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Your data matches 71 different statistics following compositions of up to 3 maps.
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Matching statistic: St000291
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00134: Standard tableaux —descent word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00134: Standard tableaux —descent word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [[1],[2]]
=> 1 => 0
[1,0,1,0]
=> [[1,3],[2,4]]
=> 101 => 1
[1,1,0,0]
=> [[1,2],[3,4]]
=> 010 => 1
[1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 10101 => 2
[1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 10010 => 2
[1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 01001 => 1
[1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> 01010 => 2
[1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 00100 => 1
[1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 1010101 => 3
[1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 1010010 => 3
[1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 1001001 => 2
[1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 1001010 => 3
[1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 1000100 => 2
[1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 0100101 => 2
[1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> 0100010 => 2
[1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 0101001 => 2
[1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> 0101010 => 3
[1,1,0,1,1,0,0,0]
=> [[1,2,4,5],[3,6,7,8]]
=> 0100100 => 2
[1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> 0010001 => 1
[1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> 0010010 => 2
[1,1,1,0,1,0,0,0]
=> [[1,2,3,5],[4,6,7,8]]
=> 0010100 => 2
[1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> 0001000 => 1
[1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> 101010101 => 4
[1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,8],[2,4,6,9,10]]
=> 101010010 => 4
[1,0,1,0,1,1,0,0,1,0]
=> [[1,3,5,6,9],[2,4,7,8,10]]
=> 101001001 => 3
[1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,6,8],[2,4,7,9,10]]
=> 101001010 => 4
[1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> 101000100 => 3
[1,0,1,1,0,0,1,0,1,0]
=> [[1,3,4,7,9],[2,5,6,8,10]]
=> 100100101 => 3
[1,0,1,1,0,0,1,1,0,0]
=> [[1,3,4,7,8],[2,5,6,9,10]]
=> 100100010 => 3
[1,0,1,1,0,1,0,0,1,0]
=> [[1,3,4,6,9],[2,5,7,8,10]]
=> 100101001 => 3
[1,0,1,1,0,1,0,1,0,0]
=> [[1,3,4,6,8],[2,5,7,9,10]]
=> 100101010 => 4
[1,0,1,1,0,1,1,0,0,0]
=> [[1,3,4,6,7],[2,5,8,9,10]]
=> 100100100 => 3
[1,0,1,1,1,0,0,0,1,0]
=> [[1,3,4,5,9],[2,6,7,8,10]]
=> 100010001 => 2
[1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> 100010100 => 3
[1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> 100001000 => 2
[1,1,0,0,1,0,1,0,1,0]
=> [[1,2,5,7,9],[3,4,6,8,10]]
=> 010010101 => 3
[1,1,0,0,1,0,1,1,0,0]
=> [[1,2,5,7,8],[3,4,6,9,10]]
=> 010010010 => 3
[1,1,0,0,1,1,0,0,1,0]
=> [[1,2,5,6,9],[3,4,7,8,10]]
=> 010001001 => 2
[1,1,0,0,1,1,0,1,0,0]
=> [[1,2,5,6,8],[3,4,7,9,10]]
=> 010001010 => 3
[1,1,0,1,0,0,1,0,1,0]
=> [[1,2,4,7,9],[3,5,6,8,10]]
=> 010100101 => 3
[1,1,0,1,0,0,1,1,0,0]
=> [[1,2,4,7,8],[3,5,6,9,10]]
=> 010100010 => 3
[1,1,0,1,0,1,0,0,1,0]
=> [[1,2,4,6,9],[3,5,7,8,10]]
=> 010101001 => 3
[1,1,0,1,0,1,0,1,0,0]
=> [[1,2,4,6,8],[3,5,7,9,10]]
=> 010101010 => 4
[1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> 010010001 => 2
[1,1,1,0,0,0,1,0,1,0]
=> [[1,2,3,7,9],[4,5,6,8,10]]
=> 001000101 => 2
[1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> 001001001 => 2
[1,1,1,0,0,1,1,0,0,0]
=> [[1,2,3,6,7],[4,5,8,9,10]]
=> 001000100 => 2
[1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> 001010001 => 2
[1,1,1,0,1,0,1,0,0,0]
=> [[1,2,3,5,7],[4,6,8,9,10]]
=> 001010100 => 3
[1,1,1,0,1,1,0,0,0,0]
=> [[1,2,3,5,6],[4,7,8,9,10]]
=> 001001000 => 2
Description
The number of descents of a binary word.
