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Your data matches 94 different statistics following compositions of up to 3 maps.
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Matching statistic: St000548
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000548: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000548: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> []
=> 0
[1,1] => [1,0,1,0]
=> [1,1,0,0]
=> []
=> 0
[2] => [1,1,0,0]
=> [1,0,1,0]
=> [1]
=> 1
[1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> 0
[1,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> 2
[2,1] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1]
=> 1
[3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> 3
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> 0
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 3
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 2
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 5
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 4
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 3
[4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 6
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 4
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 3
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 7
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 6
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 5
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 9
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 5
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 4
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 8
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 3
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 7
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 6
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 10
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> 5
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 4
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> 9
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> 3
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> 8
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> 7
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> 12
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> 2
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> 7
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1]
=> 6
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> 11
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,2,1]
=> 5
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> 10
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1]
=> 9
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> 14
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1,1]
=> 6
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> 5
Description
The number of different non-empty partial sums of an integer partition.
Matching statistic: St000391
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000391: Binary words ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 100%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000391: Binary words ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => => ? = 0
[1,1] => [1,0,1,0]
=> [1,2] => 0 => 0
[2] => [1,1,0,0]
=> [2,1] => 1 => 1
[1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 00 => 0
[1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 01 => 2
[2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 10 => 1
[3] => [1,1,1,0,0,0]
=> [3,2,1] => 11 => 3
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 000 => 0
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 001 => 3
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 010 => 2
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 011 => 5
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 100 => 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 101 => 4
[3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 110 => 3
[4] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 111 => 6
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0000 => 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0001 => 4
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 0010 => 3
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 0011 => 7
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 0100 => 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 0101 => 6
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 0110 => 5
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 0111 => 9
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1000 => 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 1001 => 5
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 1010 => 4
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1011 => 8
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1100 => 3
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1101 => 7
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 1110 => 6
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 1111 => 10
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => 00000 => 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 00001 => 5
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => 00010 => 4
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => 00011 => 9
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => 00100 => 3
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5] => 00101 => 8
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,5,4,3,6] => 00110 => 7
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,5,4,3] => 00111 => 12
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6] => 01000 => 2
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,4,6,5] => 01001 => 7
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,6] => 01010 => 6
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,6,5,4] => 01011 => 11
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,4,3,2,5,6] => 01100 => 5
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,4,3,2,6,5] => 01101 => 10
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,4,3,2,6] => 01110 => 9
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => 01111 => 14
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => 10000 => 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => 10001 => 6
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,3,5,4,6] => 10010 => 5
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3,6,5,4] => 10011 => 10
[1,3,2,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,4,3,2,6,5,7,8] => ? => ? = 10
[2,1,1,3,1] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,7,6,5,8] => ? => ? = 12
[2,1,2,1,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,5,4,6,7,8] => ? => ? = 5
[2,1,2,2,1] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6,8] => ? => ? = 11
[2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,3,6,5,4,7,8] => ? => ? = 10
[2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,7,6,5,4,8] => ? => ? = 16
[2,2,1,2,1] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,4,3,5,7,6,8] => ? => ? = 10
[2,4,1,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,1,6,5,4,3,7,8] => ? => ? = 13
[3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> [3,2,1,4,5,7,6,8] => ? => ? = 9
[3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> [3,2,1,4,6,5,7,8] => ? => ? = 8
[1,1,1,1,3,1,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,3,4,7,6,5,8,9] => ? => ? = 11
[1,1,1,2,1,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? => ? => ? = 12
