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Your data matches 162 different statistics following compositions of up to 3 maps.
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Matching statistic: St001232
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1,0]
=> 0
[1,0,1,0]
=> [1,1] => [1,0,1,0]
=> 1
[1,1,0,0]
=> [2] => [1,1,0,0]
=> 0
[1,0,1,1,0,0]
=> [1,2] => [1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0]
=> [2,1] => [1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> [2,1] => [1,1,0,0,1,0]
=> 1
[1,1,1,0,0,0]
=> [3] => [1,1,1,0,0,0]
=> 0
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3
[1,0,1,1,0,1,0,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[1,1,0,0,1,1,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,1,1,1,0,0,0,0,0]
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 5
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 5
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 5
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 5
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 5
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000738
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [[1]]
=> 1 = 0 + 1
[1,0,1,0]
=> [2,1] => [[1],[2]]
=> 2 = 1 + 1
[1,1,0,0]
=> [1,2] => [[1,2]]
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [2,3,1] => [[1,2],[3]]
=> 3 = 2 + 1
[1,1,0,0,1,0]
=> [3,1,2] => [[1,3],[2]]
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [2,1,3] => [[1,3],[2]]
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,2,3] => [[1,2,3]]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [[1,3],[2],[4]]
=> 4 = 3 + 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [[1,3],[2],[4]]
=> 4 = 3 + 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [[1,2,3],[4]]
=> 4 = 3 + 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [[1,2],[3,4]]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [[1,2,4],[3]]
=> 3 = 2 + 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [[1,3,4],[2]]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [[1,3,4],[2]]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [[1,3,4],[2]]
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [[1,2,3,4]]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [[1,2],[3,4],[5]]
=> 5 = 4 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [[1,2,4],[3],[5]]
=> 5 = 4 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [[1,3,4],[2],[5]]
=> 5 = 4 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [[1,3,4],[2],[5]]
=> 5 = 4 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [[1,3,4],[2],[5]]
=> 5 = 4 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [[1,2,3,4],[5]]
=> 5 = 4 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [[1,3],[2,5],[4]]
=> 4 = 3 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [[1,3],[2,5],[4]]
=> 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [[1,2,3],[4,5]]
=> 4 = 3 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [[1,3,5],[2],[4]]
=> 4 = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [[1,3,5],[2],[4]]
=> 4 = 3 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [[1,3,5],[2],[4]]
=> 4 = 3 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [[1,2,3,5],[4]]
=> 4 = 3 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [[1,2,5],[3,4]]
=> 3 = 2 + 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [[1,2,5],[3,4]]
=> 3 = 2 + 1
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [[1,2,4,5],[3]]
=> 3 = 2 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [[1,3,4,5],[2]]
=> 2 = 1 + 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [[1,3,4,5],[2]]
=> 2 = 1 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [[1,3,4,5],[2]]
=> 2 = 1 + 1
[1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => [[1,3,4,5],[2]]
=> 2 = 1 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [[1,2,3,4,5]]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,3,1] => [[1,3],[2,5],[4],[6]]
=> 6 = 5 + 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,6,2,3,1] => [[1,3],[2,5],[4],[6]]
=> 6 = 5 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,5,6,2,3,1] => [[1,2,3],[4,5],[6]]
=> 6 = 5 + 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [6,3,4,2,5,1] => [[1,3,5],[2],[4],[6]]
=> 6 = 5 + 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [5,3,4,2,6,1] => [[1,3,5],[2],[4],[6]]
=> 6 = 5 + 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [4,3,5,2,6,1] => [[1,3,5],[2],[4],[6]]
=> 6 = 5 + 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,4,5,2,6,1] => [[1,2,3,5],[4],[6]]
=> 6 = 5 + 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,6,2,3,4,1] => [[1,2,5],[3,4],[6]]
=> 6 = 5 + 1
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [4,5,2,3,6,1] => [[1,2,5],[3,4],[6]]
=> 6 = 5 + 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => [[1,2,4,5],[3],[6]]
=> 6 = 5 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,5,1] => [[1,3,4,5],[2],[6]]
=> 6 = 5 + 1
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [5,2,3,4,6,1] => [[1,3,4,5],[2],[6]]
=> 6 = 5 + 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => [[1,3,4,5],[2],[6]]
=> 6 = 5 + 1
Description
The first entry in the last row of a standard tableau.
For the last entry in the first row, see [[St000734]].
