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Your data matches 328 different statistics following compositions of up to 3 maps.
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Matching statistic: St000839
(load all 102 compositions to match this statistic)
(load all 102 compositions to match this statistic)
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000839: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000839: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> {{1}}
=> 1
[1,0,1,0]
=> {{1},{2}}
=> 2
[1,1,0,0]
=> {{1,2}}
=> 1
[1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 3
[1,0,1,1,0,0]
=> {{1},{2,3}}
=> 2
[1,1,0,0,1,0]
=> {{1,2},{3}}
=> 3
[1,1,0,1,0,0]
=> {{1,3},{2}}
=> 2
[1,1,1,0,0,0]
=> {{1,2,3}}
=> 1
[1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 4
[1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 3
[1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 4
[1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 3
[1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2
[1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 4
[1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 3
[1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 4
[1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> 3
[1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 2
[1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 4
[1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 2
[1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 3
[1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> 5
[1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 5
[1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 5
[1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> {{1,3},{2},{4},{5}}
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> {{1,3},{2},{4,5}}
=> 4
[1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> 5
[1,1,0,1,0,1,0,1,0,0]
=> {{1,5},{2},{3},{4}}
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> {{1,4,5},{2},{3}}
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> 5
[1,1,0,1,1,0,0,1,0,0]
=> {{1,5},{2},{3,4}}
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> {{1,3,5},{2},{4}}
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> 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: St000013
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1,0]
=> 1
[1,0,1,0]
=> [2,1] => [1,1,0,0]
=> 2
[1,1,0,0]
=> [1,2] => [1,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [3,2,1] => [1,1,1,0,0,0]
=> 3
[1,0,1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> 2
[1,1,0,0,1,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> 3
[1,1,0,1,0,0]
=> [2,1,3] => [1,1,0,0,1,0]
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 3
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 4
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 3
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 4
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 3
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 4
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 3
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 4
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 3
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> 2
Description
The height of a Dyck path.
The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.
Matching statistic: St000025
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1,0]
=> 1
[1,0,1,0]
=> [2,1] => [1,1,0,0]
=> 2
[1,1,0,0]
=> [1,2] => [1,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [3,2,1] => [1,1,1,0,0,0]
=> 3
[1,0,1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> 2
[1,1,0,0,1,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> 3
[1,1,0,1,0,0]
=> [2,1,3] => [1,1,0,0,1,0]
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 3
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 4
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 3
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 4
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 3
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 4
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 3
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 4
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 3
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> 2
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
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St000026: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00142: Dyck paths —promotion⟶ Dyck paths
St000026: 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
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 3
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,1,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 4
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 4
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 4
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 5
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 5
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 5
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 4
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 5
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 5
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
Description
The position of the first return of a Dyck path.
Matching statistic: St000093
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000093: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000093: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => ([],1)
=> 1
[1,0,1,0]
=> [1,1] => ([(0,1)],2)
=> 1
[1,1,0,0]
=> [2] => ([],2)
=> 2
[1,0,1,0,1,0]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,0,1,1,0,0]
=> [1,2] => ([(1,2)],3)
=> 2
[1,1,0,0,1,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,1,0,1,0,0]
=> [3] => ([],3)
=> 3
[1,1,1,0,0,0]
=> [3] => ([],3)
=> 3
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,0,1,1,0,0,1,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,0,1,1,0,1,0,0]
=> [1,3] => ([(2,3)],4)
=> 3
[1,0,1,1,1,0,0,0]
=> [1,3] => ([(2,3)],4)
=> 3
[1,1,0,0,1,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,0,0,1,1,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> 3
[1,1,0,1,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,1,0,1,0,1,0,0]
=> [4] => ([],4)
=> 4
[1,1,0,1,1,0,0,0]
=> [4] => ([],4)
=> 4
[1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,1,1,0,0,1,0,0]
=> [4] => ([],4)
=> 4
[1,1,1,0,1,0,0,0]
=> [4] => ([],4)
=> 4
[1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => ([(3,4)],5)
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => ([(3,4)],5)
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 4
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [5] => ([],5)
=> 5
[1,1,0,1,0,1,1,0,0,0]
=> [5] => ([],5)
=> 5
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> [5] => ([],5)
=> 5
[1,1,0,1,1,0,1,0,0,0]
=> [5] => ([],5)
=> 5
[1,1,0,1,1,1,0,0,0,0]
=> [5] => ([],5)
=> 5
Description
The cardinality of a maximal independent set of vertices of a graph.
