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Your data matches 259 different statistics following compositions of up to 3 maps.
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Matching statistic: St000013
(load all 28 compositions to match this statistic)
(load all 28 compositions to match this statistic)
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 1
[1,0,1,0]
=> 1
[1,1,0,0]
=> 2
[1,0,1,0,1,0]
=> 1
[1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0]
=> 2
[1,1,1,0,0,0]
=> 3
[1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> 3
[1,1,0,0,1,0,1,0]
=> 2
[1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> 2
[1,1,0,1,1,0,0,0]
=> 3
[1,1,1,0,0,0,1,0]
=> 3
[1,1,1,0,0,1,0,0]
=> 3
[1,1,1,0,1,0,0,0]
=> 3
[1,1,1,1,0,0,0,0]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> 4
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: St000147
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 74%●distinct values known / distinct values provided: 67%
Mp00204: Permutations —LLPS⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 74%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1] => [1]
=> 1
[1,0,1,0]
=> [1,2] => [1,1]
=> 1
[1,1,0,0]
=> [2,1] => [2]
=> 2
[1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> 1
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> 2
[1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> 2
[1,1,1,0,0,0]
=> [3,2,1] => [3]
=> 3
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,1,1,1]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,1,1]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [3,1]
=> 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,1,1]
=> 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,1,1]
=> 2
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,1]
=> 3
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,1]
=> 3
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,1]
=> 3
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [3,1]
=> 3
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,1,1,1]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,1,1]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [3,1,1]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [3,1,1]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [3,1,1]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [3,1,1]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [4,1]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,2,1]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [3,2]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [3,1,1]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,1,1]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,1,1]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,1,1]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [4,1]
=> 4
[1,0,1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,2,4,7,6,5,3,8] => ?
=> ? = 4
[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,2,5,4,3,8,7,6] => ?
=> ? = 3
[1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,2,7,4,5,6,3,8] => ?
=> ? = 3
[1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,2,8,4,5,7,6,3] => ?
=> ? = 4
[1,0,1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,2,7,4,6,5,3,8] => ?
=> ? = 4
[1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,2,7,4,6,5,8,3] => ?
=> ? = 4
[1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,2,6,5,4,3,8,7] => ?
=> ? = 4
[1,0,1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,2,6,5,4,7,3,8] => ?
=> ? = 4
[1,0,1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,2,7,5,4,6,3,8] => ?
=> ? = 4
[1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,2,7,5,6,4,3,8] => ?
=> ? = 4
[1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,2,7,5,6,4,8,3] => ?
=> ? = 4
[1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,2,8,6,5,7,4,3] => ?
=> ? = 5
[1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,3,8,4,6,7,5,2] => ?
=> ? = 4
[1,0,1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [1,3,6,5,4,2,7,8] => ?
=> ? = 4
[1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,3,8,5,6,4,7,2] => ?
=> ? = 4
[1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,4,3,2,6,5,8,7] => ?
=> ? = 3
[1,0,1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,6,3,4,5,2,7,8] => ?
=> ? = 3
[1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,6,3,4,5,7,8,2] => ?
=> ? = 3
[1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,7,3,4,5,6,2,8] => ?
=> ? = 3
[1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [1,7,3,4,6,5,8,2] => ?
=> ? = 4
[1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,8,3,4,7,6,5,2] => ?
=> ? = 5
[1,0,1,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> [1,6,3,5,4,7,2,8] => ?
=> ? = 4
[1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,6,3,5,4,7,8,2] => ?
=> ? = 4
[1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,7,3,5,6,4,8,2] => ?
=> ? = 4
[1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,8,3,6,5,7,4,2] => ?
=> ? = 5
[1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,5,4,3,2,6,8,7] => ?
=> ? = 4
[1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,7,5,4,6,3,8,2] => ?
=> ? = 5
[1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,1,3,6,5,4,8,7] => ?
=> ? = 3
[1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,7,6,5,4,8] => ?
=> ? = 4
[1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,4,3,7,6,5,8] => ?
=> ? = 3
[1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,1,5,4,3,6,8,7] => ?
=> ? = 3
[1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,1,6,5,4,3,7,8] => ?
=> ? = 4
[1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [2,3,1,7,5,6,8,4] => ?
=> ? = 3
[1,1,0,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,8,5,7,6,4] => ?
=> ? = 4
[1,1,0,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [2,3,5,4,6,8,7,1] => ?
=> ? = 3
[1,1,0,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,6,5,4,1,7,8] => ?
=> ? = 4
[1,1,0,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [2,3,6,5,4,8,7,1] => ?
=> ? = 4
[1,1,0,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [2,3,8,7,5,6,4,1] => ?
=> ? = 5
[1,1,0,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [2,4,3,7,6,5,8,1] => ?
=> ? = 4
[1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> [2,5,3,4,1,7,8,6] => ?
=> ? = 3
[1,1,0,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [2,7,3,4,5,6,8,1] => ?
=> ? = 3
[1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [2,7,3,5,6,4,1,8] => ?
=> ? = 4
[1,1,0,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [2,5,4,3,6,7,8,1] => ?
=> ? = 4
[1,1,0,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [2,5,4,3,6,8,7,1] => ?
=> ? = 4
[1,1,0,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> [2,5,4,3,7,6,8,1] => ?
=> ? = 4
[1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,3,6,1,8] => ?
=> ? = 4
[1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [2,8,4,7,5,6,3,1] => ?
=> ? = 5
[1,1,0,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [2,8,6,4,5,3,7,1] => ?
=> ? = 5
[1,1,0,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [2,8,6,4,5,7,3,1] => ?
=> ? = 5
[1,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [2,8,6,5,4,3,7,1] => ?
=> ? = 6
Description
The largest part of an integer partition.
Matching statistic: St000734
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 56% ●values known / values provided: 73%●distinct values known / distinct values provided: 56%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 56% ●values known / values provided: 73%●distinct values known / distinct values provided: 56%
Values
[1,0]
=> [1] => [1]
=> [[1]]
=> 1
[1,0,1,0]
=> [1,2] => [1,1]
=> [[1],[2]]
=> 1
[1,1,0,0]
=> [2,1] => [2]
=> [[1,2]]
=> 2
[1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> 1
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> [[1,2],[3]]
=> 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> [[1,2],[3]]
=> 2
[1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> [[1,2],[3]]
=> 2
[1,1,1,0,0,0]
=> [3,2,1] => [3]
=> [[1,2,3]]
=> 3
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [3,1]
=> [[1,2,3],[4]]
=> 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,1]
=> [[1,2,3],[4]]
=> 3
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,1]
=> [[1,2,3],[4]]
=> 3
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,1]
=> [[1,2,3],[4]]
=> 3
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [3,1]
=> [[1,2,3],[4]]
=> 3
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4]
=> [[1,2,3,4]]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [4,1]
=> [[1,2,3,4],[5]]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [3,2]
=> [[1,2,3],[4,5]]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [4,1]
=> [[1,2,3,4],[5]]
=> 4
[1,0,1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,2,4,7,6,5,3,8] => ?
=> ?
=> ? = 4
[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,2,5,4,3,8,7,6] => ?
=> ?
=> ? = 3
[1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,2,7,4,5,6,3,8] => ?
=> ?
=> ? = 3
[1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,2,8,4,5,7,6,3] => ?
=> ?
=> ? = 4
[1,0,1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,2,7,4,6,5,3,8] => ?
=> ?
=> ? = 4
[1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,2,7,4,6,5,8,3] => ?
=> ?
=> ? = 4
[1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,2,6,5,4,3,8,7] => ?
=> ?
=> ? = 4
[1,0,1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,2,6,5,4,7,3,8] => ?
=> ?
=> ? = 4
[1,0,1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,2,7,5,4,6,3,8] => ?
=> ?
=> ? = 4
[1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,2,7,5,6,4,3,8] => ?
=> ?
=> ? = 4
[1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,2,7,5,6,4,8,3] => ?
=> ?
=> ? = 4
[1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,2,8,6,5,7,4,3] => ?
=> ?
=> ? = 5
[1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,3,8,4,6,7,5,2] => ?
=> ?