[1,1,1,2,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,2,3,5,4,7,6,8,9] => ? => ? = 10
[1,1,1,3,1,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,2,3,6,5,4,7,8,9] => ? => ? = 9
[1,1,1,4,1,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? => ? => ? = 15
[1,1,1,5,1] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,2,3,8,7,6,5,4,9] => ? => ? = 22
[1,1,2,1,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,4,3,5,6,7,9,8] => ? => ? = 11
[1,1,2,2,1,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,4,3,6,5,7,8,9] => ? => ? = 8
[1,1,3,1,1,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,2,5,4,3,6,7,8,9] => ? => ? = 7
[1,1,4,1,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,2,6,5,4,3,7,8,9] => ? => ? = 12
[1,1,5,1,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,2,7,6,5,4,3,8,9] => ? => ? = 18
[1,1,6,1] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,2,8,7,6,5,4,3,9] => ? => ? = 25
[1,2,1,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,5,6,8,7,9] => ? => ? = 9
[1,2,1,1,2,1,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5,7,6,8,9] => ? => ? = 8
[1,2,1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,4,6,5,7,8,9] => ? => ? = 7
[1,4,1,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,5,4,3,2,6,7,8,9] => ? => ? = 9
[1,5,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [1,6,5,4,3,2,7,8,9] => ? => ? = 14
[1,6,1,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,7,6,5,4,3,2,8,9] => ? => ? = 20
[2,1,1,1,2,1,1] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,3,4,5,7,6,8,9] => ? => ? = 7
[2,1,1,2,1,1,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,6,5,7,8,9] => ? => ? = 6
[2,1,2,1,1,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,5,4,6,7,8,9] => ? => ? = 5
[2,2,1,1,1,2] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,4,3,5,6,7,9,8] => ? => ? = 12
[2,3,1,1,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [2,1,5,4,3,6,7,8,9] => ? => ? = 8
[2,4,1,1,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [2,1,6,5,4,3,7,8,9] => ? => ? = 13
[2,5,1,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [2,1,7,6,5,4,3,8,9] => ? => ? = 19
[3,1,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [3,2,1,4,5,6,7,9,8] => ? => ? = 11
[3,1,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [3,2,1,4,5,6,8,7,9] => ? => ? = 10
[3,1,1,2,1,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? => ? => ? = 9
[3,1,3,1,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? => ? => ? = 14
[3,4,1,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? => ? => ? = 18
[3,5,1] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [3,2,1,8,7,6,5,4,9] => ? => ? = 25
[1,1,1,1,3,1,1,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,2,3,4,7,6,5,8,9,10] => ? => ? = 11
[1,1,1,4,1,1,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? => ? => ? = 15
[1,1,3,1,1,1,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,5,4,3,6,7,8,9,10] => ? => ? = 7
[1,1,5,1,1,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [1,2,7,6,5,4,3,8,9,10] => ? => ? = 18
[1,1,7,1] => [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,2,9,8,7,6,5,4,3,10] => ? => ? = 33
[1,2,1,1,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,3,2,4,5,6,7,8,10,9] => ? => ? = 11
[1,2,1,1,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,5,6,7,9,8,10] => ? => ? = 10
[1,2,2,1,1,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,3,2,5,4,6,7,8,9,10] => ? => ? = 6
Description
The sum of the positions of the ones in a binary word.
Matching statistic: St000009
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000009: Standard tableaux ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000009: Standard tableaux ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => [[1]]
=> 0
[1,1] => [1,0,1,0]
=> [2,1] => [[1],[2]]
=> 0
[2] => [1,1,0,0]
=> [1,2] => [[1,2]]
=> 1
[1,1,1] => [1,0,1,0,1,0]
=> [3,2,1] => [[1],[2],[3]]
=> 0
[1,2] => [1,0,1,1,0,0]
=> [2,3,1] => [[1,2],[3]]
=> 2
[2,1] => [1,1,0,0,1,0]
=> [3,1,2] => [[1,3],[2]]
=> 1
[3] => [1,1,1,0,0,0]
=> [1,2,3] => [[1,2,3]]
=> 3
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [[1],[2],[3],[4]]
=> 0
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [[1,2],[3],[4]]
=> 3
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [[1,3],[2],[4]]
=> 2
[1,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [[1,2,3],[4]]
=> 5
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [[1,4],[2],[3]]
=> 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [[1,2],[3,4]]
=> 4
[3,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [[1,3,4],[2]]
=> 3
[4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [[1,2,3,4]]
=> 6
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [[1,2],[3],[4],[5]]
=> 4
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [[1,3],[2],[4],[5]]
=> 3
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [[1,2,3],[4],[5]]
=> 7
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [[1,4],[2],[3],[5]]
=> 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [[1,2],[3,4],[5]]
=> 6
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [[1,3,4],[2],[5]]
=> 5
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [[1,2,3,4],[5]]
=> 9
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [[1,5],[2],[3],[4]]
=> 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [[1,2],[3,5],[4]]
=> 5
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [[1,3],[2,5],[4]]
=> 4
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [[1,2,3],[4,5]]
=> 8
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [[1,4,5],[2],[3]]
=> 3
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [[1,2,5],[3,4]]
=> 7
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [[1,3,4,5],[2]]
=> 6
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [[1,2,3,4,5]]
=> 10
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => [[1],[2],[3],[4],[5],[6]]
=> 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,2,1] => [[1,2],[3],[4],[5],[6]]
=> 5
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,2,1] => [[1,3],[2],[4],[5],[6]]
=> 4
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,2,1] => [[1,2,3],[4],[5],[6]]
=> 9
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,2,1] => [[1,4],[2],[3],[5],[6]]
=> 3
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,2,1] => [[1,2],[3,4],[5],[6]]
=> 8
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,4,5,2,1] => [[1,3,4],[2],[5],[6]]
=> 7
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,2,1] => [[1,2,3,4],[5],[6]]
=> 12
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,3,1] => [[1,5],[2],[3],[4],[6]]
=> 2
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,3,1] => [[1,2],[3,5],[4],[6]]
=> 7
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,3,1] => [[1,3],[2,5],[4],[6]]
=> 6
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,5,6,2,3,1] => [[1,2,3],[4,5],[6]]
=> 11
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,3,4,1] => [[1,4,5],[2],[3],[6]]
=> 5
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,6,2,3,4,1] => [[1,2,5],[3,4],[6]]
=> 10
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,5,1] => [[1,3,4,5],[2],[6]]
=> 9
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [[1,2,3,4,5],[6]]
=> 14
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,2] => [[1,6],[2],[3],[4],[5]]
=> 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,1,2] => [[1,2],[3,6],[4],[5]]
=> 6
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,1,2] => [[1,3],[2,6],[4],[5]]
=> 5
[1,3,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [7,8,6,5,2,3,4,1] => ?