An independent set of a graph is a set of pairwise non-adjacent vertices. A maximum independent set is an independent set of maximum cardinality. This statistic is also called the independence number or stability number $\alpha(G)$ of $G$.
Matching statistic: St000734
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000734: 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
St000734: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [[1]]
=> 1
[1,0,1,0]
=> [1,2] => [[1,2]]
=> 2
[1,1,0,0]
=> [2,1] => [[1],[2]]
=> 1
[1,0,1,0,1,0]
=> [1,2,3] => [[1,2,3]]
=> 3
[1,0,1,1,0,0]
=> [1,3,2] => [[1,2],[3]]
=> 2
[1,1,0,0,1,0]
=> [2,1,3] => [[1,3],[2]]
=> 3
[1,1,0,1,0,0]
=> [2,3,1] => [[1,2],[3]]
=> 2
[1,1,1,0,0,0]
=> [3,2,1] => [[1],[2],[3]]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [[1,2,3,4]]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [[1,2,3],[4]]
=> 3
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [[1,2,4],[3]]
=> 4
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [[1,2,3],[4]]
=> 3
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [[1,2],[3],[4]]
=> 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [[1,3,4],[2]]
=> 4
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [[1,3],[2,4]]
=> 3
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [[1,2,4],[3]]
=> 4
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [[1,2,3],[4]]
=> 3
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [[1,2],[3],[4]]
=> 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [[1,4],[2],[3]]
=> 4
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [[1,3],[2],[4]]
=> 3
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [[1,2],[3],[4]]
=> 2
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [[1],[2],[3],[4]]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [[1,2,3,4,5]]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [[1,2,3,4],[5]]
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [[1,2,3],[4],[5]]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [[1,2,4,5],[3]]
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [[1,2,4],[3,5]]
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [[1,2,3,5],[4]]
=> 5
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [[1,2,3,4],[5]]
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [[1,2,3],[4],[5]]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [[1,2,5],[3],[4]]
=> 5
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [[1,2,4],[3],[5]]
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [[1,2,3],[4],[5]]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [[1,3,4,5],[2]]
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [[1,3,4],[2,5]]
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [[1,3,5],[2,4]]
=> 5
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [[1,3,4],[2,5]]
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [[1,3],[2,4],[5]]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [[1,2,4,5],[3]]
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [[1,2,4],[3,5]]
=> 4
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [[1,2,3,5],[4]]
=> 5
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [[1,2,3,4],[5]]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [[1,2,3],[4],[5]]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [[1,2,5],[3],[4]]
=> 5
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [[1,2,4],[3],[5]]
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [[1,2,3],[4],[5]]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [[1,2],[3],[4],[5]]
=> 2
Description
The last entry in the first row of a standard tableau.
Matching statistic: St000738
(load all 16 compositions to match this statistic)
(load all 16 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
[1,0,1,0]
=> [2,1] => [[1],[2]]
=> 2
[1,1,0,0]
=> [1,2] => [[1,2]]
=> 1
[1,0,1,0,1,0]
=> [3,2,1] => [[1],[2],[3]]
=> 3
[1,0,1,1,0,0]
=> [2,3,1] => [[1,2],[3]]
=> 3
[1,1,0,0,1,0]
=> [3,1,2] => [[1,3],[2]]
=> 2
[1,1,0,1,0,0]
=> [2,1,3] => [[1,3],[2]]
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => [[1,2,3]]
=> 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [[1],[2],[3],[4]]
=> 4
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [[1,2],[3],[4]]
=> 4
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [[1,3],[2],[4]]
=> 4
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [[1,3],[2],[4]]
=> 4
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [[1,2,3],[4]]
=> 4
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [[1,4],[2],[3]]
=> 3
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [[1,2],[3,4]]
=> 3
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [[1,4],[2],[3]]
=> 3
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [[1,4],[2],[3]]
=> 3
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [[1,2,4],[3]]
=> 3
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [[1,3,4],[2]]
=> 2
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [[1,3,4],[2]]
=> 2
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [[1,3,4],[2]]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [[1,2,3,4]]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [[1,2],[3],[4],[5]]
=> 5
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [[1,3],[2],[4],[5]]
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [[1,3],[2],[4],[5]]
=> 5
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [[1,2,3],[4],[5]]
=> 5
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [[1,4],[2],[3],[5]]
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [[1,2],[3,4],[5]]
=> 5
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [[1,4],[2],[3],[5]]
=> 5
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [[1,4],[2],[3],[5]]
=> 5
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [[1,2,4],[3],[5]]
=> 5
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [[1,3,4],[2],[5]]
=> 5
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [[1,3,4],[2],[5]]
=> 5
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [[1,3,4],[2],[5]]
=> 5
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [[1,2,3,4],[5]]
=> 5
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [[1,5],[2],[3],[4]]
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [[1,2],[3,5],[4]]
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [[1,3],[2,5],[4]]
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [[1,3],[2,5],[4]]
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [[1,2,3],[4,5]]
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [[1,5],[2],[3],[4]]
=> 4
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [[1,2],[3,5],[4]]
=> 4
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [[1,5],[2],[3],[4]]
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [[1,5],[2],[3],[4]]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [[1,2,5],[3],[4]]
=> 4
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [[1,3,5],[2],[4]]
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [[1,3,5],[2],[4]]
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [[1,3,5],[2],[4]]
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [[1,2,3,5],[4]]
=> 4
Description
The first entry in the last row of a standard tableau.