=> ? = 4
[1,0,1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [1,3,6,5,4,2,7,8] => ?
=> ?
=> ? = 4
[1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,3,8,5,6,4,7,2] => ?
=> ?
=> ? = 4
[1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,4,3,2,6,5,8,7] => ?
=> ?
=> ? = 3
[1,0,1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,6,3,4,5,2,7,8] => ?
=> ?
=> ? = 3
[1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,6,3,4,5,7,8,2] => ?
=> ?
=> ? = 3
[1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,7,3,4,5,6,2,8] => ?
=> ?
=> ? = 3
[1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [1,7,3,4,6,5,8,2] => ?
=> ?
=> ? = 4
[1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,8,3,4,7,6,5,2] => ?
=> ?
=> ? = 5
[1,0,1,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> [1,6,3,5,4,7,2,8] => ?
=> ?
=> ? = 4
[1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,6,3,5,4,7,8,2] => ?
=> ?
=> ? = 4
[1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,7,3,5,6,4,8,2] => ?
=> ?
=> ? = 4
[1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,8,3,6,5,7,4,2] => ?
=> ?
=> ? = 5
[1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,5,4,3,2,6,8,7] => ?
=> ?
=> ? = 4
[1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,7,5,4,6,3,8,2] => ?
=> ?
=> ? = 5
[1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,1,3,6,5,4,8,7] => ?
=> ?
=> ? = 3
[1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,7,6,5,4,8] => ?
=> ?
=> ? = 4
[1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,4,3,7,6,5,8] => ?
=> ?
=> ? = 3
[1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,1,5,4,3,6,8,7] => ?
=> ?
=> ? = 3
[1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,1,6,5,4,3,7,8] => ?
=> ?
=> ? = 4
[1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [2,3,1,7,5,6,8,4] => ?
=> ?
=> ? = 3
[1,1,0,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,8,5,7,6,4] => ?
=> ?
=> ? = 4
[1,1,0,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [2,3,5,4,6,8,7,1] => ?
=> ?
=> ? = 3
[1,1,0,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,6,5,4,1,7,8] => ?
=> ?
=> ? = 4
[1,1,0,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [2,3,6,5,4,8,7,1] => ?
=> ?
=> ? = 4
[1,1,0,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [2,3,8,7,5,6,4,1] => ?
=> ?
=> ? = 5
[1,1,0,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [2,4,3,7,6,5,8,1] => ?
=> ?
=> ? = 4
[1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> [2,5,3,4,1,7,8,6] => ?
=> ?
=> ? = 3
[1,1,0,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [2,7,3,4,5,6,8,1] => ?
=> ?
=> ? = 3
[1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [2,7,3,5,6,4,1,8] => ?
=> ?
=> ? = 4
[1,1,0,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [2,5,4,3,6,7,8,1] => ?
=> ?
=> ? = 4
[1,1,0,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [2,5,4,3,6,8,7,1] => ?
=> ?
=> ? = 4
[1,1,0,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> [2,5,4,3,7,6,8,1] => ?
=> ?
=> ? = 4
[1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,3,6,1,8] => ?
=> ?
=> ? = 4
[1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [2,8,4,7,5,6,3,1] => ?
=> ?
=> ? = 5
[1,1,0,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [2,8,6,4,5,3,7,1] => ?
=> ?
=> ? = 5
[1,1,0,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [2,8,6,4,5,7,3,1] => ?
=> ?
=> ? = 5
[1,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [2,8,6,5,4,3,7,1] => ?
=> ?
=> ? = 6
Description
The last entry in the first row of a standard tableau.
Matching statistic: St000010
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 72%●distinct values known / distinct values provided: 67%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 72%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1] => [1]
=> 1
[1,0,1,0]
=> [1,2] => [2]
=> 1
[1,1,0,0]
=> [2,1] => [1,1]
=> 2
[1,0,1,0,1,0]
=> [1,2,3] => [3]
=> 1
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> 2
[1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> 2
[1,1,1,0,0,0]
=> [3,2,1] => [1,1,1]
=> 3
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [3,1]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,1]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,1]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [2,1,1]
=> 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [3,1]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1]
=> 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [3,1]
=> 2
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,1,1]
=> 3
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,1,1]
=> 3
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,1,1]
=> 3
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,1,1]
=> 3
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [5]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [4,1]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [4,1]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [4,1]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,1,1]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [4,1]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [3,2]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [4,1]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [4,1]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [3,1,1]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [3,1,1]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [3,1,1]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [3,1,1]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,1,1,1]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [4,1]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [3,2]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [3,2]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,2]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,2,1]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [4,1]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [4,1]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [3,1,1]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,1,1]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,1,1]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,1,1]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,1,1,1]
=> 4
[1,0,1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,2,4,7,6,5,3,8] => ?
=> ? = 4
[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,2,5,4,3,8,7,6] => ?
=> ? = 3
[1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,2,7,4,5,6,3,8] => ?
=> ? = 3
[1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,2,8,4,5,7,6,3] => ?
=> ? = 4
[1,0,1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,2,7,4,6,5,3,8] => ?
=> ? = 4
[1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,2,7,4,6,5,8,3] => ?
=> ? = 4
[1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,2,6,5,4,3,8,7] => ?
=> ? = 4
[1,0,1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,2,6,5,4,7,3,8] => ?
=> ? = 4
[1,0,1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,2,7,5,4,6,3,8] => ?
=> ? = 4
[1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,2,7,5,6,4,3,8] => ?
=> ? = 4
[1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,2,7,5,6,4,8,3] => ?
=> ? = 4
[1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,2,8,6,5,7,4,3] => ?
=> ? = 5
[1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,3,8,4,6,7,5,2] => ?
=> ? = 4
[1,0,1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [1,3,6,5,4,2,7,8] => ?
=> ? = 4
[1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,3,8,5,6,4,7,2] => ?
=> ? = 4
[1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,4,3,2,6,5,8,7] => ?
=> ? = 3
[1,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [1,4,3,5,2,7,6,8] => ?
=> ? = 3
[1,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [1,4,3,6,5,7,2,8] => ?
=> ? = 3
[1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,5,3,4,8,6,7,2] => ?
=> ? = 3
[1,0,1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,6,3,4,5,2,7,8] => ?
=> ? = 3
[1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,6,3,4,5,7,8,2] => ?
=> ? = 3
[1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,7,3,4,5,6,2,8] => ?
=> ? = 3
[1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [1,7,3,4,6,5,8,2] => ?
=> ? = 4
[1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,8,3,4,7,6,5,2] => ?
=> ? = 5
[1,0,1,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> [1,6,3,5,4,7,2,8] => ?
=> ? = 4
[1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,6,3,5,4,7,8,2] => ?
=> ? = 4
[1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,7,3,5,6,4,8,2] => ?
=> ? = 4
[1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,8,3,6,5,7,4,2] => ?
=> ? = 5
[1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,5,4,3,2,6,8,7] => ?
=> ? = 4
[1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,6,5,4,3,7,8,2] => ?
=> ? = 5
[1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,7,5,4,6,3,8,2] => ?
=> ? = 5
[1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,1,3,6,5,4,8,7] => ?
=> ? = 3
[1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,7,6,5,4,8] => ?
=> ? = 4
[1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,4,3,7,6,5,8] => ?
=> ? = 3
[1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,1,5,4,3,6,8,7] => ?
=> ? = 3
[1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,1,6,5,4,3,7,8] => ?
=> ? = 4
[1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [2,3,1,5,8,6,7,4] => ?
=> ? = 3
[1,1,0,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [2,3,1,6,5,7,4,8] => ?
=> ? = 3
[1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [2,3,1,7,5,6,8,4] => ?
=> ? = 3
[1,1,0,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,8,5,7,6,4] => ?
=> ? = 4
[1,1,0,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [2,3,4,6,5,1,7,8] => ?
=> ? = 3
[1,1,0,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [2,3,5,4,6,8,7,1] => ?
=> ? = 3
[1,1,0,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [2,3,5,4,8,7,6,1] => ?
=> ? = 4
[1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [2,3,6,4,5,7,8,1] => ?
=> ? = 3
[1,1,0,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,6,5,4,1,7,8] => ?