=> ? = 12
[2,1,1,3,1] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [8,5,6,7,4,3,1,2] => ?
=> ? = 12
[2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [8,7,4,5,6,3,1,2] => ?
=> ? = 10
[2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [8,4,5,6,7,3,1,2] => ?
=> ? = 16
[2,2,1,2,1] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [8,6,7,5,3,4,1,2] => ?
=> ? = 10
[3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> [8,6,7,5,4,1,2,3] => ?
=> ? = 9
[3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> [8,7,5,6,4,1,2,3] => ?
=> ? = 8
[1,1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [9,7,8,6,5,4,3,2,1] => [[1,3],[2],[4],[5],[6],[7],[8],[9]]
=> ? = 7
[1,1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [9,8,6,7,5,4,3,2,1] => ?
=> ? = 6
[1,1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [9,8,7,5,6,4,3,2,1] => [[1,5],[2],[3],[4],[6],[7],[8],[9]]
=> ? = 5
[1,1,1,1,2,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [9,7,8,5,6,4,3,2,1] => ?
=> ? = 12
[1,1,1,1,3,1,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [9,8,5,6,7,4,3,2,1] => ?
=> ? = 11
[1,1,1,2,1,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [9,8,7,6,4,5,3,2,1] => [[1,6],[2],[3],[4],[5],[7],[8],[9]]
=> ? = 4
[1,1,1,2,1,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? => ?
=> ? = 12
[1,1,1,2,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [9,8,6,7,4,5,3,2,1] => ?
=> ? = 10
[1,1,1,3,1,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [9,8,7,4,5,6,3,2,1] => [[1,5,6],[2],[3],[4],[7],[8],[9]]
=> ? = 9
[1,1,1,4,1,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? => ?
=> ? = 15
[1,1,1,5,1] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [9,4,5,6,7,8,3,2,1] => ?
=> ? = 22
[1,1,2,1,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [9,8,7,6,5,3,4,2,1] => [[1,7],[2],[3],[4],[5],[6],[8],[9]]
=> ? = 3
[1,1,2,1,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [8,9,7,6,5,3,4,2,1] => ?
=> ? = 11
[1,1,2,2,1,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [9,8,7,5,6,3,4,2,1] => ?
=> ? = 8
[1,1,3,1,1,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [9,8,7,6,3,4,5,2,1] => ?
=> ? = 7
[1,1,4,1,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [9,8,7,3,4,5,6,2,1] => ?
=> ? = 12
[1,1,5,1,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [9,8,3,4,5,6,7,2,1] => ?
=> ? = 18
[1,1,6,1] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [9,3,4,5,6,7,8,2,1] => [[1,3,4,5,6,7],[2],[8],[9]]
=> ? = 25
[1,2,1,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [9,8,7,6,5,4,2,3,1] => [[1,8],[2],[3],[4],[5],[6],[7],[9]]
=> ? = 2
[1,2,1,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [8,9,7,6,5,4,2,3,1] => [[1,2],[3,8],[4],[5],[6],[7],[9]]
=> ? = 10
[1,2,1,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [9,7,8,6,5,4,2,3,1] => ?
=> ? = 9
[1,2,1,1,2,1,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [9,8,6,7,5,4,2,3,1] => ?
=> ? = 8
[1,2,1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [9,8,7,5,6,4,2,3,1] => ?
=> ? = 7
[1,2,2,1,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [9,8,7,6,4,5,2,3,1] => [[1,6],[2,8],[3],[4],[5],[7],[9]]
=> ? = 6
[1,3,1,1,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [9,8,7,6,5,2,3,4,1] => [[1,7,8],[2],[3],[4],[5],[6],[9]]
=> ? = 5
[1,4,1,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [9,8,7,6,2,3,4,5,1] => ?
=> ? = 9
[1,5,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [9,8,7,2,3,4,5,6,1] => ?
