4,5,6,7}}
=> ? = 7
[[1,5],[2,7],[3],[4],[6],[8]]
=> [8,6,4,3,2,7,1,5] => ?
=> ?
=> ? = 8
[[1,3],[2,7],[4],[5],[6],[8]]
=> [8,6,5,4,2,7,1,3] => ?
=> ?
=> ? = 8
[[1,2],[3,6],[4],[5],[7],[8]]
=> [8,7,5,4,3,6,1,2] => ?
=> ?
=> ? = 8
[[1,2],[3,5],[4],[6],[7],[8]]
=> [8,7,6,4,3,5,1,2] => ?
=> ?
=> ? = 8
[[1,3,5,7,9],[2,4,6,8,10]]
=> [2,4,6,8,10,1,3,5,7,9] => [1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> {{1,3,5,7,9,10},{2},{4},{6},{8}}
=> ? = 2
[[1,3,5,7,9,11],[2,4,6,8,10,12]]
=> [2,4,6,8,10,12,1,3,5,7,9,11] => [1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0,0,0]
=> {{1,3,5,7,9,11,12},{2},{4},{6},{8},{10}}
=> ? = 2
Description
The smallest closer of a set partition.
A closer (or right hand endpoint) of a set partition is a number that is maximal in its block. For this statistic, singletons are considered as closers.
In other words, this is the smallest among the maximal elements of the blocks.
Matching statistic: St000839
(load all 58 compositions to match this statistic)
(load all 58 compositions to match this statistic)
Mp00284: Standard tableaux —rows⟶ Set partitions
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
Mp00218: Set partitions —inverse Wachs-White-rho⟶ Set partitions
St000839: Set partitions ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
Mp00218: Set partitions —inverse Wachs-White-rho⟶ Set partitions
St000839: Set partitions ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> {{1}}
=> {{1}}
=> {{1}}
=> 1
[[1,2]]
=> {{1,2}}
=> {{1,2}}
=> {{1,2}}
=> 1
[[1],[2]]
=> {{1},{2}}
=> {{1},{2}}
=> {{1},{2}}
=> 2
[[1,2,3]]
=> {{1,2,3}}
=> {{1,2,3}}
=> {{1,2,3}}
=> 1
[[1,3],[2]]
=> {{1,3},{2}}
=> {{1},{2,3}}
=> {{1},{2,3}}
=> 2
[[1,2],[3]]
=> {{1,2},{3}}
=> {{1,2},{3}}
=> {{1,2},{3}}
=> 3
[[1],[2],[3]]
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> 3
[[1,2,3,4]]
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> 1
[[1,3,4],[2]]
=> {{1,3,4},{2}}
=> {{1,4},{2,3}}
=> {{1,3},{2,4}}
=> 2
[[1,2,4],[3]]
=> {{1,2,4},{3}}
=> {{1,2},{3,4}}
=> {{1,2},{3,4}}
=> 3
[[1,2,3],[4]]
=> {{1,2,3},{4}}
=> {{1,2,3},{4}}
=> {{1,2,3},{4}}
=> 4
[[1,3],[2,4]]
=> {{1,3},{2,4}}
=> {{1},{2,3,4}}
=> {{1},{2,3,4}}
=> 2
[[1,2],[3,4]]
=> {{1,2},{3,4}}
=> {{1,2,4},{3}}
=> {{1,2,4},{3}}
=> 3
[[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> {{1},{2},{3,4}}
=> 3
[[1,3],[2],[4]]
=> {{1,3},{2},{4}}
=> {{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> 4
[[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> 4
[[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 4