Matching statistic: St000010
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [] => []
=> 0
[1,0,1,0]
=> [2,1] => [1] => [1]
=> 1
[1,1,0,0]
=> [1,2] => [1] => [1]
=> 1
[1,0,1,0,1,0]
=> [3,2,1] => [2,1] => [1,1]
=> 2
[1,0,1,1,0,0]
=> [2,3,1] => [2,1] => [1,1]
=> 2
[1,1,0,0,1,0]
=> [3,1,2] => [1,2] => [2]
=> 1
[1,1,0,1,0,0]
=> [2,1,3] => [2,1] => [1,1]
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,2] => [2]
=> 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [3,2,1] => [1,1,1]
=> 3
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,2,1] => [1,1,1]
=> 3
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [2,3,1] => [2,1]
=> 2
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [3,2,1] => [1,1,1]
=> 3
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,3,1] => [2,1]
=> 2
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [3,1,2] => [2,1]
=> 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,1,2] => [2,1]
=> 2
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,1,3] => [2,1]
=> 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,2,1] => [1,1,1]
=> 3
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [2,3,1] => [2,1]
=> 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,3] => [3]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [3,1,2] => [2,1]
=> 2
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3] => [2,1]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3] => [3]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [4,3,2,1] => [1,1,1,1]
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [4,3,2,1] => [1,1,1,1]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [3,4,2,1] => [2,1,1]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [4,3,2,1] => [1,1,1,1]
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,4,2,1] => [2,1,1]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [4,2,3,1] => [2,1,1]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [4,2,3,1] => [2,1,1]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [3,2,4,1] => [2,1,1]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [4,3,2,1] => [1,1,1,1]
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,4,2,1] => [2,1,1]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [2,3,4,1] => [3,1]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,2,4,1] => [2,1,1]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,3,4,1] => [3,1]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [4,3,1,2] => [2,1,1]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [4,3,1,2] => [2,1,1]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [3,4,1,2] => [2,2]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [4,3,1,2] => [2,1,1]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [4,2,1,3] => [2,1,1]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [4,2,1,3] => [2,1,1]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [3,2,1,4] => [2,1,1]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [4,3,2,1] => [1,1,1,1]
=> 4
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [2,3,1,4] => [3,1]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [4,1,2,3] => [3,1]
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [3,1,2,4] => [3,1]
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,4,1,2] => [2,2]
=> 2
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [2,1,3,4] => [3,1]
=> 2
[1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => [3,2,1,4] => [2,1,1]
=> 3
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [2,3,1,4] => [3,1]
=> 2
Description
The length of the partition.
Matching statistic: St000053
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 4
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 2
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> 3
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> 2
Description
The number of valleys of the Dyck path.
Matching statistic: St000097
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [] => ([],0)
=> 0
[1,0,1,0]
=> [2,1] => [1] => ([],1)
=> 1
[1,1,0,0]
=> [1,2] => [1] => ([],1)
=> 1
[1,0,1,0,1,0]
=> [3,2,1] => [2,1] => ([(0,1)],2)
=> 2
[1,0,1,1,0,0]
=> [2,3,1] => [2,1] => ([(0,1)],2)
=> 2
[1,1,0,0,1,0]
=> [3,1,2] => [1,2] => ([],2)
=> 1
[1,1,0,1,0,0]
=> [2,1,3] => [2,1] => ([(0,1)],2)
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,2] => ([],2)
=> 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,1,3] => ([(1,2)],3)
=> 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,3] => ([],3)
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3] => ([(1,2)],3)
=> 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3] => ([],3)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> 2
[1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> 2
Description
The order of the largest clique of the graph.
A clique in a graph $G$ is a subset $U \subseteq V(G)$ such that any pair of vertices in $U$ are adjacent. I.e. the subgraph induced by $U$ is a complete graph.