=> ? = 4
[1,1,0,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [2,3,6,5,4,8,7,1] => ?
=> ? = 4
[1,1,0,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [2,3,8,7,5,6,4,1] => ?
=> ? = 5
[1,1,0,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [2,4,3,5,1,6,8,7] => ?
=> ? = 3
[1,1,0,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [2,4,3,5,1,7,6,8] => ?
=> ? = 3
[1,1,0,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,7,6,8,1] => ?
=> ? = 3
Description
The length of the partition.
Matching statistic: St000378
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 72%●distinct values known / distinct values provided: 67%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 72%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1] => [1]
=> [1]
=> 1
[1,0,1,0]
=> [1,2] => [2]
=> [1,1]
=> 1
[1,1,0,0]
=> [2,1] => [1,1]
=> [2]
=> 2
[1,0,1,0,1,0]
=> [1,2,3] => [3]
=> [1,1,1]
=> 1
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> [3]
=> 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> [3]
=> 2
[1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> [3]
=> 2
[1,1,1,0,0,0]
=> [3,2,1] => [1,1,1]
=> [2,1]
=> 3
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4]
=> [1,1,1,1]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [3,1]
=> [2,1,1]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,1]
=> [2,1,1]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,1]
=> [2,1,1]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [2,1,1]
=> [2,2]
=> 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [3,1]
=> [2,1,1]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> [4]
=> 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1]
=> [2,1,1]
=> 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [3,1]
=> [2,1,1]
=> 2
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,1,1]
=> [2,2]
=> 3
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,1,1]
=> [2,2]
=> 3
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,1,1]
=> [2,2]
=> 3
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,1,1]
=> [2,2]
=> 3
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1]
=> [3,1]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [5]
=> [1,1,1,1,1]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [4,1]
=> [2,1,1,1]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [4,1]
=> [2,1,1,1]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [4,1]
=> [2,1,1,1]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,1,1]
=> [4,1]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [4,1]
=> [2,1,1,1]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [3,2]
=> [5]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [4,1]
=> [2,1,1,1]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [4,1]
=> [2,1,1,1]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [3,1,1]
=> [4,1]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [3,1,1]
=> [4,1]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [3,1,1]
=> [4,1]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [3,1,1]
=> [4,1]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,1,1,1]
=> [3,1,1]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [4,1]
=> [2,1,1,1]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [3,2]
=> [5]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [3,2]
=> [5]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,2]
=> [5]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,2,1]
=> [2,2,1]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [4,1]
=> [2,1,1,1]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2]
=> [5]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1]
=> [2,1,1,1]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [4,1]
=> [2,1,1,1]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [3,1,1]
=> [4,1]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,1,1]
=> [4,1]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,1,1]
=> [4,1]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,1,1]
=> [4,1]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,1,1,1]
=> [3,1,1]
=> 4
[1,0,1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,2,4,7,6,5,3,8] => ?
=> ?
=> ? = 4
[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,2,5,4,3,8,7,6] => ?
=> ?
=> ? = 3
[1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,2,7,4,5,6,3,8] => ?
=> ?
=> ? = 3
[1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,2,8,4,5,7,6,3] => ?
=> ?
=> ? = 4
[1,0,1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,2,7,4,6,5,3,8] => ?
=> ?
=> ? = 4
[1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,2,7,4,6,5,8,3] => ?
=> ?
=> ? = 4
[1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,2,6,5,4,3,8,7] => ?
=> ?
=> ? = 4
[1,0,1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,2,6,5,4,7,3,8] => ?
=> ?
=> ? = 4
[1,0,1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,2,7,5,4,6,3,8] => ?
=> ?
=> ? = 4
[1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,2,7,5,6,4,3,8] => ?
=> ?
=> ? = 4
[1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,2,7,5,6,4,8,3] => ?
=> ?
=> ? = 4
[1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,2,8,6,5,7,4,3] => ?
=> ?
=> ? = 5
[1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,3,8,4,6,7,5,2] => ?
=> ?
=> ? = 4
[1,0,1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [1,3,6,5,4,2,7,8] => ?
=> ?
=> ? = 4
[1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,3,8,5,6,4,7,2] => ?
=> ?
=> ? = 4
[1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,4,3,2,6,5,8,7] => ?
=> ?
=> ? = 3
[1,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [1,4,3,5,2,7,6,8] => ?
=> ?
=> ? = 3
[1,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [1,4,3,6,5,7,2,8] => ?
=> ?
=> ? = 3
[1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,5,3,4,8,6,7,2] => ?
=> ?
=> ? = 3
[1,0,1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,6,3,4,5,2,7,8] => ?
=> ?
=> ? = 3
[1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,6,3,4,5,7,8,2] => ?
=> ?
=> ? = 3
[1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,7,3,4,5,6,2,8] => ?
=> ?
=> ? = 3
[1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [1,7,3,4,6,5,8,2] => ?
=> ?
=> ? = 4
[1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,8,3,4,7,6,5,2] => ?
=> ?
=> ? = 5
[1,0,1,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> [1,6,3,5,4,7,2,8] => ?
=> ?
=> ? = 4
[1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,6,3,5,4,7,8,2] => ?
=> ?
=> ? = 4
[1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,7,3,5,6,4,8,2] => ?
=> ?
=> ? = 4
[1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,8,3,6,5,7,4,2] => ?
=> ?
=> ? = 5
[1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,5,4,3,2,6,8,7] => ?
=> ?
=> ? = 4
[1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,6,5,4,3,7,8,2] => ?
=> ?
=> ? = 5
[1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,7,5,4,6,3,8,2] => ?
=> ?
=> ? = 5
[1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,1,3,6,5,4,8,7] => ?
=> ?
=> ? = 3
[1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,7,6,5,4,8] => ?
=> ?
=> ? = 4
[1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,4,3,7,6,5,8] => ?
=> ?
=> ? = 3
[1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,1,5,4,3,6,8,7] => ?
=> ?
=> ? = 3
[1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,1,6,5,4,3,7,8] => ?
=> ?
=> ? = 4
[1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [2,3,1,5,8,6,7,4] => ?
=> ?
=> ? = 3
[1,1,0,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [2,3,1,6,5,7,4,8] => ?
=> ?
=> ? = 3
[1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [2,3,1,7,5,6,8,4] => ?
=> ?
=> ? = 3
[1,1,0,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,8,5,7,6,4] => ?
=> ?
=> ? = 4
[1,1,0,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [2,3,4,6,5,1,7,8] => ?
=> ?
=> ? = 3
[1,1,0,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [2,3,5,4,6,8,7,1] => ?
=> ?
=> ? = 3
[1,1,0,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [2,3,5,4,8,7,6,1] => ?
=> ?
=> ? = 4
[1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [2,3,6,4,5,7,8,1] => ?
=> ?
=> ? = 3
[1,1,0,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,6,5,4,1,7,8] => ?
=> ?
=> ? = 4
[1,1,0,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [2,3,6,5,4,8,7,1] => ?
=> ?
=> ? = 4
[1,1,0,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [2,3,8,7,5,6,4,1] => ?
=> ?
=> ? = 5
[1,1,0,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [2,4,3,5,1,6,8,7] => ?
=> ?
=> ? = 3
[1,1,0,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [2,4,3,5,1,7,6,8] => ?
=> ?
=> ? = 3
[1,1,0,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,7,6,8,1] => ?
=> ?
=> ? = 3
Description
The diagonal inversion number of an integer partition.
The dinv of a partition is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \in \{0,1\}$.
See also exercise 3.19 of [2].
This statistic is equidistributed with the length of the partition, see [3].