Matching statistic: St000839
(load all 25 compositions to match this statistic)
(load all 25 compositions to match this statistic)
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00112: Set partitions —complement⟶ Set partitions
St000839: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00112: Set partitions —complement⟶ Set partitions
St000839: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> {{1}}
=> {{1}}
=> 1 = 0 + 1
[1,0,1,0]
=> {{1},{2}}
=> {{1},{2}}
=> 2 = 1 + 1
[1,1,0,0]
=> {{1,2}}
=> {{1,2}}
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> {{1},{2,3}}
=> {{1,2},{3}}
=> 3 = 2 + 1
[1,1,0,0,1,0]
=> {{1,2},{3}}
=> {{1},{2,3}}
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> {{1,3},{2}}
=> {{1,3},{2}}
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> {{1,2,3}}
=> {{1,2,3}}
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> 4 = 3 + 1
[1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> {{1,3},{2},{4}}
=> 4 = 3 + 1
[1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> {{1,2,3},{4}}
=> 4 = 3 + 1
[1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> {{1,2},{3,4}}
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> {{1,2,4},{3}}
=> 3 = 2 + 1
[1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> {{1},{2,3,4}}
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> {{1,4},{2,3}}
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> {{1,3,4},{2}}
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> {{1,2},{3,4},{5}}
=> 5 = 4 + 1
[1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> {{1,2,4},{3},{5}}
=> 5 = 4 + 1
[1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> {{1},{2,3,4},{5}}
=> 5 = 4 + 1
[1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> {{1,4},{2,3},{5}}
=> 5 = 4 + 1
[1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> {{1,3,4},{2},{5}}
=> 5 = 4 + 1
[1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> {{1,2,3,4},{5}}
=> 5 = 4 + 1
[1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> {{1},{2,3},{4,5}}
=> 4 = 3 + 1
[1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> {{1,3},{2},{4,5}}
=> 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> {{1,2,3},{4,5}}
=> 4 = 3 + 1
[1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> {{1},{2,3,5},{4}}
=> 4 = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
=> {{1,5},{2},{3,4}}
=> {{1,5},{2,3},{4}}
=> 4 = 3 + 1
[1,1,0,1,1,0,1,0,0,0]
=> {{1,3,5},{2},{4}}
=> {{1,3,5},{2},{4}}
=> 4 = 3 + 1
[1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> {{1,2,3,5},{4}}
=> 4 = 3 + 1
[1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> {{1,2},{3,4,5}}
=> 3 = 2 + 1
[1,1,1,0,0,1,1,0,0,0]
=> {{1,4,5},{2,3}}
=> {{1,2,5},{3,4}}
=> 3 = 2 + 1
[1,1,1,0,1,1,0,0,0,0]
=> {{1,2,4,5},{3}}
=> {{1,2,4,5},{3}}
=> 3 = 2 + 1
[1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> {{1},{2,3,4,5}}
=> 2 = 1 + 1
[1,1,1,1,0,0,0,1,0,0]
=> {{1,5},{2,3,4}}
=> {{1,5},{2,3,4}}
=> 2 = 1 + 1
[1,1,1,1,0,0,1,0,0,0]
=> {{1,2,5},{3,4}}
=> {{1,4,5},{2,3}}
=> 2 = 1 + 1
[1,1,1,1,0,1,0,0,0,0]
=> {{1,2,3,5},{4}}
=> {{1,3,4,5},{2}}
=> 2 = 1 + 1
[1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> {{1},{2,3},{4,5},{6}}
=> {{1},{2,3},{4,5},{6}}
=> 6 = 5 + 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> {{1},{2,3},{4,6},{5}}
=> {{1,3},{2},{4,5},{6}}
=> 6 = 5 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> {{1},{2,3},{4,5,6}}
=> {{1,2,3},{4,5},{6}}
=> 6 = 5 + 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> {{1},{2,4,5},{3},{6}}
=> {{1},{2,3,5},{4},{6}}
=> 6 = 5 + 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> {{1},{2,6},{3},{4,5}}
=> {{1,5},{2,3},{4},{6}}
=> 6 = 5 + 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> {{1},{2,4,6},{3},{5}}
=> {{1,3,5},{2},{4},{6}}
=> 6 = 5 + 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> {{1},{2,4,5,6},{3}}
=> {{1,2,3,5},{4},{6}}
=> 6 = 5 + 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> {{1},{2,3,4},{5,6}}
=> {{1,2},{3,4,5},{6}}
=> 6 = 5 + 1
[1,0,1,1,1,0,0,1,1,0,0,0]
=> {{1},{2,5,6},{3,4}}
=> {{1,2,5},{3,4},{6}}
=> 6 = 5 + 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> {{1},{2,3,5,6},{4}}
=> {{1,2,4,5},{3},{6}}
=> 6 = 5 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> {{1},{2,3,4,5},{6}}
=> {{1},{2,3,4,5},{6}}