=> ? = 14
[1,6,1,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [9,8,2,3,4,5,6,7,1] => [[1,4,5,6,7,8],[2],[3],[9]]
=> ? = 20
[1,7,1] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [9,2,3,4,5,6,7,8,1] => [[1,3,4,5,6,7,8],[2],[9]]
=> ? = 27
[2,1,1,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [8,9,7,6,5,4,3,1,2] => [[1,2],[3,9],[4],[5],[6],[7],[8]]
=> ? = 9
[2,1,1,1,1,2,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [9,7,8,6,5,4,3,1,2] => [[1,3],[2,9],[4],[5],[6],[7],[8]]
=> ? = 8
[2,1,1,1,2,1,1] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [9,8,6,7,5,4,3,1,2] => ?
=> ? = 7
[2,1,1,2,1,1,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [9,8,7,5,6,4,3,1,2] => ?
=> ? = 6
[2,1,2,1,1,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [9,8,7,6,4,5,3,1,2] => ?
=> ? = 5
[2,2,1,1,1,2] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [8,9,7,6,5,3,4,1,2] => ?
=> ? = 12
[2,3,1,1,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [9,8,7,6,3,4,5,1,2] => ?
=> ? = 8
[2,4,1,1,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [9,8,7,3,4,5,6,1,2] => ?
=> ? = 13
[2,5,1,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [9,8,3,4,5,6,7,1,2] => [[1,4,5,6,7],[2,9],[3],[8]]
=> ? = 19
[2,6,1] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [9,3,4,5,6,7,8,1,2] => [[1,3,4,5,6,7],[2,9],[8]]
=> ? = 26
[3,1,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [8,9,7,6,5,4,1,2,3] => ?
=> ? = 11
[3,1,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [9,7,8,6,5,4,1,2,3] => ?
=> ? = 10
[3,1,1,2,1,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? => ?
=> ? = 9
[3,1,3,1,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? => ?
=> ? = 14
Description
The charge of a standard tableau.
Matching statistic: St000330
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000330: Standard tableaux ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 100%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000330: Standard tableaux ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => [[1]]
=> 0
[1,1] => [1,0,1,0]
=> [1,2] => [[1,2]]
=> 0
[2] => [1,1,0,0]
=> [2,1] => [[1],[2]]
=> 1
[1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => [[1,2,3]]
=> 0
[1,2] => [1,0,1,1,0,0]
=> [1,3,2] => [[1,2],[3]]
=> 2
[2,1] => [1,1,0,0,1,0]
=> [2,1,3] => [[1,3],[2]]
=> 1
[3] => [1,1,1,0,0,0]
=> [3,2,1] => [[1],[2],[3]]
=> 3
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [[1,2,3],[4]]
=> 3
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [[1,2,4],[3]]
=> 2
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [[1,2],[3],[4]]
=> 5
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [[1,3,4],[2]]
=> 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [[1,3],[2,4]]
=> 4
[3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [[1,4],[2],[3]]
=> 3
[4] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [[1],[2],[3],[4]]
=> 6
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 4
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 3
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [[1,2,3],[4],[5]]
=> 7
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [[1,2,4,5],[3]]
=> 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [[1,2,4],[3,5]]
=> 6
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [[1,2,5],[3],[4]]
=> 5
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 9
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [[1,3,4,5],[2]]
=> 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [[1,3,4],[2,5]]
=> 5
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [[1,3,5],[2,4]]
=> 4
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [[1,3],[2,4],[5]]
=> 8
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [[1,4,5],[2],[3]]
=> 3
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [[1,4],[2,5],[3]]
=> 7
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [[1,5],[2],[3],[4]]
=> 6
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 10
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [[1,2,3,4,5,6]]
=> 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => [[1,2,3,4,5],[6]]
=> 5
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => [[1,2,3,4,6],[5]]
=> 4
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => [[1,2,3,4],[5],[6]]
=> 9
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => [[1,2,3,5,6],[4]]
=> 3
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5] => [[1,2,3,5],[4,6]]
=> 8
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,5,4,3,6] => [[1,2,3,6],[4],[5]]
=> 7
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,5,4,3] => [[1,2,3],[4],[5],[6]]
=> 12
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6] => [[1,2,4,5,6],[3]]
=> 2
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,4,6,5] => [[1,2,4,5],[3,6]]
=> 7
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,6] => [[1,2,4,6],[3,5]]
=> 6
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,6,5,4] => [[1,2,4],[3,5],[6]]
=> 11
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,4,3,2,5,6] => [[1,2,5,6],[3],[4]]
=> 5
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,4,3,2,6,5] => [[1,2,5],[3,6],[4]]
=> 10
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,4,3,2,6] => [[1,2,6],[3],[4],[5]]
=> 9
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => [[1,2],[3],[4],[5],[6]]
=> 14
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => [[1,3,4,5,6],[2]]
=> 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => [[1,3,4,5],[2,6]]
=> 6
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,3,5,4,6] => [[1,3,4,6],[2,5]]
=> 5
[1,3,2,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,4,3,2,6,5,7,8] => ?
=> ? = 10
[2,1,1,3,1] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,7,6,5,8] => ?
=> ? = 12
[2,1,2,1,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,5,4,6,7,8] => ?
