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Your data matches 84 different statistics following compositions of up to 3 maps.
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Matching statistic: St000745
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
St000745: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> 1 = 2 - 1
[[1,2]]
=> 1 = 2 - 1
[[1],[2]]
=> 2 = 3 - 1
[[1,2,3]]
=> 1 = 2 - 1
[[1,3],[2]]
=> 2 = 3 - 1
[[1,2],[3]]
=> 1 = 2 - 1
[[1],[2],[3]]
=> 3 = 4 - 1
[[1,2,3,4]]
=> 1 = 2 - 1
[[1,3,4],[2]]
=> 2 = 3 - 1
[[1,2,4],[3]]
=> 1 = 2 - 1
[[1,2,3],[4]]
=> 1 = 2 - 1
[[1,3],[2,4]]
=> 2 = 3 - 1
[[1,2],[3,4]]
=> 1 = 2 - 1
[[1,4],[2],[3]]
=> 3 = 4 - 1
[[1,3],[2],[4]]
=> 2 = 3 - 1
[[1,2],[3],[4]]
=> 1 = 2 - 1
[[1],[2],[3],[4]]
=> 4 = 5 - 1
[[1,2,3,4,5]]
=> 1 = 2 - 1
[[1,3,4,5],[2]]
=> 2 = 3 - 1
[[1,2,4,5],[3]]
=> 1 = 2 - 1
[[1,2,3,5],[4]]
=> 1 = 2 - 1
[[1,2,3,4],[5]]
=> 1 = 2 - 1
[[1,3,5],[2,4]]
=> 2 = 3 - 1
[[1,2,5],[3,4]]
=> 1 = 2 - 1
[[1,3,4],[2,5]]
=> 2 = 3 - 1
[[1,2,4],[3,5]]
=> 1 = 2 - 1
[[1,2,3],[4,5]]
=> 1 = 2 - 1
[[1,4,5],[2],[3]]
=> 3 = 4 - 1
[[1,3,5],[2],[4]]
=> 2 = 3 - 1
[[1,2,5],[3],[4]]
=> 1 = 2 - 1
[[1,3,4],[2],[5]]
=> 2 = 3 - 1
[[1,2,4],[3],[5]]
=> 1 = 2 - 1
[[1,2,3],[4],[5]]
=> 1 = 2 - 1
[[1,4],[2,5],[3]]
=> 3 = 4 - 1
[[1,3],[2,5],[4]]
=> 2 = 3 - 1
[[1,2],[3,5],[4]]
=> 1 = 2 - 1
[[1,3],[2,4],[5]]
=> 2 = 3 - 1
[[1,2],[3,4],[5]]
=> 1 = 2 - 1
[[1,5],[2],[3],[4]]
=> 4 = 5 - 1
[[1,4],[2],[3],[5]]
=> 3 = 4 - 1
[[1,3],[2],[4],[5]]
=> 2 = 3 - 1
[[1,2],[3],[4],[5]]
=> 1 = 2 - 1
[[1],[2],[3],[4],[5]]
=> 5 = 6 - 1
[[1,2,3,4,5,6]]
=> 1 = 2 - 1
[[1,3,4,5,6],[2]]
=> 2 = 3 - 1
[[1,2,4,5,6],[3]]
=> 1 = 2 - 1
[[1,2,3,5,6],[4]]
=> 1 = 2 - 1
[[1,2,3,4,6],[5]]
=> 1 = 2 - 1
[[1,2,3,4,5],[6]]
=> 1 = 2 - 1
[[1,3,5,6],[2,4]]
=> 2 = 3 - 1
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St000382
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => 1 = 2 - 1
[[1,2]]
=> [2] => 2 = 3 - 1
[[1],[2]]
=> [1,1] => 1 = 2 - 1
[[1,2,3]]
=> [3] => 3 = 4 - 1
[[1,3],[2]]
=> [1,2] => 1 = 2 - 1
[[1,2],[3]]
=> [2,1] => 2 = 3 - 1
[[1],[2],[3]]
=> [1,1,1] => 1 = 2 - 1
[[1,2,3,4]]
=> [4] => 4 = 5 - 1
[[1,3,4],[2]]
=> [1,3] => 1 = 2 - 1
[[1,2,4],[3]]
=> [2,2] => 2 = 3 - 1
[[1,2,3],[4]]
=> [3,1] => 3 = 4 - 1
[[1,3],[2,4]]
=> [1,2,1] => 1 = 2 - 1
[[1,2],[3,4]]
=> [2,2] => 2 = 3 - 1
[[1,4],[2],[3]]
=> [1,1,2] => 1 = 2 - 1
[[1,3],[2],[4]]
=> [1,2,1] => 1 = 2 - 1
[[1,2],[3],[4]]
=> [2,1,1] => 2 = 3 - 1
[[1],[2],[3],[4]]
=> [1,1,1,1] => 1 = 2 - 1
[[1,2,3,4,5]]
=> [5] => 5 = 6 - 1
[[1,3,4,5],[2]]
=> [1,4] => 1 = 2 - 1
[[1,2,4,5],[3]]
=> [2,3] => 2 = 3 - 1
[[1,2,3,5],[4]]
=> [3,2] => 3 = 4 - 1
[[1,2,3,4],[5]]
=> [4,1] => 4 = 5 - 1
[[1,3,5],[2,4]]
=> [1,2,2] => 1 = 2 - 1
[[1,2,5],[3,4]]
=> [2,3] => 2 = 3 - 1
[[1,3,4],[2,5]]
=> [1,3,1] => 1 = 2 - 1
[[1,2,4],[3,5]]
=> [2,2,1] => 2 = 3 - 1
[[1,2,3],[4,5]]
=> [3,2] => 3 = 4 - 1
[[1,4,5],[2],[3]]
=> [1,1,3] => 1 = 2 - 1
[[1,3,5],[2],[4]]
=> [1,2,2] => 1 = 2 - 1
[[1,2,5],[3],[4]]
=> [2,1,2] => 2 = 3 - 1
[[1,3,4],[2],[5]]
=> [1,3,1] => 1 = 2 - 1
[[1,2,4],[3],[5]]
=> [2,2,1] => 2 = 3 - 1
[[1,2,3],[4],[5]]
=> [3,1,1] => 3 = 4 - 1
[[1,4],[2,5],[3]]
=> [1,1,2,1] => 1 = 2 - 1
[[1,3],[2,5],[4]]
=> [1,2,2] => 1 = 2 - 1
[[1,2],[3,5],[4]]
=> [2,1,2] => 2 = 3 - 1
[[1,3],[2,4],[5]]
=> [1,2,1,1] => 1 = 2 - 1
[[1,2],[3,4],[5]]
=> [2,2,1] => 2 = 3 - 1
[[1,5],[2],[3],[4]]
=> [1,1,1,2] => 1 = 2 - 1
[[1,4],[2],[3],[5]]
=> [1,1,2,1] => 1 = 2 - 1
[[1,3],[2],[4],[5]]
=> [1,2,1,1] => 1 = 2 - 1
[[1,2],[3],[4],[5]]
=> [2,1,1,1] => 2 = 3 - 1
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => 1 = 2 - 1
[[1,2,3,4,5,6]]
=> [6] => 6 = 7 - 1
[[1,3,4,5,6],[2]]
=> [1,5] => 1 = 2 - 1
[[1,2,4,5,6],[3]]