[[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> 1
[[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> {{1,4,5},{2,3}}
=> {{1,3},{2,4,5}}
=> 2
[[1,2,4,5],[3]]
=> {{1,2,4,5},{3}}
=> {{1,2,5},{3,4}}
=> {{1,2,4},{3,5}}
=> 3
[[1,2,3,5],[4]]
=> {{1,2,3,5},{4}}
=> {{1,2,3},{4,5}}
=> {{1,2,3},{4,5}}
=> 4
[[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> {{1,2,3,4},{5}}
=> {{1,2,3,4},{5}}
=> 5
[[1,3,5],[2,4]]
=> {{1,3,5},{2,4}}
=> {{1},{2,3,4,5}}
=> {{1},{2,3,4,5}}
=> 2
[[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> {{1,2,4},{3,5}}
=> {{1,2,5},{3,4}}
=> 3
[[1,3,4],[2,5]]
=> {{1,3,4},{2,5}}
=> {{1,4},{2,3,5}}
=> {{1,3,4},{2,5}}
=> 2
[[1,2,4],[3,5]]
=> {{1,2,4},{3,5}}
=> {{1,2},{3,4,5}}
=> {{1,2},{3,4,5}}
=> 3
[[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> {{1,2,3,5},{4}}
=> {{1,2,3,5},{4}}
=> 4
[[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> {{1,5},{2},{3,4}}
=> {{1,4},{2},{3,5}}
=> 3
[[1,3,5],[2],[4]]
=> {{1,3,5},{2},{4}}
=> {{1},{2,3,5},{4}}
=> {{1},{2,3,5},{4}}
=> 4
[[1,2,5],[3],[4]]
=> {{1,2,5},{3},{4}}
=> {{1,2},{3},{4,5}}
=> {{1,2},{3},{4,5}}
=> 4
[[1,3,4],[2],[5]]
=> {{1,3,4},{2},{5}}
=> {{1,4},{2,3},{5}}
=> {{1,3},{2,4},{5}}
=> 5
[[1,2,4],[3],[5]]
=> {{1,2,4},{3},{5}}
=> {{1,2},{3,4},{5}}
=> {{1,2},{3,4},{5}}
=> 5
[[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> 5
[[1,4],[2,5],[3]]
=> {{1,4},{2,5},{3}}
=> {{1},{2},{3,4,5}}
=> {{1},{2},{3,4,5}}
=> 3
[[1,3],[2,5],[4]]
=> {{1,3},{2,5},{4}}
=> {{1},{2,3},{4,5}}
=> {{1},{2,3},{4,5}}
=> 4
[[1,2],[3,5],[4]]
=> {{1,2},{3,5},{4}}
=> {{1,2},{3,5},{4}}
=> {{1,2},{3,5},{4}}
=> 4
[[1,3],[2,4],[5]]
=> {{1,3},{2,4},{5}}
=> {{1},{2,3,4},{5}}
=> {{1},{2,3,4},{5}}
=> 5
[[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> {{1,2,4},{3},{5}}
=> {{1,2,4},{3},{5}}
=> 5
[[1,5],[2],[3],[4]]
=> {{1,5},{2},{3},{4}}
=> {{1},{2},{3},{4,5}}
=> {{1},{2},{3},{4,5}}
=> 4
[[1,4],[2],[3],[5]]
=> {{1,4},{2},{3},{5}}
=> {{1},{2},{3,4},{5}}
=> {{1},{2},{3,4},{5}}
=> 5
[[1,3],[2],[4],[5]]
=> {{1,3},{2},{4},{5}}
=> {{1},{2,3},{4},{5}}
=> {{1},{2,3},{4},{5}}
=> 5
[[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> 5
[[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> 5
[[1,2,3,4,5,6]]
=> {{1,2,3,4,5,6}}
=> {{1,2,3,4,5,6}}
=> {{1,2,3,4,5,6}}
=> 1
[[1,3,4,5,6],[2]]
=> {{1,3,4,5,6},{2}}
=> {{1,4,5,6},{2,3}}
=> {{1,3},{2,4,5,6}}
=> 2
[[1,2,4,5,6],[3]]
=> {{1,2,4,5,6},{3}}
=> {{1,2,5,6},{3,4}}
=> {{1,2,4},{3,5,6}}
=> 3
[[1,2,3,5,6],[4]]
=> {{1,2,3,5,6},{4}}
=> {{1,2,3,6},{4,5}}
=> {{1,2,3,5},{4,6}}
=> 4
[[1,2,3,4,6],[5]]
=> {{1,2,3,4,6},{5}}
=> {{1,2,3,4},{5,6}}