For the last entry in the first row, see [[St000734]].
Matching statistic: St000786
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000786: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000786: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => ([],1)
=> 1
[1,0,1,0]
=> [1,1] => ([(0,1)],2)
=> 1
[1,1,0,0]
=> [2] => ([],2)
=> 2
[1,0,1,0,1,0]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,0,1,1,0,0]
=> [1,2] => ([(1,2)],3)
=> 2
[1,1,0,0,1,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,1,0,1,0,0]
=> [3] => ([],3)
=> 3
[1,1,1,0,0,0]
=> [3] => ([],3)
=> 3
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,0,1,1,0,0,1,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,0,1,1,0,1,0,0]
=> [1,3] => ([(2,3)],4)
=> 3
[1,0,1,1,1,0,0,0]
=> [1,3] => ([(2,3)],4)
=> 3
[1,1,0,0,1,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,0,0,1,1,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> 3
[1,1,0,1,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,1,0,1,0,1,0,0]
=> [4] => ([],4)
=> 4
[1,1,0,1,1,0,0,0]
=> [4] => ([],4)
=> 4
[1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,1,1,0,0,1,0,0]
=> [4] => ([],4)
=> 4
[1,1,1,0,1,0,0,0]
=> [4] => ([],4)
=> 4
[1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => ([(3,4)],5)
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => ([(3,4)],5)
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 4
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [5] => ([],5)
=> 5
[1,1,0,1,0,1,1,0,0,0]
=> [5] => ([],5)
=> 5
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> [5] => ([],5)
=> 5
[1,1,0,1,1,0,1,0,0,0]
=> [5] => ([],5)
=> 5
[1,1,0,1,1,1,0,0,0,0]
=> [5] => ([],5)
=> 5
Description
The maximal number of occurrences of a colour in a proper colouring of a graph.
To any proper colouring with the minimal number of colours possible we associate the integer partition recording how often each colour is used. This statistic records the largest part occurring in any of these partitions.
For example, the graph on six vertices consisting of a square together with two attached triangles - ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) in the list of values - is three-colourable and admits two colouring schemes, $[2,2,2]$ and $[3,2,1]$. Therefore, the statistic on this graph is $3$.
Matching statistic: St001007
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001007: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001007: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1,0]
=> 1
[1,0,1,0]
=> [2,1] => [1,1,0,0]
=> 2
[1,1,0,0]
=> [1,2] => [1,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [3,2,1] => [1,1,1,0,0,0]
=> 3
[1,0,1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> 2
[1,1,0,0,1,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> 3
[1,1,0,1,0,0]
=> [2,1,3] => [1,1,0,0,1,0]
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 3
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 4
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 3
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 4
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 3
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 4
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 3
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 4
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 3
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> 2
Description
Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001809
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001809: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001809: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1,0]
=> 1
[1,0,1,0]
=> [2,1] => [1,1,0,0]
=> 2
[1,1,0,0]
=> [1,2] => [1,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [3,2,1] => [1,1,1,0,0,0]
=> 3
[1,0,1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> 2
[1,1,0,0,1,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> 3
[1,1,0,1,0,0]
=> [2,1,3] => [1,1,0,0,1,0]
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 3
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 4
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 3
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 4
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 3
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 4
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 3
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 4
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 3
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> 2
Description
The index of the step at the first peak of maximal height in a Dyck path.