Matching statistic: St000098
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000098: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000098: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [] => ([],0)
=> 0
[1,0,1,0]
=> [2,1] => [1] => ([],1)
=> 1
[1,1,0,0]
=> [1,2] => [1] => ([],1)
=> 1
[1,0,1,0,1,0]
=> [3,2,1] => [2,1] => ([(0,1)],2)
=> 2
[1,0,1,1,0,0]
=> [2,3,1] => [2,1] => ([(0,1)],2)
=> 2
[1,1,0,0,1,0]
=> [3,1,2] => [1,2] => ([],2)
=> 1
[1,1,0,1,0,0]
=> [2,1,3] => [2,1] => ([(0,1)],2)
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,2] => ([],2)
=> 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,1,3] => ([(1,2)],3)
=> 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,3] => ([],3)
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3] => ([(1,2)],3)
=> 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3] => ([],3)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> 2
[1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> 2
Description
The chromatic number of a graph.
The minimal number of colors needed to color the vertices of the graph such that no two vertices which share an edge have the same color.
Matching statistic: St000147
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [] => []
=> 0
[1,0,1,0]
=> [2,1] => [1] => [1]
=> 1
[1,1,0,0]
=> [1,2] => [1] => [1]
=> 1
[1,0,1,0,1,0]
=> [3,2,1] => [2,1] => [2]
=> 2
[1,0,1,1,0,0]
=> [2,3,1] => [2,1] => [2]
=> 2
[1,1,0,0,1,0]
=> [3,1,2] => [1,2] => [1,1]
=> 1
[1,1,0,1,0,0]
=> [2,1,3] => [2,1] => [2]
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,2] => [1,1]
=> 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [3,2,1] => [3]
=> 3
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,2,1] => [3]
=> 3
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [2,3,1] => [2,1]
=> 2
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [3,2,1] => [3]
=> 3
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,3,1] => [2,1]
=> 2
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [3,1,2] => [2,1]
=> 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,1,2] => [2,1]
=> 2
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,1,3] => [2,1]
=> 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,2,1] => [3]
=> 3
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [2,3,1] => [2,1]
=> 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,3] => [1,1,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [3,1,2] => [2,1]
=> 2
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3] => [2,1]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3] => [1,1,1]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [4,3,2,1] => [4]
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [4,3,2,1] => [4]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [3,4,2,1] => [3,1]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [4,3,2,1] => [4]
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,4,2,1] => [3,1]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [4,2,3,1] => [3,1]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [4,2,3,1] => [3,1]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [3,2,4,1] => [3,1]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [4,3,2,1] => [4]
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,4,2,1] => [3,1]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [2,3,4,1] => [2,1,1]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,2,4,1] => [3,1]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,3,4,1] => [2,1,1]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [4,3,1,2] => [3,1]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [4,3,1,2] => [3,1]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [3,4,1,2] => [2,1,1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [4,3,1,2] => [3,1]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [4,2,1,3] => [3,1]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [4,2,1,3] => [3,1]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [3,2,1,4] => [3,1]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [4,3,2,1] => [4]
=> 4
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [2,3,1,4] => [2,1,1]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [4,1,2,3] => [2,1,1]
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [3,1,2,4] => [2,1,1]
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,4,1,2] => [2,1,1]
=> 2
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [2,1,3,4] => [2,1,1]
=> 2
[1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => [3,2,1,4] => [3,1]
=> 3
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [2,3,1,4] => [2,1,1]
=> 2
Description
The largest part of an integer partition.
Matching statistic: St000153
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000153: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000153: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [[1]]
=> [1] => [] => 0
[1,0,1,0]
=> [[1,0],[0,1]]
=> [1,2] => [1] => 1
[1,1,0,0]
=> [[0,1],[1,0]]
=> [2,1] => [1] => 1
[1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> [1,2,3] => [1,2] => 2
[1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> [1,3,2] => [1,2] => 2
[1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> [2,1,3] => [2,1] => 1
[1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> [1,3,2] => [1,2] => 2
[1,1,1,0,0,0]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> [3,2,1] => [2,1] => 1
[1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,2,3,4] => [1,2,3] => 3
[1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,2,4,3] => [1,2,3] => 3
[1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => [1,3,2] => 2
[1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => [1,2,3] => 3
[1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => [1,3,2] => 2
[1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [2,1,3,4] => [2,1,3] => 2
[1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [2,1,4,3] => [2,1,3] => 2
[1,1,0,1,0,0,1,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => [1,3,2] => 2
[1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => [1,2,3] => 3
[1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => [1,3,2] => 2