=> {{1,2,3,4},{5,6}}
=> 5
[[1,2,3,4,5],[6]]
=> {{1,2,3,4,5},{6}}
=> {{1,2,3,4,5},{6}}
=> {{1,2,3,4,5},{6}}
=> 6
[[1,3,5,6],[2,4]]
=> {{1,3,5,6},{2,4}}
=> {{1,6},{2,3,4,5}}
=> {{1,3,5},{2,4,6}}
=> 2
[[1,3,4,7],[2,5,6,8]]
=> {{1,3,4,7},{2,5,6,8}}
=> {{1,4,6},{2,3,5,7,8}}
=> {{1,3,4,7,8},{2,5,6}}
=> ? = 2
[[1,2,4,7],[3,5,6,8]]
=> {{1,2,4,7},{3,5,6,8}}
=> {{1,2,6},{3,4,5,7,8}}
=> {{1,2,4,7,8},{3,5,6}}
=> ? = 3
[[1,2,3,7],[4,5,6,8]]
=> {{1,2,3,7},{4,5,6,8}}
=> {{1,2,3,5,6},{4,7,8}}
=> {{1,2,3,7,8},{4,5,6}}
=> ? = 4
[[1,3,5,6],[2,4,7,8]]
=> {{1,3,5,6},{2,4,7,8}}
=> {{1,6,8},{2,3,4,5,7}}
=> {{1,3,5,6,8},{2,4,7}}
=> ? = 2
[[1,2,5,6],[3,4,7,8]]
=> {{1,2,5,6},{3,4,7,8}}
=> {{1,2,4,6,8},{3,5,7}}
=> {{1,2,5,6,8},{3,4,7}}
=> ? = 3
[[1,3,4,6],[2,5,7,8]]
=> {{1,3,4,6},{2,5,7,8}}
=> {{1,4,8},{2,3,5,6,7}}
=> {{1,3,4,6,8},{2,5,7}}
=> ? = 2
[[1,2,4,6],[3,5,7,8]]
=> {{1,2,4,6},{3,5,7,8}}
=> {{1,2,8},{3,4,5,6,7}}
=> {{1,2,4,6,8},{3,5,7}}
=> ? = 3
[[1,2,3,6],[4,5,7,8]]
=> {{1,2,3,6},{4,5,7,8}}
=> {{1,2,3,5,8},{4,6,7}}
=> {{1,2,3,6,8},{4,5,7}}
=> ? = 4
[[1,5,6,8],[2],[3],[4],[7]]
=> {{1,5,6,8},{2},{3},{4},{7}}
=> {{1,6},{2,8},{3},{4,5},{7}}
=> ?
=> ? = 7
[[1,4,8],[2,5],[3,6],[7]]
=> {{1,4,8},{2,5},{3,6},{7}}
=> {{1},{2},{3,4,5,6},{7,8}}
=> {{1},{2},{3,4,5,6},{7,8}}
=> ? = 7
[[1,5,8],[2,6],[3],[4],[7]]
=> {{1,5,8},{2,6},{3},{4},{7}}
=> ?
=> ?
=> ? = 7
[[1,6,8],[2],[3],[4],[5],[7]]
=> {{1,6,8},{2},{3},{4},{5},{7}}
=> {{1},{2,8},{3},{4},{5,6},{7}}
=> ?
=> ? = 7
[[1,5,8],[2],[3],[4],[6],[7]]
=> {{1,5,8},{2},{3},{4},{6},{7}}
=> {{1},{2},{3,8},{4,5},{6},{7}}
=> ?
=> ? = 7
[[1,4,8],[2],[3],[5],[6],[7]]
=> {{1,4,8},{2},{3},{5},{6},{7}}
=> ?
=> ?
=> ? = 7
[[1,2,4],[3],[5],[6],[7],[8]]
=> {{1,2,4},{3},{5},{6},{7},{8}}
=> {{1,2},{3,4},{5},{6},{7},{8}}
=> {{1,2},{3,4},{5},{6},{7},{8}}
=> ? = 8
[[1,2,3],[4],[5],[6],[7],[8]]
=> {{1,2,3},{4},{5},{6},{7},{8}}
=> {{1,2,3},{4},{5},{6},{7},{8}}
=> {{1,2,3},{4},{5},{6},{7},{8}}
=> ? = 8
[[1,6],[2,8],[3],[4],[5],[7]]
=> {{1,6},{2,8},{3},{4},{5},{7}}
=> {{1},{2},{3},{4},{5,6},{7,8}}
=> {{1},{2},{3},{4},{5,6},{7,8}}
=> ? = 7
[[1,5],[2,8],[3],[4],[6],[7]]
=> {{1,5},{2,8},{3},{4},{6},{7}}
=> ?
=> ?
=> ? = 7
[[1,4],[2,8],[3],[5],[6],[7]]
=> {{1,4},{2,8},{3},{5},{6},{7}}
=> {{1},{2},{3,4},{5},{6},{7,8}}
=> ?
=> ? = 7
[[1,3],[2,8],[4],[5],[6],[7]]
=> {{1,3},{2,8},{4},{5},{6},{7}}
=> {{1},{2,3},{4},{5},{6},{7,8}}
=> ?
=> ? = 7
[[1,2],[3,8],[4],[5],[6],[7]]
=> {{1,2},{3,8},{4},{5},{6},{7}}
=> {{1,2},{3},{4},{5},{6,8},{7}}
=> ?
=> ? = 7
[[1,6],[2,7],[3],[4],[5],[8]]
=> {{1,6},{2,7},{3},{4},{5},{8}}
=> ?
=> ?