=> ? = 5
[2,1,2,1,2] => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4,6,8,7] => ?
=> ? = 12
[2,1,2,2,1] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6,8] => ?
=> ? = 11
[2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,3,6,5,4,7,8] => ?
=> ? = 10
[2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,7,6,5,4,8] => ?
=> ? = 16
[2,2,1,2,1] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,4,3,5,7,6,8] => ?
=> ? = 10
[2,4,1,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,1,6,5,4,3,7,8] => ?
=> ? = 13
[3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> [3,2,1,4,5,7,6,8] => ?
=> ? = 9
[3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> [3,2,1,4,6,5,7,8] => ?
=> ? = 8
[3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> [3,2,1,5,4,6,7,8] => ?
=> ? = 7
[3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,6,5,4,7,8] => ?
=> ? = 12
[1,1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,5,6,8,7,9] => [[1,2,3,4,5,6,7,9],[8]]
=> ? = 7
[1,1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,3,4,5,7,6,8,9] => [[1,2,3,4,5,6,8,9],[7]]
=> ? = 6
[1,1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,3,4,6,5,7,8,9] => [[1,2,3,4,5,7,8,9],[6]]
=> ? = 5
[1,1,1,1,2,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,3,4,6,5,8,7,9] => [[1,2,3,4,5,7,9],[6,8]]
=> ? = 12
[1,1,1,1,3,1,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,3,4,7,6,5,8,9] => ?
=> ? = 11
[1,1,1,2,1,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,2,3,5,4,6,7,8,9] => [[1,2,3,4,6,7,8,9],[5]]
=> ? = 4
[1,1,1,2,1,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? => ?
=> ? = 12
[1,1,1,2,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,2,3,5,4,7,6,8,9] => ?
=> ? = 10
[1,1,1,3,1,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,2,3,6,5,4,7,8,9] => ?
=> ? = 9
[1,1,1,4,1,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? => ?
=> ? = 15
[1,1,1,5,1] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,2,3,8,7,6,5,4,9] => ?
=> ? = 22
[1,1,2,1,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,4,3,5,6,7,8,9] => [[1,2,3,5,6,7,8,9],[4]]
=> ? = 3
[1,1,2,1,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,4,3,5,6,7,9,8] => ?
=> ? = 11
[1,1,2,2,1,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,4,3,6,5,7,8,9] => ?
=> ? = 8
[1,1,3,1,1,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,2,5,4,3,6,7,8,9] => ?
=> ? = 7
[1,1,4,1,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,2,6,5,4,3,7,8,9] => ?
=> ? = 12
[1,1,5,1,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,2,7,6,5,4,3,8,9] => ?
=> ? = 18
[1,1,6,1] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,2,8,7,6,5,4,3,9] => ?
=> ? = 25
[1,2,1,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,3,2,4,5,6,7,9,8] => [[1,2,4,5,6,7,8],[3,9]]
=> ? = 10
[1,2,1,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,5,6,8,7,9] => ?
=> ? = 9
[1,2,1,1,2,1,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5,7,6,8,9] => ?
=> ? = 8
[1,2,1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,4,6,5,7,8,9] => ?
=> ? = 7
[1,2,2,1,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,2,5,4,6,7,8,9] => [[1,2,4,6,7,8,9],[3,5]]
=> ? = 6
[1,4,1,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,5,4,3,2,6,7,8,9] => ?
=> ? = 9
[1,5,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [1,6,5,4,3,2,7,8,9] => ?
=> ? = 14
[1,6,1,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,7,6,5,4,3,2,8,9] => ?
=> ? = 20
[2,1,1,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,5,6,7,9,8] => [[1,3,4,5,6,7,8],[2,9]]
=> ? = 9
[2,1,1,1,1,2,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,3,4,5,6,8,7,9] => [[1,3,4,5,6,7,9],[2,8]]
=> ? = 8
[2,1,1,1,2,1,1] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,3,4,5,7,6,8,9] => ?
=> ? = 7
[2,1,1,2,1,1,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,6,5,7,8,9] => ?
=> ? = 6
[2,1,2,1,1,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,5,4,6,7,8,9] => ?
=> ? = 5
[2,2,1,1,1,2] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,4,3,5,6,7,9,8] => ?
=> ? = 12
[2,2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5,7,8,9] => ?
=> ? = 9
[2,3,1,1,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [2,1,5,4,3,6,7,8,9] => ?
=> ? = 8
[2,4,1,1,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [2,1,6,5,4,3,7,8,9] => ?
=> ? = 13
[2,5,1,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [2,1,7,6,5,4,3,8,9] => ?
=> ? = 19
[2,6,1] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,1,8,7,6,5,4,3,9] => [[1,3,9],[2,4],[5],[6],[7],[8]]
=> ? = 26
Description
The (standard) major index of a standard tableau.