=> [2,4] => 2 = 3 - 1
[[1,2,3,5,6],[4]]
=> [3,3] => 3 = 4 - 1
[[1,2,3,4,6],[5]]
=> [4,2] => 4 = 5 - 1
[[1,2,3,4,5],[6]]
=> [5,1] => 5 = 6 - 1
[[1,3,5,6],[2,4]]
=> [1,2,3] => 1 = 2 - 1
Description
The first part of an integer composition.
Matching statistic: St000439
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => [1,0]
=> 2
[[1,2]]
=> [2] => [1,1,0,0]
=> 3
[[1],[2]]
=> [1,1] => [1,0,1,0]
=> 2
[[1,2,3]]
=> [3] => [1,1,1,0,0,0]
=> 4
[[1,3],[2]]
=> [1,2] => [1,0,1,1,0,0]
=> 2
[[1,2],[3]]
=> [2,1] => [1,1,0,0,1,0]
=> 3
[[1],[2],[3]]
=> [1,1,1] => [1,0,1,0,1,0]
=> 2
[[1,2,3,4]]
=> [4] => [1,1,1,1,0,0,0,0]
=> 5
[[1,3,4],[2]]
=> [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[[1,2,4],[3]]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 3
[[1,2,3],[4]]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 4
[[1,3],[2,4]]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[[1,2],[3,4]]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 3
[[1,4],[2],[3]]
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
[[1,3],[2],[4]]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[[1,2],[3],[4]]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3
[[1],[2],[3],[4]]
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 2
[[1,2,3,4,5]]
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[[1,3,4,5],[2]]
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2
[[1,2,4,5],[3]]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[[1,2,3,5],[4]]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 4
[[1,2,3,4],[5]]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 5
[[1,3,5],[2,4]]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[[1,2,5],[3,4]]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[[1,3,4],[2,5]]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[[1,2,4],[3,5]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
[[1,2,3],[4,5]]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 4
[[1,4,5],[2],[3]]
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[[1,3,5],[2],[4]]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[[1,2,5],[3],[4]]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 3
[[1,3,4],[2],[5]]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[[1,2,4],[3],[5]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
[[1,2,3],[4],[5]]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 4
[[1,4],[2,5],[3]]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
[[1,3],[2,5],[4]]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[[1,2],[3,5],[4]]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 3
[[1,3],[2,4],[5]]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2
[[1,2],[3,4],[5]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
[[1,5],[2],[3],[4]]
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 2
[[1,4],[2],[3],[5]]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
[[1,3],[2],[4],[5]]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2
[[1,2],[3],[4],[5]]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 3
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 2
[[1,2,3,4,5,6]]
=> [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 7
[[1,3,4,5,6],[2]]
=> [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2
[[1,2,4,5,6],[3]]
=> [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 3
[[1,2,3,5,6],[4]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 4
[[1,2,3,4,6],[5]]
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 5
[[1,2,3,4,5],[6]]
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[[1,3,5,6],[2,4]]
=> [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 2
Description
The position of the first down step of a Dyck path.