Matching statistic: St000288
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 56% ●values known / values provided: 71%●distinct values known / distinct values provided: 56%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 56% ●values known / values provided: 71%●distinct values known / distinct values provided: 56%
Values
[1,0]
=> [1] => [1]
=> 10 => 1
[1,0,1,0]
=> [1,2] => [2]
=> 100 => 1
[1,1,0,0]
=> [2,1] => [1,1]
=> 110 => 2
[1,0,1,0,1,0]
=> [1,2,3] => [3]
=> 1000 => 1
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> 1010 => 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> 1010 => 2
[1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> 1010 => 2
[1,1,1,0,0,0]
=> [3,2,1] => [1,1,1]
=> 1110 => 3
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4]
=> 10000 => 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [3,1]
=> 10010 => 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,1]
=> 10010 => 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,1]
=> 10010 => 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [2,1,1]
=> 10110 => 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [3,1]
=> 10010 => 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> 1100 => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1]
=> 10010 => 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [3,1]
=> 10010 => 2
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,1,1]
=> 10110 => 3
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,1,1]
=> 10110 => 3
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,1,1]
=> 10110 => 3
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,1,1]
=> 10110 => 3
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1]
=> 11110 => 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [5]
=> 100000 => 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [4,1]
=> 100010 => 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [4,1]
=> 100010 => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [4,1]
=> 100010 => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,1,1]
=> 100110 => 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [4,1]
=> 100010 => 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [3,2]
=> 10100 => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [4,1]
=> 100010 => 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [4,1]
=> 100010 => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [3,1,1]
=> 100110 => 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [3,1,1]
=> 100110 => 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [3,1,1]
=> 100110 => 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [3,1,1]
=> 100110 => 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,1,1,1]
=> 101110 => 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [4,1]
=> 100010 => 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [3,2]
=> 10100 => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [3,2]
=> 10100 => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,2]
=> 10100 => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,2,1]
=> 11010 => 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [4,1]
=> 100010 => 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2]
=> 10100 => 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1]
=> 100010 => 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [4,1]
=> 100010 => 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [3,1,1]
=> 100110 => 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,1,1]
=> 100110 => 3
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,1,1]
=> 100110 => 3
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,1,1]
=> 100110 => 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,1,1,1]
=> 101110 => 4
[1,0,1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,2,4,7,6,5,3,8] => ?
=> ? => ? = 4
[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,2,5,4,3,8,7,6] => ?
=> ? => ? = 3
[1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,2,7,4,5,6,3,8] => ?
=> ? => ? = 3
[1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,2,8,4,5,7,6,3] => ?
=> ? => ? = 4
[1,0,1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,2,7,4,6,5,3,8] => ?
=> ? => ? = 4
[1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,2,7,4,6,5,8,3] => ?
=> ? => ? = 4
[1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,2,6,5,4,3,8,7] => ?
=> ? => ? = 4
[1,0,1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,2,6,5,4,7,3,8] => ?
=> ? => ? = 4
[1,0,1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,2,7,5,4,6,3,8] => ?
=> ? => ? = 4
[1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,2,7,5,6,4,3,8] => ?
=> ? => ? = 4
[1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,2,7,5,6,4,8,3] => ?
=> ? => ? = 4
[1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,2,8,6,5,7,4,3] => ?
=> ? => ? = 5
[1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,3,8,4,6,7,5,2] => ?
=> ? => ? = 4
[1,0,1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [1,3,6,5,4,2,7,8] => ?
=> ? => ? = 4
[1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,3,8,5,6,4,7,2] => ?
=> ? => ? = 4
[1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,4,3,2,6,5,8,7] => ?
=> ? => ? = 3
[1,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [1,4,3,5,2,7,6,8] => ?
=> ? => ? = 3
[1,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [1,4,3,6,5,7,2,8] => ?
=> ? => ? = 3
[1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,5,3,4,8,6,7,2] => ?
=> ? => ? = 3
[1,0,1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,6,3,4,5,2,7,8] => ?
=> ? => ? = 3
[1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,6,3,4,5,7,8,2] => ?
=> ? => ? = 3
[1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,7,3,4,5,6,2,8] => ?
=> ? => ? = 3
[1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [1,7,3,4,6,5,8,2] => ?
=> ? => ? = 4
[1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,8,3,4,7,6,5,2] => ?
=> ? => ? = 5
[1,0,1,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> [1,6,3,5,4,7,2,8] => ?
=> ? => ? = 4
[1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,6,3,5,4,7,8,2] => ?
=> ? => ? = 4
[1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,7,3,5,6,4,8,2] => ?
=> ? => ? = 4
[1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,8,3,6,5,7,4,2] => ?
=> ? => ? = 5
[1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,5,4,3,2,6,8,7] => ?
=> ? => ? = 4
[1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,6,5,4,3,7,8,2] => ?
=> ? => ? = 5
[1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,7,5,4,6,3,8,2] => ?
=> ? => ? = 5
[1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,1,3,6,5,4,8,7] => ?
=> ? => ? = 3
[1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,7,6,5,4,8] => ?
=> ? => ? = 4
[1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,4,3,7,6,5,8] => ?
=> ? => ? = 3
[1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,1,5,4,3,6,8,7] => ?
=> ? => ? = 3
[1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,1,6,5,4,3,7,8] => ?
=> ? => ? = 4
[1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [2,3,1,5,8,6,7,4] => ?
=> ? => ? = 3
[1,1,0,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [2,3,1,6,5,7,4,8] => ?
=> ? => ? = 3
[1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [2,3,1,7,5,6,8,4] => ?
=> ? => ? = 3
[1,1,0,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,8,5,7,6,4] => ?
=> ? => ? = 4
[1,1,0,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [2,3,4,6,5,1,7,8] => ?
=> ? => ? = 3
[1,1,0,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [2,3,5,4,6,8,7,1] => ?
=> ? => ? = 3
[1,1,0,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [2,3,5,4,8,7,6,1] => ?
=> ? => ? = 4
[1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [2,3,6,4,5,7,8,1] => ?
=> ? => ? = 3
[1,1,0,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,6,5,4,1,7,8] => ?
=> ? => ? = 4
[1,1,0,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [2,3,6,5,4,8,7,1] => ?
=> ? => ? = 4
[1,1,0,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [2,3,8,7,5,6,4,1] => ?
=> ? => ? = 5
[1,1,0,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [2,4,3,5,1,6,8,7] => ?
=> ? => ? = 3
[1,1,0,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [2,4,3,5,1,7,6,8] => ?
=> ? => ? = 3
[1,1,0,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,7,6,8,1] => ?
=> ? => ? = 3
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000733
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 56% ●values known / values provided: 71%●distinct values known / distinct values provided: 56%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 56% ●values known / values provided: 71%●distinct values known / distinct values provided: 56%
Values
[1,0]
=> [1] => [1]
=> [[1]]
=> 1
[1,0,1,0]
=> [1,2] => [2]
=> [[1,2]]
=> 1
[1,1,0,0]
=> [2,1] => [1,1]
=> [[1],[2]]
=> 2
[1,0,1,0,1,0]
=> [1,2,3] => [3]
=> [[1,2,3]]
=> 1
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> [[1,2],[3]]
=> 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> [[1,2],[3]]
=> 2
[1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> [[1,2],[3]]
=> 2
[1,1,1,0,0,0]
=> [3,2,1] => [1,1,1]
=> [[1],[2],[3]]
=> 3
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4]
=> [[1,2,3,4]]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [3,1]
=> [[1,2,3],[4]]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,1]
=> [[1,2,3],[4]]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,1]
=> [[1,2,3],[4]]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [3,1]
=> [[1,2,3],[4]]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1]
=> [[1,2,3],[4]]
=> 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [3,1]
=> [[1,2,3],[4]]
=> 2
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> 3
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> 3
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> 3
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> 3
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [5]
=> [[1,2,3,4,5]]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [4,1]
=> [[1,2,3,4],[5]]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [4,1]
=> [[1,2,3,4],[5]]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [4,1]
=> [[1,2,3,4],[5]]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [4,1]
=> [[1,2,3,4],[5]]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [3,2]
=> [[1,2,3],[4,5]]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [4,1]
=> [[1,2,3,4],[5]]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [4,1]
=> [[1,2,3,4],[5]]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [4,1]
=> [[1,2,3,4],[5]]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [3,2]
=> [[1,2,3],[4,5]]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [3,2]
=> [[1,2,3],[4,5]]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,2]
=> [[1,2,3],[4,5]]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [4,1]
=> [[1,2,3,4],[5]]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2]
=> [[1,2,3],[4,5]]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1]
=> [[1,2,3,4],[5]]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [4,1]
=> [[1,2,3,4],[5]]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 4
[1,0,1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,2,4,7,6,5,3,8] => ?