A descent of a standard tableau $T$ is an index $i$ such that $i+1$ appears in a row strictly below the row of $i$. The (standard) major index is the the sum of the descents.
Matching statistic: St000008
Mp00039: Integer compositions —complement⟶ Integer compositions
St000008: Integer compositions ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 72%
St000008: Integer compositions ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 72%
Values
[1] => [1] => 0
[1,1] => [2] => 0
[2] => [1,1] => 1
[1,1,1] => [3] => 0
[1,2] => [2,1] => 2
[2,1] => [1,2] => 1
[3] => [1,1,1] => 3
[1,1,1,1] => [4] => 0
[1,1,2] => [3,1] => 3
[1,2,1] => [2,2] => 2
[1,3] => [2,1,1] => 5
[2,1,1] => [1,3] => 1
[2,2] => [1,2,1] => 4
[3,1] => [1,1,2] => 3
[4] => [1,1,1,1] => 6
[1,1,1,1,1] => [5] => 0
[1,1,1,2] => [4,1] => 4
[1,1,2,1] => [3,2] => 3
[1,1,3] => [3,1,1] => 7
[1,2,1,1] => [2,3] => 2
[1,2,2] => [2,2,1] => 6
[1,3,1] => [2,1,2] => 5
[1,4] => [2,1,1,1] => 9
[2,1,1,1] => [1,4] => 1
[2,1,2] => [1,3,1] => 5
[2,2,1] => [1,2,2] => 4
[2,3] => [1,2,1,1] => 8
[3,1,1] => [1,1,3] => 3
[3,2] => [1,1,2,1] => 7
[4,1] => [1,1,1,2] => 6
[5] => [1,1,1,1,1] => 10
[1,1,1,1,1,1] => [6] => 0
[1,1,1,1,2] => [5,1] => 5
[1,1,1,2,1] => [4,2] => 4
[1,1,1,3] => [4,1,1] => 9
[1,1,2,1,1] => [3,3] => 3
[1,1,2,2] => [3,2,1] => 8
[1,1,3,1] => [3,1,2] => 7
[1,1,4] => [3,1,1,1] => 12
[1,2,1,1,1] => [2,4] => 2
[1,2,1,2] => [2,3,1] => 7
[1,2,2,1] => [2,2,2] => 6
[1,2,3] => [2,2,1,1] => 11
[1,3,1,1] => [2,1,3] => 5
[1,3,2] => [2,1,2,1] => 10
[1,4,1] => [2,1,1,2] => 9
[1,5] => [2,1,1,1,1] => 14
[2,1,1,1,1] => [1,5] => 1
[2,1,1,2] => [1,4,1] => 6
[2,1,2,1] => [1,3,2] => 5
[1,1,1,1,1,1,1,1,1] => [9] => ? = 0
[1,1,1,1,1,1,2,1] => [7,2] => ? = 7
[1,1,1,1,1,2,1,1] => [6,3] => ? = 6
[1,1,1,1,2,1,1,1] => [5,4] => ? = 5
[1,1,1,1,2,2,1] => [5,2,2] => ? = 12
[1,1,1,1,3,1,1] => [5,1,3] => ? = 11
[1,1,1,2,1,1,1,1] => [4,5] => ? = 4
[1,1,1,2,1,1,2] => [4,4,1] => ? = 12
[1,1,1,2,2,1,1] => [4,2,3] => ? = 10
[1,1,1,3,1,1,1] => [4,1,4] => ? = 9
[1,1,1,4,1,1] => [4,1,1,3] => ? = 15
[1,1,1,5,1] => [4,1,1,1,2] => ? = 22
[1,1,2,1,1,1,1,1] => [3,6] => ? = 3
[1,1,2,1,1,1,2] => [3,5,1] => ? = 11
[1,1,2,2,1,1,1] => [3,2,4] => ? = 8
[1,1,3,1,1,1,1] => [3,1,5] => ? = 7
[1,1,4,1,1,1] => [3,1,1,4] => ? = 12
[1,1,5,1,1] => [3,1,1,1,3] => ? = 18
[1,1,6,1] => [3,1,1,1,1,2] => ? = 25
[1,1,7] => [3,1,1,1,1,1,1] => ? = 33
[1,2,1,1,1,1,1,1] => [2,7] => ? = 2
[1,2,1,1,1,1,2] => [2,6,1] => ? = 10
[1,2,1,1,1,2,1] => [2,5,2] => ? = 9
[1,2,1,1,2,1,1] => [2,4,3] => ? = 8
[1,2,1,2,1,1,1] => [2,3,4] => ? = 7
[1,2,2,1,1,1,1] => [2,2,5] => ? = 6
[1,3,1,1,1,1,1] => [2,1,6] => ? = 5
[1,4,1,1,1,1] => [2,1,1,5] => ? = 9
[1,5,1,1,1] => [2,1,1,1,4] => ? = 14
[1,6,1,1] => [2,1,1,1,1,3] => ? = 20
[1,7,1] => [2,1,1,1,1,1,2] => ? = 27
[1,8] => [2,1,1,1,1,1,1,1] => ? = 35
[2,1,1,1,2,1,1] => [1,5,3] => ? = 7
[2,1,1,2,1,1,1] => [1,4,4] => ? = 6
[2,1,2,1,1,1,1] => [1,3,5] => ? = 5
[2,2,1,1,1,1,1] => [1,2,6] => ? = 4
[2,2,2,1,1,1] => [1,2,2,4] => ? = 9
[2,3,1,1,1,1] => [1,2,1,5] => ? = 8
[2,4,1,1,1] => [1,2,1,1,4] => ? = 13
[2,5,1,1] => [1,2,1,1,1,3] => ? = 19
[2,6,1] => [1,2,1,1,1,1,2] => ? = 26
[2,7] => [1,2,1,1,1,1,1,1] => ? = 34
[3,1,1,1,1,1,1] => [1,1,7] => ? = 3
[3,1,1,1,2,1] => [1,1,5,2] => ? = 10
[3,1,1,2,1,1] => [1,1,4,3] => ? = 9
[3,1,3,1,1] => [1,1,3,1,3] => ? = 14
[3,2,1,1,1,1] => [1,1,2,5] => ? = 7
[3,3,1,1,1] => [1,1,2,1,4] => ? = 12
[3,4,1,1] => [1,1,2,1,1,3] => ? = 18
[3,5,1] => [1,1,2,1,1,1,2] => ? = 25
Description
The major index of the composition.
The descents of a composition $[c_1,c_2,\dots,c_k]$ are the partial sums $c_1, c_1+c_2,\dots, c_1+\dots+c_{k-1}$, excluding the sum of all parts. The major index of a composition is the sum of its descents.
For details about the major index see [[Permutations/Descents-Major]].
Matching statistic: St000228
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 24% ●values known / values provided: 57%●distinct values known / distinct values provided: 24%
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 24% ●values known / values provided: 57%●distinct values known / distinct values provided: 24%
Values
[1] => [1,0]
=> [1,0]
=> []
=> 0
[1,1] => [1,0,1,0]
=> [1,1,0,0]
=> []
=> 0
[2] => [1,1,0,0]
=> [1,0,1,0]
=> [1]
=> 1
[1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> 0
[1,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> 2
[2,1] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1]
=> 1
[3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> 3
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> 0
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 3
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 2
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 5
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 4
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 3
[4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 6
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 4
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 3
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 7
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 6
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 5
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 9
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 5
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 4
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 8
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 3
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 7
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 6
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 10