Matching statistic: St000025
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => [1,0]
=> 1 = 2 - 1
[[1,2]]
=> [2] => [1,1,0,0]
=> 2 = 3 - 1
[[1],[2]]
=> [1,1] => [1,0,1,0]
=> 1 = 2 - 1
[[1,2,3]]
=> [3] => [1,1,1,0,0,0]
=> 3 = 4 - 1
[[1,3],[2]]
=> [1,2] => [1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,2],[3]]
=> [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[[1],[2],[3]]
=> [1,1,1] => [1,0,1,0,1,0]
=> 1 = 2 - 1
[[1,2,3,4]]
=> [4] => [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[[1,3,4],[2]]
=> [1,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[[1,2,4],[3]]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[1,2,3],[4]]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[[1,3],[2,4]]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[[1,2],[3,4]]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[1,4],[2],[3]]
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,3],[2],[4]]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[[1,2],[3],[4]]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[[1],[2],[3],[4]]
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[[1,2,3,4,5]]
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[[1,3,4,5],[2]]
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[1,2,4,5],[3]]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[1,2,3,5],[4]]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3 = 4 - 1
[[1,2,3,4],[5]]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4 = 5 - 1
[[1,3,5],[2,4]]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[[1,2,5],[3,4]]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[1,3,4],[2,5]]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[[1,2,4],[3,5]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[1,2,3],[4,5]]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3 = 4 - 1
[[1,4,5],[2],[3]]
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[[1,3,5],[2],[4]]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[[1,2,5],[3],[4]]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[1,3,4],[2],[5]]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[[1,2,4],[3],[5]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[1,2,3],[4],[5]]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3 = 4 - 1
[[1,4],[2,5],[3]]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[[1,3],[2,5],[4]]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[[1,2],[3,5],[4]]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[1,3],[2,4],[5]]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[[1,2],[3,4],[5]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[1,5],[2],[3],[4]]
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,4],[2],[3],[5]]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[[1,3],[2],[4],[5]]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[[1,2],[3],[4],[5]]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2 = 3 - 1
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[[1,2,3,4,5,6]]
=> [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6 = 7 - 1
[[1,3,4,5,6],[2]]
=> [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
[[1,2,4,5,6],[3]]
=> [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[[1,2,3,5,6],[4]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[[1,2,3,4,6],[5]]
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4 = 5 - 1
[[1,2,3,4,5],[6]]
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 5 = 6 - 1
[[1,3,5,6],[2,4]]
=> [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
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 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000026: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000026: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => [1,0]
=> 1 = 2 - 1
[[1,2]]
=> [2] => [1,1,0,0]
=> 2 = 3 - 1
[[1],[2]]
=> [1,1] => [1,0,1,0]
=> 1 = 2 - 1
[[1,2,3]]
=> [3] => [1,1,1,0,0,0]
=> 3 = 4 - 1
[[1,3],[2]]
=> [1,2] => [1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,2],[3]]
=> [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[[1],[2],[3]]
=> [1,1,1] => [1,0,1,0,1,0]
=> 1 = 2 - 1
[[1,2,3,4]]
=> [4] => [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[[1,3,4],[2]]
=> [1,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[[1,2,4],[3]]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[1,2,3],[4]]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[[1,3],[2,4]]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[[1,2],[3,4]]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[1,4],[2],[3]]
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,3],[2],[4]]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[[1,2],[3],[4]]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[[1],[2],[3],[4]]
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[[1,2,3,4,5]]
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[[1,3,4,5],[2]]
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[1,2,4,5],[3]]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[1,2,3,5],[4]]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3 = 4 - 1
[[1,2,3,4],[5]]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4 = 5 - 1
[[1,3,5],[2,4]]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[[1,2,5],[3,4]]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[1,3,4],[2,5]]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[[1,2,4],[3,5]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[1,2,3],[4,5]]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3 = 4 - 1
[[1,4,5],[2],[3]]
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[[1,3,5],[2],[4]]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[[1,2,5],[3],[4]]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[1,3,4],[2],[5]]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[[1,2,4],[3],[5]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[1,2,3],[4],[5]]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3 = 4 - 1
[[1,4],[2,5],[3]]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[[1,3],[2,5],[4]]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[[1,2],[3,5],[4]]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[1,3],[2,4],[5]]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[[1,2],[3,4],[5]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[1,5],[2],[3],[4]]
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,4],[2],[3],[5]]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[[1,3],[2],[4],[5]]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[[1,2],[3],[4],[5]]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2 = 3 - 1
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[[1,2,3,4,5,6]]
=> [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6 = 7 - 1
[[1,3,4,5,6],[2]]
=> [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
[[1,2,4,5,6],[3]]
=> [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[[1,2,3,5,6],[4]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[[1,2,3,4,6],[5]]
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4 = 5 - 1
[[1,2,3,4,5],[6]]
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 5 = 6 - 1
[[1,3,5,6],[2,4]]
=> [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
Description
The position of the first return of a Dyck path.