=> 6 = 5 + 1
[1,0,1,1,1,1,0,0,0,1,0,0]
=> {{1},{2,6},{3,4,5}}
=> {{1,5},{2,3,4},{6}}
=> 6 = 5 + 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> {{1},{2,3,6},{4,5}}
=> {{1,4,5},{2,3},{6}}
=> 6 = 5 + 1
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: St000009
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000009: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000009: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [[1]]
=> 0
[1,0,1,0]
=> [1,2] => [1,2] => [[1,2]]
=> 1
[1,1,0,0]
=> [2,1] => [2,1] => [[1],[2]]
=> 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [[1,2],[3]]
=> 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [[1,3],[2]]
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => [[1,3],[2]]
=> 1
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => [[1],[2],[3]]
=> 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,4,3,2] => [[1,2],[3],[4]]
=> 3
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => [[1,2],[3],[4]]
=> 3
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,3,2] => [[1,2],[3],[4]]
=> 3
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [[1,3],[2,4]]
=> 2
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,4,3,1] => [[1,3],[2],[4]]
=> 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => [[1,4],[2],[3]]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => [[1,4],[2],[3]]
=> 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,4,2,1] => [[1,4],[2],[3]]
=> 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,5,4,3] => [[1,3],[2,4],[5]]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [[1,3],[2,4],[5]]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,5,4,3] => [[1,3],[2,4],[5]]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,5,4,1,3] => [[1,3],[2,4],[5]]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,5,4,3,1] => [[1,3],[2],[4],[5]]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [2,5,4,3,1] => [[1,3],[2],[4],[5]]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,5,4,3,1] => [[1,3],[2],[4],[5]]
=> 3
[1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [3,2,1,5,4] => [[1,4],[2,5],[3]]
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => [3,2,5,4,1] => [[1,4],[2,5],[3]]
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [3,5,4,2,1] => [3,5,4,2,1] => [[1,4],[2],[3],[5]]
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [4,3,2,1,5] => [[1,5],[2],[3],[4]]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => [4,3,2,5,1] => [[1,5],[2],[3],[4]]
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [4,3,5,2,1] => [4,3,5,2,1] => [[1,5],[2],[3],[4]]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [4,5,3,2,1] => [4,5,3,2,1] => [[1,5],[2],[3],[4]]
=> 1
[1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 0
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,6] => [1,6,5,4,3,2] => [[1,2],[3],[4],[5],[6]]
=> 5
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,3,2,5,6,4] => [1,6,5,4,3,2] => [[1,2],[3],[4],[5],[6]]
=> 5
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,6,5,4] => [1,6,5,4,3,2] => [[1,2],[3],[4],[5],[6]]
=> 5
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,3,5,4,2,6] => [1,6,5,4,3,2] => [[1,2],[3],[4],[5],[6]]
=> 5
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,3,5,4,6,2] => [1,6,5,4,3,2] => [[1,2],[3],[4],[5],[6]]
=> 5
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,3,5,6,4,2] => [1,6,5,4,3,2] => [[1,2],[3],[4],[5],[6]]
=> 5
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,3,6,5,4,2] => [1,6,5,4,3,2] => [[1,2],[3],[4],[5],[6]]
=> 5
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,4,3,2,6,5] => [1,6,5,4,3,2] => [[1,2],[3],[4],[5],[6]]
=> 5
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,4,3,6,5,2] => [1,6,5,4,3,2] => [[1,2],[3],[4],[5],[6]]
=> 5
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,4,6,5,3,2] => [1,6,5,4,3,2] => [[1,2],[3],[4],[5],[6]]
=> 5
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,4,3,2,6] => [1,6,5,4,3,2] => [[1,2],[3],[4],[5],[6]]
=> 5
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,5,4,3,6,2] => [1,6,5,4,3,2] => [[1,2],[3],[4],[5],[6]]
=> 5
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,5,4,6,3,2] => [1,6,5,4,3,2] => [[1,2],[3],[4],[5],[6]]
=> 5
Description
The charge of a standard tableau.