The following 318 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000024The number of double up and double down steps of a Dyck path. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000439The position of the first down step of a Dyck path. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. 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. St000010The length of the partition. St000011The number of touch points (or returns) of a Dyck path. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000147The largest part of an integer partition. St000288The number of ones in a binary word. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000383The last part of an integer composition. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000505The biggest entry in the block containing the 1. St000507The number of ascents of a standard tableau. St000676The number of odd rises of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000808The number of up steps of the associated bargraph. St000971The smallest closer of a set partition. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001494The Alon-Tarsi number of a graph. St001581The achromatic number of a graph. St001733The number of weak left to right maxima of a Dyck path. St000009The charge of a standard tableau. St000053The number of valleys of the Dyck path. St000171The degree of the graph. St000211The rank of the set partition. St000377The dinv defect of an integer partition. St000521The number of distinct subtrees of an ordered tree. St000691The number of changes of a binary word. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001176The size of a partition minus its first part. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001479The number of bridges of a graph. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St000444The length of the maximal rise 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. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000503The maximal difference between two elements in a common block. St000728The dimension of a set partition. St000932The number of occurrences of the pattern UDU in a Dyck path. St001721The degree of a binary word. St000172The Grundy number of a graph. St001029The size of the core of a graph. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000308The height of the tree associated to a permutation. St001725The harmonious chromatic number of a graph. St000384The maximal part of the shifted composition of an integer partition. St000784The maximum of the length and the largest part of the integer partition. St000245The number of ascents of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000054The first entry of the permutation. St000451The length of the longest pattern of the form k 1 2. 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. St001268The size of the largest ordinal summand in the poset. St000996The number of exclusive left-to-right maxima of a permutation. St000740The last entry of a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000161The sum of the sizes of the right subtrees of a binary tree. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St000209Maximum difference of elements in cycles. 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. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St001497The position of the largest weak excedence of a permutation. St000653The last descent of a permutation. St000702The number of weak deficiencies of a permutation. St000662The staircase size of the code of a permutation. St000058The order of a permutation. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000019The cardinality of the support of a permutation. St000306The bounce count of a Dyck path. St000378The diagonal inversion number of an integer partition. St000028The number of stack-sorts needed to sort a permutation. St000012The area of a Dyck path. St000984The number of boxes below precisely one peak. 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. St000007The number of saliances of the permutation. St000081The number of edges of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001963The tree-depth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St000246The number of non-inversions of a permutation. St001298The number of repeated entries in the Lehmer code 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. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000454The largest eigenvalue of a graph if it is integral. St000446The disorder of a permutation. St000809The reduced reflection length of the permutation. St001330The hat guessing number of a graph. St001280The number of parts of an integer partition that are at least two. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000297The number of leading ones in a binary word. St000392The length of the longest run of ones in a binary word. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000982The length of the longest constant subword. St001372The length of a longest cyclic run of ones of a binary word. St000332The positive inversions of an alternating sign matrix. St001046The maximal number of arcs nesting a given arc of a perfect matching. St000651The maximal size of a rise in a permutation. St000485The length of the longest cycle of a permutation. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St000067The inversion number of the alternating sign matrix. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000497The lcb statistic of a set partition. St000572The dimension exponent of a set partition. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000031The number of cycles in the cycle decomposition of a permutation. St000225Difference between largest and smallest parts in a partition. St000470The number of runs in a permutation. St000501The size of the first part in the decomposition of a permutation. St000542The number of left-to-right-minima of a permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St000259The diameter of a connected graph. St000864The number of circled entries of the shifted recording tableau of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St000240The number of indices that are not small excedances. St000316The number of non-left-to-right-maxima of a permutation. St000443The number of long tunnels of a Dyck path. 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. St000051The size of the left subtree of a binary tree. St000133The "bounce" of a permutation. St000204The number of internal nodes of a binary tree. St000304The load of a permutation. St000356The number of occurrences of the pattern 13-2. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St000015The number of peaks of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000166The depth minus 1 of an ordered tree. St000325The width of the tree associated to a permutation. St000822The Hadwiger number of the graph. St000991The number of right-to-left minima of a permutation. 