Matching statistic: St000297
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00094: Integer compositions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => 1 => 1 = 2 - 1
[[1,2]]
=> [2] => 10 => 1 = 2 - 1
[[1],[2]]
=> [1,1] => 11 => 2 = 3 - 1
[[1,2,3]]
=> [3] => 100 => 1 = 2 - 1
[[1,3],[2]]
=> [1,2] => 110 => 2 = 3 - 1
[[1,2],[3]]
=> [2,1] => 101 => 1 = 2 - 1
[[1],[2],[3]]
=> [1,1,1] => 111 => 3 = 4 - 1
[[1,2,3,4]]
=> [4] => 1000 => 1 = 2 - 1
[[1,3,4],[2]]
=> [1,3] => 1100 => 2 = 3 - 1
[[1,2,4],[3]]
=> [2,2] => 1010 => 1 = 2 - 1
[[1,2,3],[4]]
=> [3,1] => 1001 => 1 = 2 - 1
[[1,3],[2,4]]
=> [1,2,1] => 1101 => 2 = 3 - 1
[[1,2],[3,4]]
=> [2,2] => 1010 => 1 = 2 - 1
[[1,4],[2],[3]]
=> [1,1,2] => 1110 => 3 = 4 - 1
[[1,3],[2],[4]]
=> [1,2,1] => 1101 => 2 = 3 - 1
[[1,2],[3],[4]]
=> [2,1,1] => 1011 => 1 = 2 - 1
[[1],[2],[3],[4]]
=> [1,1,1,1] => 1111 => 4 = 5 - 1
[[1,2,3,4,5]]
=> [5] => 10000 => 1 = 2 - 1
[[1,3,4,5],[2]]
=> [1,4] => 11000 => 2 = 3 - 1
[[1,2,4,5],[3]]
=> [2,3] => 10100 => 1 = 2 - 1
[[1,2,3,5],[4]]
=> [3,2] => 10010 => 1 = 2 - 1
[[1,2,3,4],[5]]
=> [4,1] => 10001 => 1 = 2 - 1
[[1,3,5],[2,4]]
=> [1,2,2] => 11010 => 2 = 3 - 1
[[1,2,5],[3,4]]
=> [2,3] => 10100 => 1 = 2 - 1
[[1,3,4],[2,5]]
=> [1,3,1] => 11001 => 2 = 3 - 1
[[1,2,4],[3,5]]
=> [2,2,1] => 10101 => 1 = 2 - 1
[[1,2,3],[4,5]]
=> [3,2] => 10010 => 1 = 2 - 1
[[1,4,5],[2],[3]]
=> [1,1,3] => 11100 => 3 = 4 - 1
[[1,3,5],[2],[4]]
=> [1,2,2] => 11010 => 2 = 3 - 1
[[1,2,5],[3],[4]]
=> [2,1,2] => 10110 => 1 = 2 - 1
[[1,3,4],[2],[5]]
=> [1,3,1] => 11001 => 2 = 3 - 1
[[1,2,4],[3],[5]]
=> [2,2,1] => 10101 => 1 = 2 - 1
[[1,2,3],[4],[5]]
=> [3,1,1] => 10011 => 1 = 2 - 1
[[1,4],[2,5],[3]]
=> [1,1,2,1] => 11101 => 3 = 4 - 1
[[1,3],[2,5],[4]]
=> [1,2,2] => 11010 => 2 = 3 - 1
[[1,2],[3,5],[4]]
=> [2,1,2] => 10110 => 1 = 2 - 1
[[1,3],[2,4],[5]]
=> [1,2,1,1] => 11011 => 2 = 3 - 1
[[1,2],[3,4],[5]]
=> [2,2,1] => 10101 => 1 = 2 - 1
[[1,5],[2],[3],[4]]
=> [1,1,1,2] => 11110 => 4 = 5 - 1
[[1,4],[2],[3],[5]]
=> [1,1,2,1] => 11101 => 3 = 4 - 1
[[1,3],[2],[4],[5]]
=> [1,2,1,1] => 11011 => 2 = 3 - 1
[[1,2],[3],[4],[5]]
=> [2,1,1,1] => 10111 => 1 = 2 - 1
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => 11111 => 5 = 6 - 1
[[1,2,3,4,5,6]]
=> [6] => 100000 => 1 = 2 - 1
[[1,3,4,5,6],[2]]
=> [1,5] => 110000 => 2 = 3 - 1
[[1,2,4,5,6],[3]]
=> [2,4] => 101000 => 1 = 2 - 1
[[1,2,3,5,6],[4]]
=> [3,3] => 100100 => 1 = 2 - 1
[[1,2,3,4,6],[5]]
=> [4,2] => 100010 => 1 = 2 - 1
[[1,2,3,4,5],[6]]
=> [5,1] => 100001 => 1 = 2 - 1
[[1,3,5,6],[2,4]]
=> [1,2,3] => 110100 => 2 = 3 - 1
Description
The number of leading ones in a binary word.