Matching statistic: St000024
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1,0]
=> 0
[1,0,1,0]
=> [2,1] => [2,1] => [1,1,0,0]
=> 1
[1,1,0,0]
=> [1,2] => [1,2] => [1,0,1,0]
=> 0
[1,0,1,1,0,0]
=> [2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> 2
[1,1,0,0,1,0]
=> [3,1,2] => [2,3,1] => [1,1,0,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 3
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 3
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 3
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 3
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,3,1] => [6,4,5,2,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 5
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,6,2,3,1] => [6,4,5,2,1,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 5
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,5,6,2,3,1] => [6,4,5,1,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 5
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [6,3,4,2,5,1] => [6,4,2,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 5
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [5,3,4,2,6,1] => [6,4,2,3,1,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 5
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [4,3,5,2,6,1] => [6,4,2,1,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 5
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,4,5,2,6,1] => [6,4,1,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 5
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,6,2,3,4,1] => [6,3,4,5,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 5
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [4,5,2,3,6,1] => [6,3,4,1,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 5
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => [6,3,1,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 5
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,5,1] => [6,2,3,4,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 5
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [5,2,3,4,6,1] => [6,2,3,4,1,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 5
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => [6,2,3,1,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 5
Description
The number of double up and double down steps of a Dyck path.
In other words, this is the number of double rises (and, equivalently, the number of double falls) of a Dyck path.
Matching statistic: St000171
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000171: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000171: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => ([],1)
=> 0
[1,0,1,0]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[1,1,0,0]
=> [1,2] => [1,2] => ([],2)
=> 0
[1,0,1,1,0,0]
=> [2,3,1] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => ([(1,2)],3)
=> 1
[1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => ([],3)
=> 0
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,4,3] => ([(2,3)],4)
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
=> 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [5,1,2,4,3] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,3,1] => [6,1,5,2,4,3] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,6,2,3,1] => [6,1,5,3,2,4] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,5,6,2,3,1] => [6,1,5,2,3,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [6,3,4,2,5,1] => [6,4,1,2,5,3] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [5,3,4,2,6,1] => [6,4,1,3,2,5] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [4,3,5,2,6,1] => [6,4,2,1,3,5] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,4,5,2,6,1] => [6,4,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,6,2,3,4,1] => [6,1,2,5,3,4] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [4,5,2,3,6,1] => [6,1,4,2,3,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => [6,3,1,2,4,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,5,1] => [6,1,2,3,5,4] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [5,2,3,4,6,1] => [6,1,2,4,3,5] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => [6,1,3,2,4,5] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
Description
The degree of the graph.
This is the maximal vertex degree of a graph.
Matching statistic: St000394
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => [1,0]
=> 0
[1,0,1,0]
=> [1,1,0,0]
=> [2,1] => [1,1,0,0]
=> 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,2] => [1,0,1,0]
=> 0
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> 2
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => [1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => [1,0,1,1,0,0]
=> 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => [1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 3
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 3
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 3
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> 3
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [5,3,2,4,6,1] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> 5
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [5,2,4,3,6,1] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> 5
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [5,2,3,4,6,1] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> 5
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,6,4,5,1] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> 5
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,6,4,3,5,1] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,6,3,5,4,1] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [2,6,3,4,5,1] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,2,3,5,6,1] => [1,1,1,1,0,0,0,1,0,1,0,0]
=> 5
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [2,5,3,4,6,1] => [1,1,0,1,1,1,0,0,0,1,0,0]
=> 5
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,6,4,5,1] => [1,1,0,1,0,1,1,1,0,0,0,0]
=> 5
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,1] => [1,1,1,0,0,1,0,1,0,1,0,0]
=> 5
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,4,3,5,6,1] => [1,1,0,1,1,0,0,1,0,1,0,0]
=> 5
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,5,4,6,1] => [1,1,0,1,0,1,1,0,0,1,0,0]
=> 5
Description
The sum of the heights of the peaks of a Dyck path minus the number of peaks.
Matching statistic: St000645
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000645: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000645: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1,0]
=> 0
[1,0,1,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[1,1,0,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 0
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 0
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 3
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 3
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 3
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> 5
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> 5
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> 5
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> 5
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> 5
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> 5
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> 5
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> 5
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> 5
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> 5
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 5
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 5
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 5
Description
The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between.