[1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [3,2,1,4] => [3,2,1] => 1
[1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> [2,1,4,3] => [2,1,3] => 2
[1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [1,4,3,2] => [1,3,2] => 2
[1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [4,3,2,1] => [3,2,1] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,2,3,4,5] => [1,2,3,4] => 4
[1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => [1,2,3,4] => 4
[1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => [1,2,4,3] => 3
[1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => [1,2,3,4] => 4
[1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [1,2,4,3] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => [1,3,2,4] => 3
[1,0,1,1,0,0,1,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [1,3,2,4] => 3
[1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => [1,2,4,3] => 3
[1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => [1,2,3,4] => 4
[1,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [1,2,4,3] => 3
[1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,4,3,2,5] => [1,4,3,2] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [1,2,4,3] => 3
[1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [1,5,4,3,2] => [1,4,3,2] => 2
[1,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [2,1,3,4,5] => [2,1,3,4] => 3
[1,1,0,0,1,0,1,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [2,1,3,5,4] => [2,1,3,4] => 3
[1,1,0,0,1,1,0,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [2,1,4,3,5] => [2,1,4,3] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [2,1,3,5,4] => [2,1,3,4] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => [1,3,2,4] => 3
[1,1,0,1,0,0,1,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [1,3,2,4] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => [1,2,4,3] => 3
[1,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => [1,2,3,4] => 4
[1,1,0,1,1,0,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,4,3,2,5] => [1,4,3,2] => 2
[1,1,1,0,0,0,1,0,1,0]
=> [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [3,2,1,4,5] => [3,2,1,4] => 2
[1,1,1,0,0,1,0,0,1,0]
=> [[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [2,1,4,3,5] => [2,1,4,3] => 2
[1,1,1,0,0,1,1,0,0,0]
=> [[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [2,1,5,4,3] => [2,1,4,3] => 2
[1,1,1,0,1,0,0,0,1,0]
=> [[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,4,3,2,5] => [1,4,3,2] => 2
[1,1,1,0,1,0,1,0,0,0]
=> [[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [1,2,4,3] => 3
[1,1,1,0,1,1,0,0,0,0]
=> [[0,0,1,0,0],[0,1,-1,0,1],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [1,5,4,3,2] => [1,4,3,2] => 2
Description
The number of adjacent cycles of a permutation.
This is the number of cycles of the permutation of the form (i,i+1,i+2,...i+k) which includes the fixed points (i).
Matching statistic: St000245
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => [1] => 0
[1,0,1,0]
=> [1,1,0,0]
=> [1,2] => [1,2] => 1
[1,1,0,0]
=> [1,0,1,0]
=> [2,1] => [1,2] => 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 2
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,1,3] => [1,2,3] => 2
[1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [3,2,1] => [1,3,2] => 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => [1,2,3] => 2
[1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 3
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [1,2,3,4] => 3
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,3,2,4] => 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,2,3,4] => 3
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,3,2,4] => 2
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,4,2,3] => 2
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,4,2,3] => 2
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,3,4,2] => 2
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => 3
[1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,3,2,4] => 2
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,4,3,2] => 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,4,2,3] => 2
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,3,2,4] => 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,4,3,2] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 4
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => [1,2,3,4,5] => 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => [1,3,2,4,5] => 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [1,2,3,4,5] => 4
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [1,3,2,4,5] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [1,4,2,3,5] => 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [1,4,2,3,5] => 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [1,3,4,2,5] => 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [1,2,3,4,5] => 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [1,3,2,4,5] => 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => [1,4,3,2,5] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [1,3,2,4,5] => 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [1,4,3,2,5] => 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,5,2,3,4] => 3
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,5,2,3,4] => 3
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,5,2,4,3] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,5,2,3,4] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,4,5,2,3] => 3
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,4,5,2,3] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [1,3,4,5,2] => 3
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 4
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,4,2,5,3] => 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,5,4,2,3] => 2
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,5,2,4,3] => 2
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,5,3,2,4] => 2
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,4,2,5,3] => 2
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,3,5,2,4] => 3
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [1,4,2,5,3] => 2
Description
The number of ascents of a permutation.