Matching statistic: St000383
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> => [1] => 1 = 2 - 1
[[1,2]]
=> 0 => [2] => 2 = 3 - 1
[[1],[2]]
=> 1 => [1,1] => 1 = 2 - 1
[[1,2,3]]
=> 00 => [3] => 3 = 4 - 1
[[1,3],[2]]
=> 10 => [1,2] => 2 = 3 - 1
[[1,2],[3]]
=> 01 => [2,1] => 1 = 2 - 1
[[1],[2],[3]]
=> 11 => [1,1,1] => 1 = 2 - 1
[[1,2,3,4]]
=> 000 => [4] => 4 = 5 - 1
[[1,3,4],[2]]
=> 100 => [1,3] => 3 = 4 - 1
[[1,2,4],[3]]
=> 010 => [2,2] => 2 = 3 - 1
[[1,2,3],[4]]
=> 001 => [3,1] => 1 = 2 - 1
[[1,3],[2,4]]
=> 101 => [1,2,1] => 1 = 2 - 1
[[1,2],[3,4]]
=> 010 => [2,2] => 2 = 3 - 1
[[1,4],[2],[3]]
=> 110 => [1,1,2] => 2 = 3 - 1
[[1,3],[2],[4]]
=> 101 => [1,2,1] => 1 = 2 - 1
[[1,2],[3],[4]]
=> 011 => [2,1,1] => 1 = 2 - 1
[[1],[2],[3],[4]]
=> 111 => [1,1,1,1] => 1 = 2 - 1
[[1,2,3,4,5]]
=> 0000 => [5] => 5 = 6 - 1
[[1,3,4,5],[2]]
=> 1000 => [1,4] => 4 = 5 - 1
[[1,2,4,5],[3]]
=> 0100 => [2,3] => 3 = 4 - 1
[[1,2,3,5],[4]]
=> 0010 => [3,2] => 2 = 3 - 1
[[1,2,3,4],[5]]
=> 0001 => [4,1] => 1 = 2 - 1
[[1,3,5],[2,4]]
=> 1010 => [1,2,2] => 2 = 3 - 1
[[1,2,5],[3,4]]
=> 0100 => [2,3] => 3 = 4 - 1
[[1,3,4],[2,5]]
=> 1001 => [1,3,1] => 1 = 2 - 1
[[1,2,4],[3,5]]
=> 0101 => [2,2,1] => 1 = 2 - 1
[[1,2,3],[4,5]]
=> 0010 => [3,2] => 2 = 3 - 1
[[1,4,5],[2],[3]]
=> 1100 => [1,1,3] => 3 = 4 - 1
[[1,3,5],[2],[4]]
=> 1010 => [1,2,2] => 2 = 3 - 1
[[1,2,5],[3],[4]]
=> 0110 => [2,1,2] => 2 = 3 - 1
[[1,3,4],[2],[5]]
=> 1001 => [1,3,1] => 1 = 2 - 1
[[1,2,4],[3],[5]]
=> 0101 => [2,2,1] => 1 = 2 - 1
[[1,2,3],[4],[5]]
=> 0011 => [3,1,1] => 1 = 2 - 1
[[1,4],[2,5],[3]]
=> 1101 => [1,1,2,1] => 1 = 2 - 1
[[1,3],[2,5],[4]]
=> 1010 => [1,2,2] => 2 = 3 - 1
[[1,2],[3,5],[4]]
=> 0110 => [2,1,2] => 2 = 3 - 1
[[1,3],[2,4],[5]]
=> 1011 => [1,2,1,1] => 1 = 2 - 1
[[1,2],[3,4],[5]]
=> 0101 => [2,2,1] => 1 = 2 - 1
[[1,5],[2],[3],[4]]
=> 1110 => [1,1,1,2] => 2 = 3 - 1
[[1,4],[2],[3],[5]]
=> 1101 => [1,1,2,1] => 1 = 2 - 1
[[1,3],[2],[4],[5]]
=> 1011 => [1,2,1,1] => 1 = 2 - 1
[[1,2],[3],[4],[5]]
=> 0111 => [2,1,1,1] => 1 = 2 - 1
[[1],[2],[3],[4],[5]]
=> 1111 => [1,1,1,1,1] => 1 = 2 - 1
[[1,2,3,4,5,6]]
=> 00000 => [6] => 6 = 7 - 1
[[1,3,4,5,6],[2]]
=> 10000 => [1,5] => 5 = 6 - 1
[[1,2,4,5,6],[3]]
=> 01000 => [2,4] => 4 = 5 - 1
[[1,2,3,5,6],[4]]
=> 00100 => [3,3] => 3 = 4 - 1
[[1,2,3,4,6],[5]]
=> 00010 => [4,2] => 2 = 3 - 1
[[1,2,3,4,5],[6]]
=> 00001 => [5,1] => 1 = 2 - 1
[[1,3,5,6],[2,4]]
=> 10100 => [1,2,3] => 3 = 4 - 1
Description
The last part of an integer composition.