=> ?
=> ? = 4
[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,2,5,4,3,8,7,6] => ?
=> ?
=> ? = 3
[1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,2,7,4,5,6,3,8] => ?
=> ?
=> ? = 3
[1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,2,8,4,5,7,6,3] => ?
=> ?
=> ? = 4
[1,0,1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,2,7,4,6,5,3,8] => ?
=> ?
=> ? = 4
[1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,2,7,4,6,5,8,3] => ?
=> ?
=> ? = 4
[1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,2,6,5,4,3,8,7] => ?
=> ?
=> ? = 4
[1,0,1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,2,6,5,4,7,3,8] => ?
=> ?
=> ? = 4
[1,0,1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,2,7,5,4,6,3,8] => ?
=> ?
=> ? = 4
[1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,2,7,5,6,4,3,8] => ?
=> ?
=> ? = 4
[1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,2,7,5,6,4,8,3] => ?
=> ?
=> ? = 4
[1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,2,8,6,5,7,4,3] => ?
=> ?
=> ? = 5
[1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,3,8,4,6,7,5,2] => ?
=> ?
=> ? = 4
[1,0,1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [1,3,6,5,4,2,7,8] => ?
=> ?
=> ? = 4
[1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,3,8,5,6,4,7,2] => ?
=> ?
=> ? = 4
[1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,4,3,2,6,5,8,7] => ?
=> ?
=> ? = 3
[1,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [1,4,3,5,2,7,6,8] => ?
=> ?
=> ? = 3
[1,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [1,4,3,6,5,7,2,8] => ?
=> ?
=> ? = 3
[1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,5,3,4,8,6,7,2] => ?
=> ?
=> ? = 3
[1,0,1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,6,3,4,5,2,7,8] => ?
=> ?
=> ? = 3
[1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,6,3,4,5,7,8,2] => ?
=> ?
=> ? = 3
[1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,7,3,4,5,6,2,8] => ?
=> ?
=> ? = 3
[1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [1,7,3,4,6,5,8,2] => ?
=> ?
=> ? = 4
[1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,8,3,4,7,6,5,2] => ?
=> ?
=> ? = 5
[1,0,1,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> [1,6,3,5,4,7,2,8] => ?
=> ?
=> ? = 4
[1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,6,3,5,4,7,8,2] => ?
=> ?
=> ? = 4
[1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,7,3,5,6,4,8,2] => ?
=> ?
=> ? = 4
[1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,8,3,6,5,7,4,2] => ?
=> ?
=> ? = 5
[1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,5,4,3,2,6,8,7] => ?
=> ?
=> ? = 4
[1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,6,5,4,3,7,8,2] => ?
=> ?
=> ? = 5
[1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,7,5,4,6,3,8,2] => ?
=> ?
=> ? = 5
[1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,1,3,6,5,4,8,7] => ?
=> ?
=> ? = 3
[1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,7,6,5,4,8] => ?
=> ?
=> ? = 4
[1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,4,3,7,6,5,8] => ?
=> ?
=> ? = 3
[1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,1,5,4,3,6,8,7] => ?
=> ?
=> ? = 3
[1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,1,6,5,4,3,7,8] => ?
=> ?
=> ? = 4
[1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [2,3,1,5,8,6,7,4] => ?
=> ?
=> ? = 3
[1,1,0,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [2,3,1,6,5,7,4,8] => ?
=> ?
=> ? = 3
[1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [2,3,1,7,5,6,8,4] => ?
=> ?
=> ? = 3
[1,1,0,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,8,5,7,6,4] => ?
=> ?
=> ? = 4
[1,1,0,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [2,3,4,6,5,1,7,8] => ?
=> ?
=> ? = 3
[1,1,0,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [2,3,5,4,6,8,7,1] => ?
=> ?
=> ? = 3
[1,1,0,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [2,3,5,4,8,7,6,1] => ?
=> ?
=> ? = 4
[1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [2,3,6,4,5,7,8,1] => ?
=> ?
=> ? = 3
[1,1,0,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,6,5,4,1,7,8] => ?
=> ?
=> ? = 4
[1,1,0,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [2,3,6,5,4,8,7,1] => ?
=> ?
=> ? = 4
[1,1,0,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [2,3,8,7,5,6,4,1] => ?
=> ?
=> ? = 5
[1,1,0,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [2,4,3,5,1,6,8,7] => ?
=> ?
=> ? = 3
[1,1,0,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [2,4,3,5,1,7,6,8] => ?
=> ?
=> ? = 3
[1,1,0,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,7,6,8,1] => ?
=> ?
=> ? = 3
Description
The row containing the largest entry of a standard tableau.
Matching statistic: St000394
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 61% ●values known / values provided: 69%●distinct values known / distinct values provided: 61%
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 61% ●values known / values provided: 69%●distinct values known / distinct values provided: 61%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 6 - 1
[1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 5 - 1
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 6 - 1
[1,1,0,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 5 - 1
[1,1,0,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 5 - 1
[1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 5 - 1
[1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 5 - 1
[1,1,0,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 5 - 1
[1,1,0,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 5 - 1
[1,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 5 - 1
[1,1,0,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 5 - 1
[1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 6 - 1
[1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> ? = 6 - 1
[1,1,0,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 5 - 1
[1,1,0,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 5 - 1
[1,1,0,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 5 - 1
[1,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 6 - 1
[1,1,0,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> ? = 6 - 1
[1,1,1,0,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 5 - 1
[1,1,1,0,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 5 - 1
[1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 6 - 1
[1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 6 - 1
[1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 5 - 1
[1,1,1,0,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 6 - 1
[1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 6 - 1
[1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 6 - 1
[1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 6 - 1
[1,1,1,1,0,1,1,1,0,0,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> ? = 6 - 1
[1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 6 - 1
[1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 5 - 1
[1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 5 - 1
[1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 5 - 1
[1,1,1,1,1,0,0,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 5 - 1
[1,1,1,1,1,0,0,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 5 - 1
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 7 - 1
[1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 6 - 1
[1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 6 - 1
[1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 6 - 1
[1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 6 - 1
[1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> ? = 6 - 1
[1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 6 - 1
[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 7 - 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 7 - 1
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 7 - 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 10 - 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 8 - 1
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 7 - 1
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 7 - 1
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 7 - 1
[1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 7 - 1
Description
The sum of the heights of the peaks of a Dyck path minus the number of peaks.
Matching statistic: St001389
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001389: Integer partitions ⟶ ℤResult quality: 61% ●values known / values provided: 67%●distinct values known / distinct values provided: 61%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001389: Integer partitions ⟶ ℤResult quality: 61% ●values known / values provided: 67%●distinct values known / distinct values provided: 61%
Values
[1,0]
=> [1] => [1] => [1]
=> 1
[1,0,1,0]
=> [1,2] => [1,2] => [1,1]
=> 1
[1,1,0,0]
=> [2,1] => [2,1] => [2]
=> 2
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,1,1]
=> 1
[1,0,1,1,0,0]
=> [1,3,2] => [3,1,2] => [2,1]
=> 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,1]
=> 2
[1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => [2,1]
=> 2
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => [3]
=> 3
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [4,1,2,3] => [2,1,1]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,1,2,4] => [2,1,1]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,4,1,2] => [2,1,1]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [4,3,1,2] => [3,1]
=> 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,1]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,4,1,3] => [2,1,1]
=> 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1,4] => [2,1,1]
=> 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,3,4,1] => [2,1,1]
=> 2
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,2,3,1] => [3,1]
=> 3
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => [3,1]
=> 3
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => [3,1]
=> 3
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,4,2,1] => [3,1]
=> 3
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [4]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [5,1,2,3,4] => [2,1,1,1]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [4,1,2,3,5] => [2,1,1,1]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [4,5,1,2,3] => [2,1,1,1]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [5,4,1,2,3] => [3,1,1]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [3,1,2,4,5] => [2,1,1,1]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [3,5,1,2,4] => [2,1,1,1]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [3,4,1,2,5] => [2,1,1,1]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [3,4,5,1,2] => [2,1,1,1]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [5,3,4,1,2] => [3,1,1]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [4,3,1,2,5] => [3,1,1]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [4,3,5,1,2] => [3,1,1]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [4,5,3,1,2] => [3,1,1]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [5,4,3,1,2] => [4,1]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,1,1]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,5,1,3,4] => [2,1,1,1]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,4,1,3,5] => [2,1,1,1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,4,5,1,3] => [2,1,1,1]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [5,2,4,1,3] => [3,1,1]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3,1,4,5] => [2,1,1,1]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,3,5,1,4] => [2,1,1,1]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,3,4,1,5] => [2,1,1,1]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,3,4,5,1] => [2,1,1,1]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [5,2,3,4,1] => [3,1,1]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,2,3,1,5] => [3,1,1]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [4,2,3,5,1] => [3,1,1]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [4,5,2,3,1] => [3,1,1]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,4,2,3,1] => [4,1]
=> 4
[1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,2,5,7,6,4,8,3] => [7,5,6,4,8,1,2,3] => ?