=> ? = 8
[[1,5],[2,7],[3],[4],[6],[8]]
=> {{1,5},{2,7},{3},{4},{6},{8}}
=> ?
=> ?
=> ? = 8
[[1,4],[2,7],[3],[5],[6],[8]]
=> {{1,4},{2,7},{3},{5},{6},{8}}
=> ?
=> ?
=> ? = 8
[[1,3],[2,7],[4],[5],[6],[8]]
=> {{1,3},{2,7},{4},{5},{6},{8}}
=> ?
=> ?
=> ? = 8
[[1,2],[3,7],[4],[5],[6],[8]]
=> {{1,2},{3,7},{4},{5},{6},{8}}
=> ?
=> ?
=> ? = 8
[[1,5],[2,6],[3],[4],[7],[8]]
=> {{1,5},{2,6},{3},{4},{7},{8}}
=> {{1},{2},{3},{4,5,6},{7},{8}}
=> {{1},{2},{3},{4,5,6},{7},{8}}
=> ? = 8
[[1,4],[2,6],[3],[5],[7],[8]]
=> {{1,4},{2,6},{3},{5},{7},{8}}
=> ?
=> ?
=> ? = 8
[[1,3],[2,6],[4],[5],[7],[8]]
=> {{1,3},{2,6},{4},{5},{7},{8}}
=> ?
=> ?
=> ? = 8
[[1,2],[3,6],[4],[5],[7],[8]]
=> {{1,2},{3,6},{4},{5},{7},{8}}
=> {{1,2},{3},{4,6},{5},{7},{8}}
=> ?
=> ? = 8
[[1,4],[2,5],[3],[6],[7],[8]]
=> {{1,4},{2,5},{3},{6},{7},{8}}
=> {{1},{2},{3,4,5},{6},{7},{8}}
=> ?
=> ? = 8
[[1,3],[2,5],[4],[6],[7],[8]]
=> {{1,3},{2,5},{4},{6},{7},{8}}
=> ?
=> ?
=> ? = 8
[[1,2],[3,5],[4],[6],[7],[8]]
=> {{1,2},{3,5},{4},{6},{7},{8}}
=> {{1,2},{3,5},{4},{6},{7},{8}}
=> ?
=> ? = 8
[[1,3],[2,4],[5],[6],[7],[8]]
=> {{1,3},{2,4},{5},{6},{7},{8}}
=> {{1},{2,3,4},{5},{6},{7},{8}}
=> {{1},{2,3,4},{5},{6},{7},{8}}
=> ? = 8
[[1,2],[3,4],[5],[6],[7],[8]]
=> {{1,2},{3,4},{5},{6},{7},{8}}
=> {{1,2,4},{3},{5},{6},{7},{8}}
=> {{1,2,4},{3},{5},{6},{7},{8}}
=> ? = 8
[[1,7],[2],[3],[4],[5],[6],[8]]
=> {{1,7},{2},{3},{4},{5},{6},{8}}
=> {{1},{2},{3},{4},{5},{6,7},{8}}
=> {{1},{2},{3},{4},{5},{6,7},{8}}
=> ? = 8
[[1,6],[2],[3],[4],[5],[7],[8]]
=> {{1,6},{2},{3},{4},{5},{7},{8}}
=> {{1},{2},{3},{4},{5,6},{7},{8}}
=> ?
=> ? = 8
[[1,5],[2],[3],[4],[6],[7],[8]]
=> {{1,5},{2},{3},{4},{6},{7},{8}}
=> {{1},{2},{3},{4,5},{6},{7},{8}}
=> {{1},{2},{3},{4,5},{6},{7},{8}}
=> ? = 8
[[1,4],[2],[3],[5],[6],[7],[8]]
=> {{1,4},{2},{3},{5},{6},{7},{8}}
=> {{1},{2},{3,4},{5},{6},{7},{8}}
=> ?
=> ? = 8
[[1,3],[2],[4],[5],[6],[7],[8]]
=> {{1,3},{2},{4},{5},{6},{7},{8}}
=> {{1},{2,3},{4},{5},{6},{7},{8}}
=> {{1},{2,3},{4},{5},{6},{7},{8}}
=> ? = 8
[[1,2],[3],[4],[5],[6],[7],[8]]
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 8
[[1,3,5,7,9,11],[2,4,6,8,10,12]]
=> {{1,3,5,7,9,11},{2,4,6,8,10,12}}
=> {{1},{2,3,4,5,6,7,8,9,10,11,12}}
=> ?
=> ? = 2
Description
The largest opener of a set partition.
An opener (or left hand endpoint) of a set partition is a number that is minimal in its block. For this statistic, singletons are considered as openers.