Matching statistic: St000292
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00134: Standard tableaux —descent word⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00134: Standard tableaux —descent word⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [[1],[2]]
=> 1 => 0
[1,0,1,0]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 101 => 1
[1,1,0,0]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> 010 => 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 10101 => 2
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 01001 => 2
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 10010 => 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> 01010 => 2
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 00100 => 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 1010101 => 3
[1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 0100101 => 3
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 1001001 => 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 0101001 => 3
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> 0010001 => 2
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 1010010 => 2
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> 0100010 => 2
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 1001010 => 2
[1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> 0101010 => 3
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> 0010010 => 2
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 1000100 => 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [[1,2,4,5],[3,6,7,8]]
=> 0100100 => 2
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [[1,2,3,5],[4,6,7,8]]
=> 0010100 => 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> 0001000 => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> 101010101 => 4
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[1,2,5,7,9],[3,4,6,8,10]]
=> 010010101 => 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[1,3,4,7,9],[2,5,6,8,10]]
=> 100100101 => 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[1,2,4,7,9],[3,5,6,8,10]]
=> 010100101 => 4
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[1,2,3,7,9],[4,5,6,8,10]]
=> 001000101 => 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[1,3,5,6,9],[2,4,7,8,10]]
=> 101001001 => 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[1,2,5,6,9],[3,4,7,8,10]]
=> 010001001 => 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[1,3,4,6,9],[2,5,7,8,10]]
=> 100101001 => 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[1,2,4,6,9],[3,5,7,8,10]]
=> 010101001 => 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> 001001001 => 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[1,3,4,5,9],[2,6,7,8,10]]
=> 100010001 => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> 001010001 => 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> 000100001 => 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,8],[2,4,6,9,10]]
=> 101010010 => 3
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[1,2,5,7,8],[3,4,6,9,10]]
=> 010010010 => 3
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[1,3,4,7,8],[2,5,6,9,10]]
=> 100100010 => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[1,2,4,7,8],[3,5,6,9,10]]
=> 010100010 => 3
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,6,8],[2,4,7,9,10]]
=> 101001010 => 3
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[1,2,5,6,8],[3,4,7,9,10]]
=> 010001010 => 3
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[1,3,4,6,8],[2,5,7,9,10]]
=> 100101010 => 3
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[1,2,4,6,8],[3,5,7,9,10]]
=> 010101010 => 4
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> 100010010 => 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> 101000100 => 2
[1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[1,3,4,6,7],[2,5,8,9,10]]
=> 100100100 => 2
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[1,2,3,6,7],[4,5,8,9,10]]
=> 001000100 => 2
[1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> 100010100 => 2
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[1,2,3,5,7],[4,6,8,9,10]]
=> 001010100 => 3
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[1,2,3,4,7],[5,6,8,9,10]]
=> 000100100 => 2
Description
The number of ascents of a binary word.
Matching statistic: St000552
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
Mp00015: Binary trees —to ordered tree: right child = right brother⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
St000552: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00015: Binary trees —to ordered tree: right child = right brother⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
St000552: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [.,.]
=> [[]]
=> ([(0,1)],2)
=> 0
[1,0,1,0]
=> [[.,.],.]
=> [[[]]]
=> ([(0,2),(1,2)],3)
=> 1
[1,1,0,0]
=> [.,[.,.]]
=> [[],[]]
=> ([(0,2),(1,2)],3)
=> 1
[1,0,1,0,1,0]
=> [[[.,.],.],.]
=> [[[[]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,0,1,1,0,0]
=> [[.,.],[.,.]]
=> [[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> [[[],[]]]
=> ([(0,3),(1,3),(2,3)],4)
=> 1
[1,1,0,1,0,0]
=> [.,[[.,.],.]]
=> [[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,1,0,0,0]
=> [.,[.,[.,.]]]
=> [[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> 1
[1,0,1,0,1,0,1,0]
=> [[[[.,.],.],.],.]
=> [[[[[]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 3
[1,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,.]]
=> [[[[]]],[]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 3
[1,0,1,1,0,0,1,0]
=> [[[.,.],[.,.]],.]
=> [[[[]],[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,1,0,1,0,0]
=> [[.,.],[[.,.],.]]
=> [[[]],[[]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 3
[1,0,1,1,1,0,0,0]
=> [[.,.],[.,[.,.]]]
=> [[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[1,1,0,0,1,0,1,0]
=> [[[.,[.,.]],.],.]
=> [[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[1,1,0,0,1,1,0,0]
=> [[.,[.,.]],[.,.]]
=> [[[],[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[1,1,0,1,0,0,1,0]
=> [[.,[[.,.],.]],.]
=> [[[],[[]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[1,1,0,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> [[],[[[]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 3
[1,1,0,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> [[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[1,1,1,0,0,0,1,0]
=> [[.,[.,[.,.]]],.]