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. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001481The minimal height of a peak of a Dyck path. St001498The normalised height of a Nakayama algebra with magnitude 1. St001806The upper middle entry of a permutation. St000021The number of descents of a permutation. St000030The sum of the descent differences of a permutations. St000094The depth of an ordered tree. St000120The number of left tunnels of a Dyck path. St000155The number of exceedances (also excedences) of a permutation. St000168The number of internal nodes of an ordered tree. St000238The number of indices that are not small weak excedances. St000305The inverse major index of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000331The number of upper interactions of a Dyck path. 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. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000795The mad of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000931The number of occurrences of the pattern UUU in a Dyck path. St001077The prefix exchange distance of a permutation. St001117The game chromatic index of a graph. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001841The number of inversions of a set partition. St001869The maximum cut size of a graph. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001480The number of simple summands of the module J^2/J^3. St000692Babson and Steingrímsson's statistic of a permutation. St000216The absolute length of a permutation. St000061The number of nodes on the left branch of a binary tree. 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. St000083The number of left oriented leafs of a binary tree except the first one. St001674The number of vertices of the largest induced star graph in the graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001323The independence gap of a graph. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001645The pebbling number of a connected graph. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000143The largest repeated part of a partition. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. 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. St001717The largest size of an interval in a poset. St001812The biclique partition number of a graph. St001082The number of boxed occurrences of 123 in a permutation. St000898The number of maximal entries in the last diagonal of the monotone triangle. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000144The pyramid weight of the Dyck path. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St000840The number of closers smaller than the largest opener in a perfect matching. St001090The number of pop-stack-sorts needed to sort a permutation. St001136The largest label with larger sister in the leaf labelled binary unordered tree associated with the perfect matching. St001726The number of visible inversions of a permutation. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001115The number of even descents of a permutation. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000299The number of nonisomorphic vertex-induced subtrees. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000833The comajor index of a permutation. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000235The number of indices that are not cyclical small weak excedances. St000673The number of non-fixed points of a permutation. St000060The greater neighbor of the maximum. St000213The number of weak exceedances (also weak excedences) of a permutation. St001589The nesting number of a perfect matching. St000005The bounce statistic of a Dyck path. St000029The depth of a permutation. St000224The sorting index of a permutation. St000242The number of indices that are not cyclical small weak excedances. St000358The number of occurrences of the pattern 31-2. St000800The number of occurrences of the vincular pattern |231 in a permutation. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. 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. St001434The number of negative sum pairs of a signed permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001584The area statistic between a Dyck path and its bounce path. St001742The difference of the maximal and the minimal degree in a graph. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St000327The number of cover relations in a poset. St000744The length of the path to the largest entry in a standard Young tableau. St000528The height of a poset. St000912The number of maximal antichains in a poset. St001343The dimension of the reduced incidence algebra of a poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St000080The rank of the poset. St001782The order of rowmotion on the set of order ideals of a poset. St000680The Grundy value for Hackendot on posets. St000906The length of the shortest maximal chain in a poset. St000186The sum of the first row in a Gelfand-Tsetlin pattern. St000907The number of maximal antichains of minimal length in a poset. St000455The second largest eigenvalue of a graph if it is integral. St000896The number of zeros on the main diagonal of an alternating sign matrix. St001861The number of Bruhat lower covers of a permutation. St000264The girth of a graph, which is not a tree. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001060The distinguishing index of a graph. St000041The number of nestings of a perfect matching. St001394The genus of a permutation. St001596The number of two-by-two squares inside a skew partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001246The maximal difference between two consecutive entries of a permutation. St001769The reflection length of a signed permutation. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000115The single entry in the last row. St000052The number of valleys of a Dyck path not on the x-axis. St000422The energy of a graph, if it is integral. St000516The number of stretching pairs of a permutation. St000665The number of rafts of a permutation. St000871The number of very big ascents of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001209The pmaj statistic of a parking function. St001555The order of a signed permutation. St001557The number of inversions of the second entry of a permutation. St001760The number of prefix or suffix reversals needed to sort a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001877Number of indecomposable injective modules with projective dimension 2. St001960The number of descents of a permutation minus one if its first entry is not one. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000075The orbit size of a standard tableau under promotion. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St000742The number of big ascents of a permutation after prepending zero.
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