Matching statistic: St000025
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 78% ●values known / values provided: 87%●distinct values known / distinct values provided: 78%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 78% ●values known / values provided: 87%●distinct values known / distinct values provided: 78%
Values
[[1]]
=> [1] => [1,0]
=> 1
[[1,2]]
=> [1,2] => [1,0,1,0]
=> 1
[[1],[2]]
=> [2,1] => [1,1,0,0]
=> 2
[[1,2,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> 1
[[1,3],[2]]
=> [2,1,3] => [1,1,0,0,1,0]
=> 2
[[1,2],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 3
[[1],[2],[3]]
=> [3,2,1] => [1,1,1,0,0,0]
=> 3
[[1,2,3,4]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1
[[1,3,4],[2]]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2
[[1,2,4],[3]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 3
[[1,2,3],[4]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 4
[[1,3],[2,4]]
=> [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 2
[[1,2],[3,4]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 3
[[1,4],[2],[3]]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 3
[[1,3],[2],[4]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 4
[[1,2],[3],[4]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 4
[[1],[2],[3],[4]]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 1
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 2
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> 2
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> 3
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> 2
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> 3
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
=> 3
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 4
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> 4
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 2
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> 3
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 4
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 5
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [1,1,0,1,1,0,0,0,1,0,1,0]
=> 2
[[1,3,5,7],[2,4,6,8]]
=> [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 2
[[1,2,5,7],[3,4,6,8]]
=> [3,4,6,8,1,2,5,7] => [1,1,1,0,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 3
[[1,3,4,7],[2,5,6,8]]
=> [2,5,6,8,1,3,4,7] => [1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 2
[[1,2,4,7],[3,5,6,8]]
=> [3,5,6,8,1,2,4,7] => [1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 3
[[1,2,3,7],[4,5,6,8]]
=> [4,5,6,8,1,2,3,7] => [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 4
[[1,3,5,6],[2,4,7,8]]
=> [2,4,7,8,1,3,5,6] => [1,1,0,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> ? = 2
[[1,2,5,6],[3,4,7,8]]
=> [3,4,7,8,1,2,5,6] => [1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0]
=> ? = 3
[[1,3,4,6],[2,5,7,8]]
=> [2,5,7,8,1,3,4,6] => [1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 2
[[1,2,4,6],[3,5,7,8]]
=> [3,5,7,8,1,2,4,6] => [1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 3
[[1,2,3,6],[4,5,7,8]]
=> [4,5,7,8,1,2,3,6] => [1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 4
[[1,3,4,5],[2,6,7,8]]
=> [2,6,7,8,1,3,4,5] => [1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 2
[[1,2,4,5],[3,6,7,8]]
=> [3,6,7,8,1,2,4,5] => [1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 3
[[1,2,3,5],[4,6,7,8]]
=> [4,6,7,8,1,2,3,5] => [1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 4
[[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 5
[[1,5,6,8],[2],[3],[4],[7]]
=> [7,4,3,2,1,5,6,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7
[[1,4,8],[2,5],[3,6],[7]]
=> [7,3,6,2,5,1,4,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7
[[1,5,8],[2,6],[3],[4],[7]]
=> [7,4,3,2,6,1,5,8] => ?
=> ? = 7
[[1,6,8],[2],[3],[4],[5],[7]]
=> [7,5,4,3,2,1,6,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7
[[1,5,8],[2],[3],[4],[6],[7]]
=> [7,6,4,3,2,1,5,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7
[[1,4,8],[2],[3],[5],[6],[7]]
=> [7,6,5,3,2,1,4,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7
[[1,2,4],[3],[5],[6],[7],[8]]
=> [8,7,6,5,3,1,2,4] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[1,2,3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,1,2,3] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[1,7],[2,8],[3],[4],[5],[6]]
=> [6,5,4,3,2,8,1,7] => [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> ? = 6
[[1,6],[2,8],[3],[4],[5],[7]]
=> [7,5,4,3,2,8,1,6] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> ? = 7
[[1,5],[2,8],[3],[4],[6],[7]]
=> [7,6,4,3,2,8,1,5] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> ? = 7
[[1,4],[2,8],[3],[5],[6],[7]]
=> [7,6,5,3,2,8,1,4] => ?
=> ? = 7
[[1,3],[2,8],[4],[5],[6],[7]]
=> [7,6,5,4,2,8,1,3] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> ? = 7
[[1,2],[3,8],[4],[5],[6],[7]]
=> [7,6,5,4,3,8,1,2] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> ? = 7
[[1,6],[2,7],[3],[4],[5],[8]]
=> [8,5,4,3,2,7,1,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[1,5],[2,7],[3],[4],[6],[8]]
=> [8,6,4,3,2,7,1,5] => ?