Matching statistic: St001050
(load all 26 compositions to match this statistic)
(load all 26 compositions to match this statistic)
Mp00284: Standard tableaux —rows⟶ Set partitions
Mp00112: Set partitions —complement⟶ Set partitions
St001050: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00112: Set partitions —complement⟶ Set partitions
St001050: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> {{1}}
=> {{1}}
=> 1 = 2 - 1
[[1,2]]
=> {{1,2}}
=> {{1,2}}
=> 1 = 2 - 1
[[1],[2]]
=> {{1},{2}}
=> {{1},{2}}
=> 2 = 3 - 1
[[1,2,3]]
=> {{1,2,3}}
=> {{1,2,3}}
=> 1 = 2 - 1
[[1,3],[2]]
=> {{1,3},{2}}
=> {{1,3},{2}}
=> 2 = 3 - 1
[[1,2],[3]]
=> {{1,2},{3}}
=> {{1},{2,3}}
=> 1 = 2 - 1
[[1],[2],[3]]
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> 3 = 4 - 1
[[1,2,3,4]]
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> 1 = 2 - 1
[[1,3,4],[2]]
=> {{1,3,4},{2}}
=> {{1,2,4},{3}}
=> 2 = 3 - 1
[[1,2,4],[3]]
=> {{1,2,4},{3}}
=> {{1,3,4},{2}}
=> 1 = 2 - 1
[[1,2,3],[4]]
=> {{1,2,3},{4}}
=> {{1},{2,3,4}}
=> 1 = 2 - 1
[[1,3],[2,4]]
=> {{1,3},{2,4}}
=> {{1,3},{2,4}}
=> 2 = 3 - 1
[[1,2],[3,4]]
=> {{1,2},{3,4}}
=> {{1,2},{3,4}}
=> 1 = 2 - 1
[[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> 3 = 4 - 1
[[1,3],[2],[4]]
=> {{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> 2 = 3 - 1
[[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> {{1},{2},{3,4}}
=> 1 = 2 - 1
[[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 4 = 5 - 1
[[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> 1 = 2 - 1
[[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> {{1,2,3,5},{4}}
=> 2 = 3 - 1
[[1,2,4,5],[3]]
=> {{1,2,4,5},{3}}
=> {{1,2,4,5},{3}}
=> 1 = 2 - 1
[[1,2,3,5],[4]]
=> {{1,2,3,5},{4}}
=> {{1,3,4,5},{2}}
=> 1 = 2 - 1
[[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> {{1},{2,3,4,5}}
=> 1 = 2 - 1
[[1,3,5],[2,4]]
=> {{1,3,5},{2,4}}
=> {{1,3,5},{2,4}}
=> 2 = 3 - 1
[[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> {{1,4,5},{2,3}}
=> 1 = 2 - 1
[[1,3,4],[2,5]]
=> {{1,3,4},{2,5}}
=> {{1,4},{2,3,5}}
=> 2 = 3 - 1
[[1,2,4],[3,5]]
=> {{1,2,4},{3,5}}
=> {{1,3},{2,4,5}}
=> 1 = 2 - 1
[[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> {{1,2},{3,4,5}}
=> 1 = 2 - 1
[[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> {{1,2,5},{3},{4}}
=> 3 = 4 - 1
[[1,3,5],[2],[4]]
=> {{1,3,5},{2},{4}}
=> {{1,3,5},{2},{4}}
=> 2 = 3 - 1
[[1,2,5],[3],[4]]
=> {{1,2,5},{3},{4}}
=> {{1,4,5},{2},{3}}
=> 1 = 2 - 1
[[1,3,4],[2],[5]]
=> {{1,3,4},{2},{5}}
=> {{1},{2,3,5},{4}}
=> 2 = 3 - 1
[[1,2,4],[3],[5]]
=> {{1,2,4},{3},{5}}
=> {{1},{2,4,5},{3}}
=> 1 = 2 - 1
[[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> {{1},{2},{3,4,5}}
=> 1 = 2 - 1
[[1,4],[2,5],[3]]
=> {{1,4},{2,5},{3}}
=> {{1,4},{2,5},{3}}
=> 3 = 4 - 1
[[1,3],[2,5],[4]]
=> {{1,3},{2,5},{4}}
=> {{1,4},{2},{3,5}}
=> 2 = 3 - 1
[[1,2],[3,5],[4]]
=> {{1,2},{3,5},{4}}
=> {{1,3},{2},{4,5}}
=> 1 = 2 - 1
[[1,3],[2,4],[5]]
=> {{1,3},{2,4},{5}}
=> {{1},{2,4},{3,5}}
=> 2 = 3 - 1
[[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> {{1},{2,3},{4,5}}
=> 1 = 2 - 1
[[1,5],[2],[3],[4]]
=> {{1,5},{2},{3},{4}}
=> {{1,5},{2},{3},{4}}
=> 4 = 5 - 1
[[1,4],[2],[3],[5]]
=> {{1,4},{2},{3},{5}}
=> {{1},{2,5},{3},{4}}
=> 3 = 4 - 1
[[1,3],[2],[4],[5]]
=> {{1,3},{2},{4},{5}}
=> {{1},{2},{3,5},{4}}
=> 2 = 3 - 1
[[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> {{1},{2},{3},{4,5}}
=> 1 = 2 - 1
[[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> 5 = 6 - 1
[[1,2,3,4,5,6]]
=> {{1,2,3,4,5,6}}
=> {{1,2,3,4,5,6}}
=> 1 = 2 - 1
[[1,3,4,5,6],[2]]
=> {{1,3,4,5,6},{2}}
=> {{1,2,3,4,6},{5}}
=> 2 = 3 - 1
[[1,2,4,5,6],[3]]
=> {{1,2,4,5,6},{3}}
=> {{1,2,3,5,6},{4}}
=> 1 = 2 - 1
[[1,2,3,5,6],[4]]
=> {{1,2,3,5,6},{4}}
=> {{1,2,4,5,6},{3}}
=> 1 = 2 - 1
[[1,2,3,4,6],[5]]
=> {{1,2,3,4,6},{5}}
=> {{1,3,4,5,6},{2}}
=> 1 = 2 - 1
[[1,2,3,4,5],[6]]
=> {{1,2,3,4,5},{6}}
=> {{1},{2,3,4,5,6}}
=> 1 = 2 - 1
[[1,3,5,6],[2,4]]
=> {{1,3,5,6},{2,4}}
=> {{1,2,4,6},{3,5}}
=> 2 = 3 - 1
Description
The number of terminal closers of a set partition.
A closer of a set partition is a number that is maximal in its block. In particular, a singleton is a closer. This statistic counts the number of terminal closers. In other words, this is the number of closers such that all larger elements are also closers.
Matching statistic: St000326
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => 1 => 0 => 2
[[1,2]]
=> [2] => 10 => 01 => 2
[[1],[2]]
=> [1,1] => 11 => 00 => 3
[[1,2,3]]
=> [3] => 100 => 011 => 2
[[1,3],[2]]
=> [1,2] => 110 => 001 => 3
[[1,2],[3]]
=> [2,1] => 101 => 010 => 2
[[1],[2],[3]]
=> [1,1,1] => 111 => 000 => 4
[[1,2,3,4]]
=> [4] => 1000 => 0111 => 2
[[1,3,4],[2]]