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> 5
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 4
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> 9
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> 3
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> 8
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> 7
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> ? = 12
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> 2
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> 7
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1]
=> 6
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> ? = 11
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,2,1]
=> 5
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> 10
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1]
=> 9
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> ? = 14
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1,1]
=> 6
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> 5
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> 10
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,1,1]
=> 4
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1,1]
=> 9
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> ? = 13
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> ? = 12
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> ? = 11
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> ? = 15
[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2,2,1]
=> ? = 11
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,2,1]
=> ? = 15
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,1]
=> ? = 12
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,2,1]
=> ? = 18
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,3,2,2,2,1]
=> ? = 13
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [3,3,2,2,1,1]
=> ? = 12
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [3,3,2,2,1]
=> ? = 11
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [3,3,2,1,1,1]
=> ? = 11
[1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [4,4,3,2,1]
=> ? = 14
[1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2,1]
=> ? = 20
[2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,2,2,2,1]
=> ? = 12
[2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [3,2,2,2,1,1]
=> ? = 11
[2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [4,3,3,3,2,1]
=> ? = 16
[2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [4,3,3,2,1]
=> ? = 13
[2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2,1]
=> ? = 19
[3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [4,3,2,2,2,1]
=> ? = 14
[3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [4,3,2,2,1]
=> ? = 12
[3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,2,1]
=> ? = 18
[4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [4,3,2,1,1,1]
=> ? = 12
[4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [4,3,2,1,1]
=> ? = 11
[4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,2,1]
=> ? = 17
[5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1,1]
=> ? = 16
[6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1]
=> ? = 15
[7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1]
=> ? = 21
[1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,2,2,2,2,1,1]
=> ? = 12
[1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,2,2,2,2,1]
=> ? = 11
[1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [2,2,2,2,1,1,1]
=> ? = 11
[1,1,1,4,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,2,1]
=> ? = 15
[1,1,1,5] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,4,4,4,3,2,1]
=> ? = 22
[1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> [3,3,3,2,1]
=> ? = 12
[1,1,5,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [4,4,4,3,2,1]
=> ? = 18
[1,1,6] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,5,5,4,3,2,1]
=> ? = 25
[1,2,1,3,1] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [3,3,2,2,2,1]
=> ? = 13
[1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [3,3,2,2,1,1]
=> ? = 12
[1,2,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,0,1,1,0,0,0,0,0]
=> [3,3,2,2,1]
=> ? = 11
[1,3,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [3,3,2,1,1,1,1]
=> ? = 12
[1,3,1,2,1] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [3,3,2,1,1,1]
=> ? = 11
[1,5,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [4,4,3,2,1]
=> ? = 14
[1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [5,5,4,3,2,1]
=> ? = 20
[1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,6,5,4,3,2,1]
=> ? = 27
[2,1,1,3,1] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,2,2,2,1]
=> ? = 12
[2,1,2,1,2] => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [3,2,2,2,1,1,1]
=> ? = 12
[2,1,2,2,1] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [3,2,2,2,1,1]
=> ? = 11
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St001161
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001161: Dyck paths ⟶ ℤResult quality: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 63%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001161: Dyck paths ⟶ ℤResult quality: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 63%
Values
[1] => [1] => [1,0]
=> 0
[1,1] => [2] => [1,1,0,0]
=> 0
[2] => [1,1] => [1,0,1,0]
=> 1
[1,1,1] => [3] => [1,1,1,0,0,0]
=> 0
[1,2] => [2,1] => [1,1,0,0,1,0]
=> 2
[2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[3] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,3] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 5
[2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
[3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3