=> ? = 8
[[1,4],[2,7],[3],[5],[6],[8]]
=> [8,6,5,3,2,7,1,4] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[1,3],[2,7],[4],[5],[6],[8]]
=> [8,6,5,4,2,7,1,3] => ?
=> ? = 8
[[1,2],[3,7],[4],[5],[6],[8]]
=> [8,6,5,4,3,7,1,2] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[1,5],[2,6],[3],[4],[7],[8]]
=> [8,7,4,3,2,6,1,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[1,4],[2,6],[3],[5],[7],[8]]
=> [8,7,5,3,2,6,1,4] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[1,3],[2,6],[4],[5],[7],[8]]
=> [8,7,5,4,2,6,1,3] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[1,2],[3,6],[4],[5],[7],[8]]
=> [8,7,5,4,3,6,1,2] => ?
=> ? = 8
[[1,4],[2,5],[3],[6],[7],[8]]
=> [8,7,6,3,2,5,1,4] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[1,3],[2,5],[4],[6],[7],[8]]
=> [8,7,6,4,2,5,1,3] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[1,2],[3,5],[4],[6],[7],[8]]
=> [8,7,6,4,3,5,1,2] => ?
=> ? = 8
[[1,3],[2,4],[5],[6],[7],[8]]
=> [8,7,6,5,2,4,1,3] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[1,2],[3,4],[5],[6],[7],[8]]
=> [8,7,6,5,3,4,1,2] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[1,8],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7
[[1,7],[2],[3],[4],[5],[6],[8]]
=> [8,6,5,4,3,2,1,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[1,6],[2],[3],[4],[5],[7],[8]]
=> [8,7,5,4,3,2,1,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[1,5],[2],[3],[4],[6],[7],[8]]
=> [8,7,6,4,3,2,1,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[1,4],[2],[3],[5],[6],[7],[8]]
=> [8,7,6,5,3,2,1,4] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[1,3],[2],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,2,1,3] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[1,2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,1,2] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[1],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
Description
The number of initial rises of a Dyck path.
In other words, this is the height of the first peak of $D$.
Matching statistic: St000026
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000026: Dyck paths ⟶ ℤResult quality: 78% ●values known / values provided: 87%●distinct values known / distinct values provided: 78%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000026: Dyck paths ⟶ ℤResult quality: 78% ●values known / values provided: 87%●distinct values known / distinct values provided: 78%
Values
[[1]]
=> [1] => [.,.]
=> [1,0]
=> 1
[[1,2]]
=> [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 1
[[1],[2]]
=> [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 2
[[1,2,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 1
[[1,3],[2]]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 2
[[1,2],[3]]
=> [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 3
[[1],[2],[3]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 3
[[1,2,3,4]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 1
[[1,3,4],[2]]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 2
[[1,2,4],[3]]
=> [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 3
[[1,2,3],[4]]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 4
[[1,3],[2,4]]
=> [2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[[1,2],[3,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 3
[[1,4],[2],[3]]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 3
[[1,3],[2],[4]]
=> [4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 4
[[1,2],[3],[4]]
=> [4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> 4
[[1],[2],[3],[4]]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> 4
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 4
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> 5
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 4
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> 4
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [[[.,.],[.,[.,.]]],.]
=> [1,1,1,0,0,1,0,1,0,0]
=> 5
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [[[.,[.,.]],[.,.]],.]
=> [1,1,1,0,1,0,0,1,0,0]
=> 5
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,1,0,1,0,0,0]
=> 5
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> 4
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [[[.,.],[[.,.],.]],.]
=> [1,1,1,0,0,1,1,0,0,0]
=> 5
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [[[.,[.,.]],[.,.]],.]
=> [1,1,1,0,1,0,0,1,0,0]
=> 5
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [[[[.,.],.],[.,.]],.]
=> [1,1,1,1,0,0,0,1,0,0]
=> 5
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [[[[.,.],[.,.]],.],.]
=> [1,1,1,1,0,0,1,0,0,0]
=> 5
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [[[[.,[.,.]],.],.],.]
=> [1,1,1,1,0,1,0,0,0,0]
=> 5
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [[.,.],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 2
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [[.,[.,.]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 3
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [[.,[.,[.,.]]],[.,[.,.]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 4
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [[.,[.,[.,[.,.]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 5
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [[.,[.,[.,[.,[.,.]]]]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 6
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [[.,.],[[.,.],[.,[.,.]]]]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 2
[[1,3,5,7],[2,4,6,8]]
=> [2,4,6,8,1,3,5,7] => [[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 2
[[1,2,5,7],[3,4,6,8]]
=> [3,4,6,8,1,2,5,7] => [[.,[.,.]],[.,[[.,.],[[.,.],.]]]]