=> ? = 4
[1,0,1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,2,6,5,4,7,3,8] => ? => ?
=> ? = 4
[1,0,1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,2,6,5,7,4,3,8] => ? => ?
=> ? = 4
[1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,2,6,7,5,4,3,8] => [6,7,5,4,1,2,3,8] => ?
=> ? = 4
[1,0,1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [1,3,6,5,4,2,7,8] => [6,5,3,4,1,2,7,8] => ?
=> ? = 4
[1,0,1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [1,4,5,3,2,6,7,8] => [4,5,3,1,2,6,7,8] => ?
=> ? = 3
[1,0,1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [1,4,5,3,6,7,2,8] => [4,5,3,6,7,1,2,8] => ?
=> ? = 3
[1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,4,5,3,7,8,6,2] => [4,5,7,8,3,6,1,2] => ?
=> ? = 3
[1,0,1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,4,5,6,3,2,7,8] => [4,5,6,3,1,2,7,8] => ?
=> ? = 3
[1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [1,4,5,7,6,3,8,2] => [7,4,5,6,3,8,1,2] => ?
=> ? = 4
[1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,4,5,7,8,6,3,2] => [7,8,4,5,6,3,1,2] => ?
=> ? = 4
[1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,4,5,8,7,6,3,2] => [8,7,4,5,6,3,1,2] => ?
=> ? = 5
[1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,4,6,5,3,7,8,2] => [6,4,5,3,7,8,1,2] => ?
=> ? = 4
[1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [1,4,7,6,5,3,8,2] => [7,6,4,5,3,8,1,2] => ?
=> ? = 5
[1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,4,7,6,8,5,3,2] => [7,6,8,4,5,3,1,2] => ?
=> ? = 5
[1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,5,4,3,2,6,8,7] => [5,4,3,8,1,2,6,7] => ?
=> ? = 4
[1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,5,4,3,2,7,6,8] => [5,4,3,7,1,2,6,8] => ?
=> ? = 4
[1,0,1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [1,5,4,6,3,2,7,8] => [5,4,6,3,1,2,7,8] => ?
=> ? = 4
[1,0,1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [1,5,6,4,3,2,7,8] => [5,6,4,3,1,2,7,8] => ?
=> ? = 4
[1,0,1,1,1,1,0,1,0,0,0,0,1,1,0,0]
=> [1,5,6,4,3,2,8,7] => [5,6,4,3,8,1,2,7] => ?
=> ? = 4
[1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,1,3,4,5,8,7,6] => [8,2,7,1,3,4,5,6] => ?
=> ? = 3
[1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,3,5,4,8,7,6] => [8,2,5,7,1,3,4,6] => ?
=> ? = 3
[1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,1,3,6,5,4,8,7] => [6,2,5,8,1,3,4,7] => ?
=> ? = 3
[1,1,0,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,1,3,6,7,8,5,4] => [6,7,8,2,5,1,3,4] => ?
=> ? = 3
[1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,7,6,5,4,8] => [7,6,2,5,1,3,4,8] => ?
=> ? = 4
[1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,4,3,5,8,7,6] => [8,2,4,7,1,3,5,6] => ?
=> ? = 3
[1,1,0,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,1,4,5,8,7,6,3] => [8,7,2,4,5,6,1,3] => ?
=> ? = 4
[1,1,0,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,1,5,4,6,7,8,3] => [5,2,4,6,7,8,1,3] => ?
=> ? = 3
[1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,1,6,5,4,3,7,8] => ? => ?
=> ? = 4
[1,1,0,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [2,1,6,8,7,5,4,3] => [8,6,7,5,2,4,1,3] => ?
=> ? = 5
[1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,1,7,6,5,4,3,8] => [7,6,5,2,4,1,3,8] => ?
=> ? = 5
[1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,1,7,8,6,5,4,3] => [7,8,6,5,2,4,1,3] => ?
=> ? = 5
[1,1,0,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [2,3,1,6,5,4,7,8] => [6,2,3,5,1,4,7,8] => ?
=> ? = 3
[1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [2,3,1,6,7,5,8,4] => [6,7,2,3,5,8,1,4] => ?
=> ? = 3
[1,1,0,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,6,8,7,5,4] => [8,6,7,2,3,5,1,4] => ?
=> ? = 4
[1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,1,8,7,6,5,4] => [8,7,6,2,3,5,1,4] => ?
=> ? = 5
[1,1,0,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [2,3,4,1,5,8,7,6] => [8,2,3,4,7,1,5,6] => ?
=> ? = 3
[1,1,0,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> [2,3,4,1,7,6,5,8] => [7,2,3,4,6,1,5,8] => ?
=> ? = 3
[1,1,0,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [2,3,4,6,5,8,7,1] => [6,8,2,3,4,5,7,1] => ?
=> ? = 3
[1,1,0,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [2,3,5,4,6,8,7,1] => ? => ?
=> ? = 3
[1,1,0,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [2,3,5,4,8,7,6,1] => [8,5,7,2,3,4,6,1] => ?
=> ? = 4
[1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [2,3,5,6,4,7,8,1] => [5,6,2,3,4,7,8,1] => ?
=> ? = 3
[1,1,0,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [2,3,6,5,4,8,7,1] => ? => ?
=> ? = 4
[1,1,0,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [2,3,7,8,6,5,4,1] => [7,8,6,5,2,3,4,1] => ?
=> ? = 5
[1,1,0,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [2,4,3,5,1,7,6,8] => [4,2,3,5,7,1,6,8] => ?
=> ? = 3
[1,1,0,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,7,6,8,1] => [4,7,2,3,5,6,8,1] => ?
=> ? = 3
[1,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [2,4,3,5,8,7,6,1] => [8,4,7,2,3,5,6,1] => ?
=> ? = 4
[1,1,0,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [2,4,3,6,5,7,8,1] => [4,6,2,3,5,7,8,1] => ?
=> ? = 3
[1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [2,4,3,7,6,8,5,1] => [7,4,6,8,2,3,5,1] => ?
=> ? = 4
[1,1,0,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [2,4,3,7,8,6,5,1] => [7,8,4,6,2,3,5,1] => ?
=> ? = 4
Description
The number of partitions of the same length below the given integer partition.