For a Dyck path $D = D_1 \cdots D_{2n}$ with peaks in positions $i_1 < \ldots < i_k$ and valleys in positions $j_1 < \ldots < j_{k-1}$, this statistic is given by
$$
\sum_{a=1}^{k-1} (j_a-i_a)(i_{a+1}-j_a)
$$
Matching statistic: St000987
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000987: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000987: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => ([],1)
=> 0
[1,0,1,0]
=> [1,1,0,0]
=> [2,1] => ([(0,1)],2)
=> 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,2] => ([],2)
=> 0
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => ([(1,2)],3)
=> 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => ([(1,2)],3)
=> 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => ([],3)
=> 0
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 3
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 3
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
[1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => ([(2,3)],4)
=> 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(2,3)],4)
=> 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => ([],4)
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => ([(3,4)],5)
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => ([(3,4)],5)
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => ([(3,4)],5)
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => ([(3,4)],5)
=> 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => ([],5)
=> 0
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [4,1,5,2,6,3] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 5
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [3,5,1,2,6,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 5
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [3,4,5,1,6,2] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 5
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,1,5,6,2,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 5
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,5,1,6,3,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 5
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 5
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [2,4,5,6,1,3] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 5
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [3,4,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 5
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [2,4,5,1,6,3] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 5
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,5,6,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 5
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,4,5,6,2] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 5
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 5
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,5,1,6,4] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 5
Description
The number of positive eigenvalues of the Laplacian matrix of the graph.
This is the number of vertices minus the number of connected components of the graph.
Matching statistic: St001721
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St001721: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St001721: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 1 => 0
[1,0,1,0]
=> [1,1] => [2] => 10 => 1
[1,1,0,0]
=> [2] => [1,1] => 11 => 0
[1,0,1,1,0,0]
=> [1,2] => [2,1] => 101 => 2
[1,1,0,0,1,0]
=> [2,1] => [1,2] => 110 => 1
[1,1,0,1,0,0]
=> [2,1] => [1,2] => 110 => 1
[1,1,1,0,0,0]
=> [3] => [1,1,1] => 111 => 0
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,2] => 1010 => 3
[1,0,1,1,0,1,0,0]
=> [1,2,1] => [2,2] => 1010 => 3
[1,0,1,1,1,0,0,0]
=> [1,3] => [2,1,1] => 1011 => 3
[1,1,0,0,1,1,0,0]
=> [2,2] => [1,2,1] => 1101 => 2
[1,1,0,1,1,0,0,0]
=> [2,2] => [1,2,1] => 1101 => 2
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,1,2] => 1110 => 1
[1,1,1,0,0,1,0,0]
=> [3,1] => [1,1,2] => 1110 => 1
[1,1,1,0,1,0,0,0]
=> [3,1] => [1,1,2] => 1110 => 1
[1,1,1,1,0,0,0,0]
=> [4] => [1,1,1,1] => 1111 => 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1] => 10101 => 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [2,2,1] => 10101 => 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [2,1,2] => 10110 => 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [2,1,2] => 10110 => 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [2,1,2] => 10110 => 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [2,1,1,1] => 10111 => 4
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [1,2,2] => 11010 => 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [1,2,2] => 11010 => 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,2,1,1] => 11011 => 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [1,2,2] => 11010 => 3
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [1,2,2] => 11010 => 3
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [1,2,2] => 11010 => 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [1,2,1,1] => 11011 => 3
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [1,1,2,1] => 11101 => 2
[1,1,1,0,0,1,1,0,0,0]
=> [3,2] => [1,1,2,1] => 11101 => 2
[1,1,1,0,1,1,0,0,0,0]
=> [3,2] => [1,1,2,1] => 11101 => 2
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [1,1,1,2] => 11110 => 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,1] => [1,1,1,2] => 11110 => 1
[1,1,1,1,0,0,1,0,0,0]
=> [4,1] => [1,1,1,2] => 11110 => 1
[1,1,1,1,0,1,0,0,0,0]
=> [4,1] => [1,1,1,2] => 11110 => 1
[1,1,1,1,1,0,0,0,0,0]
=> [5] => [1,1,1,1,1] => 11111 => 0
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,2,1] => [2,2,2] => 101010 => 5
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,2,1] => [2,2,2] => 101010 => 5
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,3] => [2,2,1,1] => 101011 => 5
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,2,2,1] => [2,2,2] => 101010 => 5
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,2,1] => [2,2,2] => 101010 => 5
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,2,1] => [2,2,2] => 101010 => 5
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,3] => [2,2,1,1] => 101011 => 5
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,3,2] => [2,1,2,1] => 101101 => 5
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,3,2] => [2,1,2,1] => 101101 => 5
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,3,2] => [2,1,2,1] => 101101 => 5
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,4,1] => [2,1,1,2] => 101110 => 5
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,4,1] => [2,1,1,2] => 101110 => 5
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,4,1] => [2,1,1,2] => 101110 => 5
Description
The degree of a binary word.