=> [1,3] => 1100 => 0011 => 3
[[1,2,4],[3]]
=> [2,2] => 1010 => 0101 => 2
[[1,2,3],[4]]
=> [3,1] => 1001 => 0110 => 2
[[1,3],[2,4]]
=> [1,2,1] => 1101 => 0010 => 3
[[1,2],[3,4]]
=> [2,2] => 1010 => 0101 => 2
[[1,4],[2],[3]]
=> [1,1,2] => 1110 => 0001 => 4
[[1,3],[2],[4]]
=> [1,2,1] => 1101 => 0010 => 3
[[1,2],[3],[4]]
=> [2,1,1] => 1011 => 0100 => 2
[[1],[2],[3],[4]]
=> [1,1,1,1] => 1111 => 0000 => 5
[[1,2,3,4,5]]
=> [5] => 10000 => 01111 => 2
[[1,3,4,5],[2]]
=> [1,4] => 11000 => 00111 => 3
[[1,2,4,5],[3]]
=> [2,3] => 10100 => 01011 => 2
[[1,2,3,5],[4]]
=> [3,2] => 10010 => 01101 => 2
[[1,2,3,4],[5]]
=> [4,1] => 10001 => 01110 => 2
[[1,3,5],[2,4]]
=> [1,2,2] => 11010 => 00101 => 3
[[1,2,5],[3,4]]
=> [2,3] => 10100 => 01011 => 2
[[1,3,4],[2,5]]
=> [1,3,1] => 11001 => 00110 => 3
[[1,2,4],[3,5]]
=> [2,2,1] => 10101 => 01010 => 2
[[1,2,3],[4,5]]
=> [3,2] => 10010 => 01101 => 2
[[1,4,5],[2],[3]]
=> [1,1,3] => 11100 => 00011 => 4
[[1,3,5],[2],[4]]
=> [1,2,2] => 11010 => 00101 => 3
[[1,2,5],[3],[4]]
=> [2,1,2] => 10110 => 01001 => 2
[[1,3,4],[2],[5]]
=> [1,3,1] => 11001 => 00110 => 3
[[1,2,4],[3],[5]]
=> [2,2,1] => 10101 => 01010 => 2
[[1,2,3],[4],[5]]
=> [3,1,1] => 10011 => 01100 => 2
[[1,4],[2,5],[3]]
=> [1,1,2,1] => 11101 => 00010 => 4
[[1,3],[2,5],[4]]
=> [1,2,2] => 11010 => 00101 => 3
[[1,2],[3,5],[4]]
=> [2,1,2] => 10110 => 01001 => 2
[[1,3],[2,4],[5]]
=> [1,2,1,1] => 11011 => 00100 => 3
[[1,2],[3,4],[5]]
=> [2,2,1] => 10101 => 01010 => 2
[[1,5],[2],[3],[4]]
=> [1,1,1,2] => 11110 => 00001 => 5
[[1,4],[2],[3],[5]]
=> [1,1,2,1] => 11101 => 00010 => 4
[[1,3],[2],[4],[5]]
=> [1,2,1,1] => 11011 => 00100 => 3
[[1,2],[3],[4],[5]]
=> [2,1,1,1] => 10111 => 01000 => 2
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => 11111 => 00000 => 6
[[1,2,3,4,5,6]]
=> [6] => 100000 => 011111 => 2
[[1,3,4,5,6],[2]]
=> [1,5] => 110000 => 001111 => 3
[[1,2,4,5,6],[3]]
=> [2,4] => 101000 => 010111 => 2
[[1,2,3,5,6],[4]]
=> [3,3] => 100100 => 011011 => 2
[[1,2,3,4,6],[5]]
=> [4,2] => 100010 => 011101 => 2
[[1,2,3,4,5],[6]]
=> [5,1] => 100001 => 011110 => 2
[[1,3,5,6],[2,4]]
=> [1,2,3] => 110100 => 001011 => 3
Description
The position of the first one in a binary word after appending a 1 at the end.
Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Matching statistic: St001505
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001505: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001505: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => [1] => [1,0]
=> 2
[[1,2]]
=> [1,2] => [1,2] => [1,0,1,0]
=> 3
[[1],[2]]
=> [2,1] => [2,1] => [1,1,0,0]
=> 2
[[1,2,3]]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 4
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 3
[[1,2],[3]]
=> [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
=> 2
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 2
[[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 5
[[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 4
[[1,2,4],[3]]
=> [3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 3
[[1,2,3],[4]]
=> [4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 2
[[1,3],[2,4]]
=> [2,4,1,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 3
[[1,2],[3,4]]
=> [3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 2
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 3
[[1,3],[2],[4]]
=> [4,2,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 2
[[1,2],[3],[4]]
=> [4,3,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 2
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 2
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 6
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 5
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 4
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 3
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 2
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> 4
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 3
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> 3
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 3
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 2
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 4
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 3
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 3
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 2
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 2
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 2
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 3
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 2
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 2
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 2