For a partition $\lambda_1 \geq \dots \lambda_k > 0$, this number is
$$ \det\left( \binom{\lambda_{k+1-i}}{j-i+1} \right)_{1 \le i,j \le k}.$$
Matching statistic: St001918
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001918: Integer partitions ⟶ ℤResult quality: 61% ●values known / values provided: 67%●distinct values known / distinct values provided: 61%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001918: Integer partitions ⟶ ℤResult quality: 61% ●values known / values provided: 67%●distinct values known / distinct values provided: 61%
Values
[1,0]
=> [1] => [1] => [1]
=> 0 = 1 - 1
[1,0,1,0]
=> [1,2] => [1,2] => [1,1]
=> 0 = 1 - 1
[1,1,0,0]
=> [2,1] => [2,1] => [2]
=> 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,1,1]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [3,1,2] => [2,1]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,1]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => [2,1]
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => [3]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [4,1,2,3] => [2,1,1]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,1,2,4] => [2,1,1]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,4,1,2] => [2,1,1]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [4,3,1,2] => [3,1]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,1]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,4,1,3] => [2,1,1]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1,4] => [2,1,1]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,3,4,1] => [2,1,1]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,2,3,1] => [3,1]
=> 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => [3,1]
=> 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => [3,1]
=> 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,4,2,1] => [3,1]
=> 2 = 3 - 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [4]
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [5,1,2,3,4] => [2,1,1,1]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [4,1,2,3,5] => [2,1,1,1]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [4,5,1,2,3] => [2,1,1,1]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [5,4,1,2,3] => [3,1,1]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [3,1,2,4,5] => [2,1,1,1]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [3,5,1,2,4] => [2,1,1,1]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [3,4,1,2,5] => [2,1,1,1]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [3,4,5,1,2] => [2,1,1,1]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [5,3,4,1,2] => [3,1,1]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [4,3,1,2,5] => [3,1,1]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [4,3,5,1,2] => [3,1,1]
=> 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [4,5,3,1,2] => [3,1,1]
=> 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [5,4,3,1,2] => [4,1]
=> 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,1,1]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,5,1,3,4] => [2,1,1,1]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,4,1,3,5] => [2,1,1,1]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,4,5,1,3] => [2,1,1,1]
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [5,2,4,1,3] => [3,1,1]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3,1,4,5] => [2,1,1,1]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,3,5,1,4] => [2,1,1,1]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,3,4,1,5] => [2,1,1,1]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,3,4,5,1] => [2,1,1,1]
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [5,2,3,4,1] => [3,1,1]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,2,3,1,5] => [3,1,1]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [4,2,3,5,1] => [3,1,1]
=> 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [4,5,2,3,1] => [3,1,1]
=> 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,4,2,3,1] => [4,1]
=> 3 = 4 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,2,5,7,6,4,8,3] => [7,5,6,4,8,1,2,3] => ?
=> ? = 4 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,2,6,5,4,7,3,8] => ? => ?
=> ? = 4 - 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,2,6,5,7,4,3,8] => ? => ?
=> ? = 4 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,2,6,7,5,4,3,8] => [6,7,5,4,1,2,3,8] => ?
=> ? = 4 - 1
[1,0,1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [1,3,6,5,4,2,7,8] => [6,5,3,4,1,2,7,8] => ?
=> ? = 4 - 1
[1,0,1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [1,4,5,3,2,6,7,8] => [4,5,3,1,2,6,7,8] => ?
=> ? = 3 - 1
[1,0,1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [1,4,5,3,6,7,2,8] => [4,5,3,6,7,1,2,8] => ?
=> ? = 3 - 1
[1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,4,5,3,7,8,6,2] => [4,5,7,8,3,6,1,2] => ?
=> ? = 3 - 1
[1,0,1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,4,5,6,3,2,7,8] => [4,5,6,3,1,2,7,8] => ?
=> ? = 3 - 1
[1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [1,4,5,7,6,3,8,2] => [7,4,5,6,3,8,1,2] => ?
=> ? = 4 - 1
[1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,4,5,7,8,6,3,2] => [7,8,4,5,6,3,1,2] => ?
=> ? = 4 - 1
[1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,4,5,8,7,6,3,2] => [8,7,4,5,6,3,1,2] => ?
=> ? = 5 - 1
[1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,4,6,5,3,7,8,2] => [6,4,5,3,7,8,1,2] => ?
=> ? = 4 - 1
[1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [1,4,7,6,5,3,8,2] => [7,6,4,5,3,8,1,2] => ?
=> ? = 5 - 1
[1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,4,7,6,8,5,3,2] => [7,6,8,4,5,3,1,2] => ?
=> ? = 5 - 1
[1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,5,4,3,2,6,8,7] => [5,4,3,8,1,2,6,7] => ?
=> ? = 4 - 1
[1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,5,4,3,2,7,6,8] => [5,4,3,7,1,2,6,8] => ?
=> ? = 4 - 1
[1,0,1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [1,5,4,6,3,2,7,8] => [5,4,6,3,1,2,7,8] => ?
=> ? = 4 - 1
[1,0,1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [1,5,6,4,3,2,7,8] => [5,6,4,3,1,2,7,8] => ?
=> ? = 4 - 1
[1,0,1,1,1,1,0,1,0,0,0,0,1,1,0,0]
=> [1,5,6,4,3,2,8,7] => [5,6,4,3,8,1,2,7] => ?
=> ? = 4 - 1
[1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,1,3,4,5,8,7,6] => [8,2,7,1,3,4,5,6] => ?
=> ? = 3 - 1
[1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,3,5,4,8,7,6] => [8,2,5,7,1,3,4,6] => ?
=> ? = 3 - 1
[1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,1,3,6,5,4,8,7] => [6,2,5,8,1,3,4,7] => ?
=> ? = 3 - 1
[1,1,0,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,1,3,6,7,8,5,4] => [6,7,8,2,5,1,3,4] => ?
=> ? = 3 - 1
[1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,7,6,5,4,8] => [7,6,2,5,1,3,4,8] => ?
=> ? = 4 - 1
[1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,4,3,5,8,7,6] => [8,2,4,7,1,3,5,6] => ?
=> ? = 3 - 1
[1,1,0,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,1,4,5,8,7,6,3] => [8,7,2,4,5,6,1,3] => ?
=> ? = 4 - 1
[1,1,0,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,1,5,4,6,7,8,3] => [5,2,4,6,7,8,1,3] => ?
=> ? = 3 - 1
[1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,1,6,5,4,3,7,8] => ? => ?
=> ? = 4 - 1
[1,1,0,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [2,1,6,8,7,5,4,3] => [8,6,7,5,2,4,1,3] => ?
=> ? = 5 - 1
[1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,1,7,6,5,4,3,8] => [7,6,5,2,4,1,3,8] => ?
=> ? = 5 - 1
[1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,1,7,8,6,5,4,3] => [7,8,6,5,2,4,1,3] => ?
=> ? = 5 - 1
[1,1,0,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [2,3,1,6,5,4,7,8] => [6,2,3,5,1,4,7,8] => ?
=> ? = 3 - 1
[1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [2,3,1,6,7,5,8,4] => [6,7,2,3,5,8,1,4] => ?
=> ? = 3 - 1
[1,1,0,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,6,8,7,5,4] => [8,6,7,2,3,5,1,4] => ?
=> ? = 4 - 1
[1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,1,8,7,6,5,4] => [8,7,6,2,3,5,1,4] => ?
=> ? = 5 - 1
[1,1,0,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [2,3,4,1,5,8,7,6] => [8,2,3,4,7,1,5,6] => ?
=> ? = 3 - 1
[1,1,0,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> [2,3,4,1,7,6,5,8] => [7,2,3,4,6,1,5,8] => ?
=> ? = 3 - 1
[1,1,0,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [2,3,4,6,5,8,7,1] => [6,8,2,3,4,5,7,1] => ?
=> ? = 3 - 1
[1,1,0,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [2,3,5,4,6,8,7,1] => ? => ?
=> ? = 3 - 1
[1,1,0,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [2,3,5,4,8,7,6,1] => [8,5,7,2,3,4,6,1] => ?
=> ? = 4 - 1
[1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [2,3,5,6,4,7,8,1] => [5,6,2,3,4,7,8,1] => ?
=> ? = 3 - 1
[1,1,0,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [2,3,6,5,4,8,7,1] => ? => ?
=> ? = 4 - 1
[1,1,0,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [2,3,7,8,6,5,4,1] => [7,8,6,5,2,3,4,1] => ?
=> ? = 5 - 1
[1,1,0,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [2,4,3,5,1,7,6,8] => [4,2,3,5,7,1,6,8] => ?
=> ? = 3 - 1
[1,1,0,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,7,6,8,1] => [4,7,2,3,5,6,8,1] => ?
=> ? = 3 - 1
[1,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [2,4,3,5,8,7,6,1] => [8,4,7,2,3,5,6,1] => ?
=> ? = 4 - 1
[1,1,0,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [2,4,3,6,5,7,8,1] => [4,6,2,3,5,7,8,1] => ?