=> [[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,1,1,0,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> [[],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[1,1,1,0,1,0,0,0]
=> [.,[.,[[.,.],.]]]
=> [[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[1,1,1,1,0,0,0,0]
=> [.,[.,[.,[.,.]]]]
=> [[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [[[[[.,.],.],.],.],.]
=> [[[[[[]]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [[[[.,.],.],.],[.,.]]
=> [[[[[]]]],[]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [[[[.,.],.],[.,.]],.]
=> [[[[[]]],[]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> [[[[]]],[[]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [[[.,.],.],[.,[.,.]]]
=> [[[[]]],[],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [[[[.,.],[.,.]],.],.]
=> [[[[[]],[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [[[.,.],[.,.]],[.,.]]
=> [[[[]],[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [[[.,.],[[.,.],.]],.]
=> [[[[]],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [[.,.],[[[.,.],.],.]]
=> [[[]],[[[]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [[.,.],[[.,.],[.,.]]]
=> [[[]],[[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [[[.,.],[.,[.,.]]],.]
=> [[[[]],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> [[[]],[],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [[.,.],[.,[.,[.,.]]]]
=> [[[]],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [[[[.,[.,.]],.],.],.]
=> [[[[[],[]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [[[.,[.,.]],.],[.,.]]
=> [[[[],[]]],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [[[.,[.,.]],[.,.]],.]
=> [[[[],[]],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> [[[],[]],[[]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [[[.,[[.,.],.]],.],.]
=> [[[[],[[]]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [[.,[[.,.],.]],[.,.]]
=> [[[],[[]]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [[.,[[[.,.],.],.]],.]
=> [[[],[[[]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [.,[[[[.,.],.],.],.]]
=> [[],[[[[]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 4
[1,1,0,1,1,0,0,0,1,0]
=> [[.,[[.,.],[.,.]]],.]
=> [[[],[[]],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [[[.,[.,[.,.]]],.],.]
=> [[[[],[],[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [[.,[[.,[.,.]],.]],.]
=> [[[],[[],[]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [.,[[.,[.,.]],[.,.]]]
=> [[],[[],[]],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,1,1,0,1,0,0,0,1,0]
=> [[.,[.,[[.,.],.]]],.]
=> [[[],[],[[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 2
[1,1,1,0,1,0,1,0,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> [[],[],[[[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3
[1,1,1,0,1,1,0,0,0,0]
=> [.,[.,[[.,.],[.,.]]]]
=> [[],[],[[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 2
Description
The number of cut vertices of a graph.
A cut vertex is one whose deletion increases the number of connected components.
The following 61 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001692The number of vertices with higher degree than the average degree in a graph. St000203The number of external nodes of a binary tree. St000390The number of runs of ones in a binary word. St000507The number of ascents of a standard tableau. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St000619The number of cyclic descents of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St000056The decomposition (or block) number of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000068The number of minimal elements in a poset. St000172The Grundy number of a graph. St000239The number of small weak excedances. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000527The width of the poset. St000991The number of right-to-left minima of a permutation. St001029The size of the core of a graph. St001304The number of maximally independent sets of vertices of a graph. St001389The number of partitions of the same length below the given integer partition. St001581The achromatic number of a graph. St000141The maximum drop size of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000234The number of global ascents of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000632The jump number of the poset. St001427The number of descents of a signed permutation. St001812The biclique partition number of a graph. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000354The number of recoils of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000061The number of nodes on the left branch of a binary tree. St000668The least common multiple of the parts of the partition. St000702The number of weak deficiencies of a permutation. St000708The product of the parts of an integer partition. St000288The number of ones in a binary word. St000829The Ulam distance of a permutation to the identity permutation. St001896The number of right descents of a signed permutations. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St000015The number of peaks of a Dyck path. St001649The length of a longest trail in a graph. St001180Number of indecomposable injective modules with projective dimension at most 1. St001960The number of descents of a permutation minus one if its first entry is not one. St000389The number of runs of ones of odd length in a binary word. St000035The number of left outer peaks of a permutation. St000648The number of 2-excedences of a permutation. St000201The number of leaf nodes in a binary tree. St001712The number of natural descents of a standard Young tableau. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000834The number of right outer peaks of a permutation. St000522The number of 1-protected nodes of a rooted tree. St000521The number of distinct subtrees of an ordered tree.
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