=> [1,1,0,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3
[[1,3,4,7],[2,5,6,8]]
=> [2,5,6,8,1,3,4,7] => [[.,.],[[.,[.,.]],[.,[[.,.],.]]]]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> ? = 2
[[1,2,4,7],[3,5,6,8]]
=> [3,5,6,8,1,2,4,7] => [[.,[.,.]],[[.,.],[.,[[.,.],.]]]]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3
[[1,2,3,7],[4,5,6,8]]
=> [4,5,6,8,1,2,3,7] => [[.,[.,[.,.]]],[.,[.,[[.,.],.]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 4
[[1,3,5,6],[2,4,7,8]]
=> [2,4,7,8,1,3,5,6] => [[.,.],[[.,.],[[.,[.,.]],[.,.]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 2
[[1,2,5,6],[3,4,7,8]]
=> [3,4,7,8,1,2,5,6] => [[.,[.,.]],[.,[[.,[.,.]],[.,.]]]]
=> [1,1,0,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> ? = 3
[[1,3,4,6],[2,5,7,8]]
=> [2,5,7,8,1,3,4,6] => [[.,.],[[.,[.,.]],[[.,.],[.,.]]]]
=> [1,1,0,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 2
[[1,2,4,6],[3,5,7,8]]
=> [3,5,7,8,1,2,4,6] => [[.,[.,.]],[[.,.],[[.,.],[.,.]]]]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3
[[1,2,3,6],[4,5,7,8]]
=> [4,5,7,8,1,2,3,6] => [[.,[.,[.,.]]],[.,[[.,.],[.,.]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 4
[[1,3,4,5],[2,6,7,8]]
=> [2,6,7,8,1,3,4,5] => [[.,.],[[.,[.,[.,.]]],[.,[.,.]]]]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 2
[[1,2,4,5],[3,6,7,8]]
=> [3,6,7,8,1,2,4,5] => [[.,[.,.]],[[.,[.,.]],[.,[.,.]]]]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 3
[[1,2,3,5],[4,6,7,8]]
=> [4,6,7,8,1,2,3,5] => [[.,[.,[.,.]]],[[.,.],[.,[.,.]]]]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 4
[[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 5
[[1,5,6,8],[2],[3],[4],[7]]
=> [7,4,3,2,1,5,6,8] => [[[[[.,.],.],.],[.,[.,.]]],[.,.]]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
=> ? = 7
[[1,4,8],[2,5],[3,6],[7]]
=> [7,3,6,2,5,1,4,8] => [[[[.,.],.],[[[.,.],.],.]],[.,.]]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0,1,0]
=> ? = 7
[[1,5,8],[2,6],[3],[4],[7]]
=> [7,4,3,2,6,1,5,8] => ?
=> ?
=> ? = 7
[[1,6,8],[2],[3],[4],[5],[7]]
=> [7,5,4,3,2,1,6,8] => [[[[[[.,.],.],.],.],[.,.]],[.,.]]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> ? = 7
[[1,5,8],[2],[3],[4],[6],[7]]
=> [7,6,4,3,2,1,5,8] => [[[[[[.,.],.],.],[.,.]],.],[.,.]]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> ? = 7
[[1,4,8],[2],[3],[5],[6],[7]]
=> [7,6,5,3,2,1,4,8] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> ? = 7
[[1,2,4],[3],[5],[6],[7],[8]]
=> [8,7,6,5,3,1,2,4] => [[[[[[.,[.,.]],[.,.]],.],.],.],.]
=> [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> ? = 8
[[1,2,3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,1,2,3] => [[[[[[.,[.,[.,.]]],.],.],.],.],.]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 8
[[1,7],[2,8],[3],[4],[5],[6]]
=> [6,5,4,3,2,8,1,7] => [[[[[[.,.],.],.],.],.],[[.,.],.]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 6
[[1,6],[2,8],[3],[4],[5],[7]]
=> [7,5,4,3,2,8,1,6] => [[[[[[.,.],.],.],.],[.,.]],[.,.]]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> ? = 7
[[1,5],[2,8],[3],[4],[6],[7]]
=> [7,6,4,3,2,8,1,5] => ?
=> ?
=> ? = 7
[[1,4],[2,8],[3],[5],[6],[7]]
=> [7,6,5,3,2,8,1,4] => ?