A valley in a binary word is a letter $0$ which is not immediately followed by a $1$. A peak is a letter $1$ which is not immediately followed by a $0$.
Let $f$ be the map that replaces every valley with a peak. The degree of a binary word $w$ is the number of times $f$ has to be applied to obtain a binary word without zeros.
The following 152 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000013The height of a Dyck path. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000093The cardinality of a maximal independent set of vertices of a graph. St000734The last entry in the first row of a standard tableau. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001809The index of the step at the first peak of maximal height in a Dyck path. St000439The position of the first down step of a Dyck path. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000730The maximal arc length of a set partition. St000874The position of the last double rise in a Dyck path. St000444The length of the maximal rise of a Dyck path. St000010The length of the partition. St000147The largest part of an integer partition. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001118The acyclic chromatic index of a graph. St000378The diagonal inversion number of an integer partition. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St000806The semiperimeter of the associated bargraph. St001725The harmonious chromatic 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. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St000054The first entry of the permutation. St000451The length of the longest pattern of the form k 1 2. St000996The number of exclusive left-to-right maxima of a permutation. St000288The number of ones in a binary word. St000733The row containing the largest entry of a standard tableau. 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. St000209Maximum difference of elements in cycles. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000845The maximal number of elements covered by an element in a poset. St000161The sum of the sizes of the right subtrees of a binary tree. St000676The number of odd rises of a Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St000675The number of centered multitunnels of a Dyck path. St000653The last descent of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000028The number of stack-sorts needed to sort a permutation. St001280The number of parts of an integer partition that are at least two. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000740The last entry of a permutation. St001497The position of the largest weak excedence of a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000007The number of saliances of the permutation. St000507The number of ascents of a standard tableau. St001777The number of weak descents in an integer composition. St000446The disorder of a permutation. St001480The number of simple summands of the module J^2/J^3. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001726The number of visible inversions of a permutation. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St000053The number of valleys of the Dyck path. St000019The cardinality of the support of a permutation. St000678The number of up steps after the last double rise of a Dyck path. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St000006The dinv of a Dyck path. St000651The maximal size of a rise in a permutation. St000809The reduced reflection length of the permutation. St001462The number of factors of a standard tableaux under concatenation. St000067The inversion number of the alternating sign matrix. St000015The number of peaks of a Dyck path. St000155The number of exceedances (also excedences) of a permutation. St000331The number of upper interactions of a Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St000216The absolute length of a permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000051The size of the left subtree of a binary tree. St000204The number of internal nodes of a binary tree. St000304The load of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000356The number of occurrences of the pattern 13-2. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St000240The number of indices that are not small excedances. St000692Babson and Steingrímsson's statistic of a permutation. St000030The sum of the descent differences of a permutations. St000133The "bounce" of a permutation. St000305The inverse major index of a permutation. St001117The game chromatic index of a graph. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001584The area statistic between a Dyck path and its bounce path. St000443The number of long tunnels of a Dyck path. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. 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. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001481The minimal height of a peak of a Dyck path. St000082The number of elements smaller than a binary tree in Tamari order. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St000105The number of blocks in the set partition. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000925The number of topologically connected components of a set partition. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St001152The number of pairs with even minimum in a perfect matching. St000354The number of recoils of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000831The number of indices that are either descents or recoils. St001061The number of indices that are both descents and recoils of a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000652The maximal difference between successive positions of a permutation. St001246The maximal difference between two consecutive entries of a permutation. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St000223The number of nestings in the permutation. St001136The largest label with larger sister in the leaf labelled binary unordered tree associated with the perfect matching. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000840The number of closers smaller than the largest opener in a perfect matching. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. 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. St000029The depth of a permutation. St000224The sorting index of a permutation. St001082The number of boxed occurrences of 123 in a permutation. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001434The number of negative sum pairs of a signed permutation. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001727The number of invisible inversions of a permutation. St000060The greater neighbor of the maximum. St000327The number of cover relations in a poset. St001330The hat guessing number of a graph. St000031The number of cycles in the cycle decomposition of a permutation. St001645The pebbling number of a connected graph. St001769The reflection length of a signed permutation. St001861The number of Bruhat lower covers of a permutation. St001894The depth of a signed permutation. St000898The number of maximal entries in the last diagonal of the monotone triangle. St000896The number of zeros on the main diagonal of an alternating sign matrix. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001866The nesting alignments of a signed permutation.
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