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 2
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 3
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 2
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 2
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 2
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 2
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 7
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 6
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [3,2,1,4,5,6] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> 5
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [4,2,3,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 4
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [5,2,3,4,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 3
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [6,2,3,4,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 2
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [3,4,1,2,5,6] => [1,1,1,0,1,0,0,0,1,0,1,0]
=> 5
Description
The number of elements generated by the Dyck path as a map in the full transformation monoid.
We view the resolution quiver of a Dyck path (corresponding to an LNakayamaalgebra) as a transformation and associate to it the submonoid generated by this map in the full transformation monoid.
The following 74 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000011The number of touch points (or returns) of a Dyck path. St000505The biggest entry in the block containing the 1. St000759The smallest missing part in an integer partition. St000971The smallest closer of a set partition. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001733The number of weak left to right maxima of a Dyck path. St000053The number of valleys of the Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000504The cardinality of the first block of a set partition. St000823The number of unsplittable factors of the set partition. St000363The number of minimal vertex covers of a graph. St000273The domination number of a graph. St000544The cop number of a graph. St000916The packing number of a graph. St001829The common independence number of a graph. St001322The size of a minimal independent dominating set in a graph. St001316The domatic number of a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St000617The number of global maxima of a Dyck path. St000054The first entry of the permutation. St000989The number of final rises of a permutation. St000234The number of global ascents of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000007The number of saliances of the permutation. St000501The size of the first part in the decomposition of a permutation. St000068The number of minimal elements in a poset. St000542The number of left-to-right-minima of a permutation. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000654The first descent of a permutation. St000990The first ascent of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000237The number of small exceedances. St000740The last entry of a permutation. St001060The distinguishing index of a graph. St000546The number of global descents of a permutation. St000738The first entry in the last row of a standard tableau. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000314The number of left-to-right-maxima of a permutation. St000991The number of right-to-left minima of a permutation. 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. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000015The number of peaks of a Dyck path. St000056The decomposition (or block) number of a permutation. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000335The difference of lower and upper interactions. 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. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001481The minimal height of a peak of a Dyck path. St000051The size of the left subtree of a binary tree. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex 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. St000061The number of nodes on the left branch of a binary tree. St000260The radius of a connected graph. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000264The girth of a graph, which is not a tree. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000338The number of pixed points of a permutation. St001948The number of augmented double ascents of a permutation. St000454The largest eigenvalue of a graph if it is integral. St001889The size of the connectivity set of a signed permutation. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices.
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