=> ?
=> ? = 7
[[1,3],[2,8],[4],[5],[6],[7]]
=> [7,6,5,4,2,8,1,3] => [[[[[[.,.],[.,.]],.],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> ? = 7
[[1,2],[3,8],[4],[5],[6],[7]]
=> [7,6,5,4,3,8,1,2] => [[[[[[.,[.,.]],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 7
[[1,6],[2,7],[3],[4],[5],[8]]
=> [8,5,4,3,2,7,1,6] => [[[[[[.,.],.],.],.],[[.,.],.]],.]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> ? = 8
[[1,5],[2,7],[3],[4],[6],[8]]
=> [8,6,4,3,2,7,1,5] => ?
=> ?
=> ? = 8
[[1,4],[2,7],[3],[5],[6],[8]]
=> [8,6,5,3,2,7,1,4] => ?
=> ?
=> ? = 8
[[1,3],[2,7],[4],[5],[6],[8]]
=> [8,6,5,4,2,7,1,3] => ?
=> ?
=> ? = 8
[[1,2],[3,7],[4],[5],[6],[8]]
=> [8,6,5,4,3,7,1,2] => [[[[[[.,[.,.]],.],.],.],[.,.]],.]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
=> ? = 8
[[1,5],[2,6],[3],[4],[7],[8]]
=> [8,7,4,3,2,6,1,5] => [[[[[[.,.],.],.],[[.,.],.]],.],.]
=> [1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
=> ? = 8
[[1,4],[2,6],[3],[5],[7],[8]]
=> [8,7,5,3,2,6,1,4] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0]
=> ? = 8
[[1,3],[2,6],[4],[5],[7],[8]]
=> [8,7,5,4,2,6,1,3] => [[[[[[.,.],[.,.]],.],[.,.]],.],.]
=> [1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0]
=> ? = 8
[[1,2],[3,6],[4],[5],[7],[8]]
=> [8,7,5,4,3,6,1,2] => ?
=> ?
=> ? = 8
[[1,4],[2,5],[3],[6],[7],[8]]
=> [8,7,6,3,2,5,1,4] => [[[[[[.,.],.],[[.,.],.]],.],.],.]
=> [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> ? = 8
[[1,3],[2,5],[4],[6],[7],[8]]
=> [8,7,6,4,2,5,1,3] => [[[[[[.,.],[.,.]],[.,.]],.],.],.]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> ? = 8
[[1,2],[3,5],[4],[6],[7],[8]]
=> [8,7,6,4,3,5,1,2] => ?
=> ?
=> ? = 8
[[1,3],[2,4],[5],[6],[7],[8]]
=> [8,7,6,5,2,4,1,3] => [[[[[[.,.],[[.,.],.]],.],.],.],.]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> ? = 8
[[1,2],[3,4],[5],[6],[7],[8]]
=> [8,7,6,5,3,4,1,2] => [[[[[[.,[.,.]],[.,.]],.],.],.],.]
=> [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> ? = 8
[[1,8],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7
[[1,7],[2],[3],[4],[5],[6],[8]]
=> [8,6,5,4,3,2,1,7] => [[[[[[[.,.],.],.],.],.],[.,.]],.]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 8
[[1,6],[2],[3],[4],[5],[7],[8]]
=> [8,7,5,4,3,2,1,6] => [[[[[[[.,.],.],.],.],[.,.]],.],.]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> ? = 8
[[1,5],[2],[3],[4],[6],[7],[8]]
=> [8,7,6,4,3,2,1,5] => [[[[[[[.,.],.],.],[.,.]],.],.],.]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> ? = 8
[[1,4],[2],[3],[5],[6],[7],[8]]
=> [8,7,6,5,3,2,1,4] => [[[[[[[.,.],.],[.,.]],.],.],.],.]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> ? = 8
[[1,3],[2],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,2,1,3] => [[[[[[[.,.],[.,.]],.],.],.],.],.]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 8
[[1,2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,1,2] => [[[[[[[.,[.,.]],.],.],.],.],.],.]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 8
[[1],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],.]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
Description
The position of the first return of a Dyck path.
The following 23 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000054The first entry of the permutation. St000740The last entry of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000989The number of final rises of a permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000297The number of leading ones in a binary word. St000542The number of left-to-right-minima of a permutation. St001497The position of the largest weak excedence of a permutation. St000051The size of the left subtree of a binary tree. St000991The number of right-to-left minima of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. 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. St000061The number of nodes on the left branch of a binary tree. St000739The first entry in the last row of a semistandard tableau. St000736The last entry in the first row of a semistandard tableau. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one.
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