=> ? = 3 - 1
[1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [2,4,3,7,6,8,5,1] => [7,4,6,8,2,3,5,1] => ?
=> ? = 4 - 1
[1,1,0,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [2,4,3,7,8,6,5,1] => [7,8,4,6,2,3,5,1] => ?
=> ? = 4 - 1
Description
The degree of the cyclic sieving polynomial corresponding to an integer partition.
Let $\lambda$ be an integer partition of $n$ and let $N$ be the least common multiple of the parts of $\lambda$. Fix an arbitrary permutation $\pi$ of cycle type $\lambda$. Then $\pi$ induces a cyclic action of order $N$ on $\{1,\dots,n\}$.
The corresponding character can be identified with the cyclic sieving polynomial $C_\lambda(q)$ of this action, modulo $q^N-1$. Explicitly, it is
$$
\sum_{p\in\lambda} [p]_{q^{N/p}},
$$
where $[p]_q = 1+\dots+q^{p-1}$ is the $q$-integer.
This statistic records the degree of $C_\lambda(q)$. Equivalently, it equals
$$
\left(1 - \frac{1}{\lambda_1}\right) N,
$$
where $\lambda_1$ is the largest part of $\lambda$.
The statistic is undefined for the empty partition.
The following 249 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000507The number of ascents of a standard tableau. St000676The number of odd rises of a Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000984The number of boxes below precisely one peak. St001058The breadth of the ordered tree. St000012The area of a Dyck path. St000011The number of touch points (or returns) of a Dyck path. St000439The position of the first down step of a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000097The order of the largest clique of the graph. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. 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. St000093The cardinality of a maximal independent set of vertices of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000098The chromatic number of a graph. St000675The number of centered multitunnels of a Dyck path. St000340The number of non-final maximal constant sub-paths of length greater than one. St000306The bounce count of a Dyck path. St001720The minimal length of a chain of small intervals in a lattice. St000504The cardinality of the first block of a set partition. St001062The maximal size of a block of a set partition. St000167The number of leaves of an ordered tree. St000925The number of topologically connected components of a set partition. St000730The maximal arc length of a set partition. St001809The index of the step at the first peak of maximal height in a Dyck path. St000444The length of the maximal rise of a Dyck path. St001494The Alon-Tarsi number of a graph. St000874The position of the last double rise in a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000025The number of initial rises of a 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:
St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St000053The number of valleys of the Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension 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 $eA$. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St000105The number of blocks in the set partition. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000808The number of up steps of the associated bargraph. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St000234The number of global ascents of a permutation. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001777The number of weak descents in an integer composition. St001963The tree-depth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001962The proper pathwidth of a graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001029The size of the core of a graph. St001108The 2-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001580The acyclic chromatic number of a graph. St000272The treewidth of a graph. St000536The pathwidth of a graph. St000172The Grundy number of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001270The bandwidth of a graph. St000031The number of cycles in the cycle decomposition of a permutation. St000007The number of saliances of the permutation. St000528The height of a poset. St000912The number of maximal antichains in a poset. St000527The width of the poset. St001343The dimension of the reduced incidence algebra of a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000744The length of the path to the largest entry in a standard Young tableau. St000822The Hadwiger number of the graph. St000153The number of adjacent cycles of a permutation. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St000643The size of the largest orbit of antichains under Panyushev complementation. St001717The largest size of an interval in a poset. St000068The number of minimal elements in a poset. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000632The jump number of the poset. St001644The dimension of a graph. St000141The maximum drop size of a permutation. St000308The height of the tree associated to a permutation. St000741The Colin de Verdière graph invariant. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001726The number of visible inversions of a permutation. St000809The reduced reflection length of the permutation. St000957The number of Bruhat lower covers of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001461The number of topologically connected components of the chord diagram of a permutation. St000019The cardinality of the support of a permutation. St000245The number of ascents of a permutation. St000470The number of runs in a permutation. St000354The number of recoils of a permutation. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001330The hat guessing number of a graph. St000702The number of weak deficiencies of a permutation. St000157The number of descents of a standard tableau. St000164The number of short pairs. St000291The number of descents of a binary word. St000390The number of runs of ones in a binary word. St000843The decomposition number of a perfect matching. St000292The number of ascents of a binary word. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000651The maximal size of a rise in a permutation. St000451The length of the longest pattern of the form k 1 2. St001427The number of descents of a signed permutation. St000542The number of left-to-right-minima of a permutation. St000300The number of independent sets of vertices of a graph. St001489The maximum of the number of descents and the number of inverse descents. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000062The length of the longest increasing subsequence of the permutation. St000166The depth minus 1 of an ordered tree. St000094The depth of an ordered tree. St000015The number of peaks of a Dyck path. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000877The depth of the binary word interpreted as a 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. St001498The normalised height of a Nakayama algebra with magnitude 1. St000080The rank of the poset. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001047The maximal number of arcs crossing a given arc of a perfect matching. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. 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 $eA$. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of 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. St000056The decomposition (or block) number of a permutation. St000084The number of subtrees. St000087The number of induced subgraphs. St000149The number of cells of the partition whose leg is zero and arm is odd. St000213The number of weak exceedances (also weak excedences) of a permutation. St000239The number of small weak excedances. St000286The number of connected components of the complement of a graph. St000328The maximum number of child nodes in a tree. St000363The number of minimal vertex covers of a graph. St000443The number of long tunnels of a Dyck path. St000469The distinguishing number of a graph. St000636The hull number of a graph. St000722The number of different neighbourhoods in a graph. St000926The clique-coclique number of a graph. St000991The number of right-to-left minima of a permutation. St001110The 3-dynamic chromatic number of a graph. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001316The domatic number of a graph. St001342The number of vertices in the center of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001581The achromatic number of a graph. St001587Half of the largest even part of an integer partition. St001645The pebbling number of a connected graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001670The connected partition number of a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001725The harmonious chromatic number of a graph. St001746The coalition number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001883The mutual visibility number of a graph. St000021The number of descents of a permutation. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000155The number of exceedances (also excedences) of a permutation. St000171The degree of the graph. St000224The sorting index of a permutation. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000301The number of facets of the stable set polytope of a graph. St000310The minimal degree of a vertex of a graph. St000331The number of upper interactions of a Dyck path. St000362The size of a minimal vertex cover of a graph. St000454The largest eigenvalue of a graph if it is integral. St000778The metric dimension of a graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000989The number of final rises of a permutation. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001180Number of indecomposable injective modules with projective dimension at most 1. 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. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001357The maximal degree of a regular spanning subgraph of a graph. St001391The disjunction number of a graph. 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. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001949The rigidity index of a graph. St000061The number of nodes on the left branch of a binary tree. St000083The number of left oriented leafs of a binary tree except the first one. St000216The absolute length of a permutation. St001480The number of simple summands of the module J^2/J^3. St001674The number of vertices of the largest induced star graph in the graph. St001812The biclique partition number of a graph. St000662The staircase size of the code of a permutation. St000652The maximal difference between successive positions of a permutation. 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. St000150The floored half-sum of the multiplicities of a partition. St000028The number of stack-sorts needed to sort 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. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St000316The number of non-left-to-right-maxima of a permutation. St000299The number of nonisomorphic vertex-induced subtrees. St001589The nesting number of a perfect matching. St001590The crossing number of a perfect matching. St000619The number of cyclic descents of a permutation. St000732The number of double deficiencies of a 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. St000914The sum of the values of the Möbius function of a poset. St000455The second largest eigenvalue of a graph if it is integral. St001890The maximum magnitude of the Möbius function of a poset. St000983The length of the longest alternating subword. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000317The cycle descent number of a permutation. St000628The balance of a binary word. St001870The number of positive entries followed by a negative entry in a signed permutation. St001893The flag descent of a signed permutation. St000942The number of critical left to right maxima of the parking functions. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001462The number of factors of a standard tableaux under concatenation. St001712The number of natural descents of a standard Young tableau. St001935The number of ascents in a parking function. St001946The number of descents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St001820The size of the image of the pop stack sorting operator. St001624The breadth of a lattice.
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