[4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 6
[1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,1,2,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,1,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 7
[1,2,1,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 6
[1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 5
[1,4] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 9
[2,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 5
[2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4
[2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 8
[3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3
[3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 7
[4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 6
[5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 10
[1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 5
[1,1,1,2,1] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4
[1,1,1,3] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 9
[1,1,2,1,1] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
[1,1,2,2] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 8
[1,1,3,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> 7
[1,1,4] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> 12
[1,2,1,1,1] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
[1,2,1,2] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 7
[1,2,2,1] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 6
[1,2,3] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> 11
[1,3,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 5
[1,3,2] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 10
[1,4,1] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 9
[1,5] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 14
[2,1,1,1,1] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[2,1,1,2] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 6
[2,1,2,1] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 5
[1,1,1,1,1,1,1,1] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,1,1,1,1,1,2] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7
[1,1,1,1,1,2,1] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 6
[1,1,1,1,2,1,1] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 5
[1,1,1,1,2,2] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 12
[1,1,1,1,3,1] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 11
[1,1,1,2,1,1,1] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[1,1,1,2,1,2] => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 11
[1,1,1,2,2,1] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 10
[1,1,1,3,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 9
[1,1,1,4,1] => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 15
[1,1,1,5] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 22
[1,1,2,1,1,1,1] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[1,1,2,1,1,2] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 10
[1,1,2,1,2,1] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 9
[1,1,2,2,1,1] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 8
[1,1,3,1,1,1] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 7
[1,1,4,1,1] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 12
[1,1,5,1] => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 18
[1,1,6] => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 25
[1,1,1,1,1,1,1,1,1] => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 0
[1,1,1,1,1,1,1,2] => [8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 8
[1,1,1,1,1,1,2,1] => [7,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> ? = 7
[1,1,1,1,1,2,1,1] => [6,3] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 6
[1,1,1,1,2,1,1,1] => [5,4] => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 5
[1,1,1,1,2,2,1] => [5,2,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 12
[1,1,1,1,3,1,1] => [5,1,3] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 11
[1,1,1,2,1,1,1,1] => [4,5] => [1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 4
[1,1,1,2,1,1,2] => [4,4,1] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 12
[1,1,1,2,2,1,1] => [4,2,3] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 10
[1,1,1,3,1,1,1] => [4,1,4] => [1,1,1,1,0,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 9
[1,1,1,4,1,1] => [4,1,1,3] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 15
[1,1,1,5,1] => [4,1,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 22
[1,1,2,1,1,1,1,1] => [3,6] => [1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3
[1,1,2,1,1,1,2] => [3,5,1] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 11
[1,1,2,2,1,1,1] => [3,2,4] => [1,1,1,0,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 8
[1,1,3,1,1,1,1] => [3,1,5] => [1,1,1,0,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 7
[1,1,4,1,1,1] => [3,1,1,4] => [1,1,1,0,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 12
[1,1,5,1,1] => [3,1,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 18
[1,1,6,1] => [3,1,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 25
[1,1,7] => [3,1,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 33
[1,2,1,1,1,1,1,1] => [2,7] => [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2
[1,2,1,1,1,1,2] => [2,6,1] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 10
[1,2,1,1,1,2,1] => [2,5,2] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 9
[1,2,1,1,2,1,1] => [2,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 8
[1,2,1,2,1,1,1] => [2,3,4] => [1,1,0,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 7
[1,2,2,1,1,1,1] => [2,2,5] => [1,1,0,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 6
[1,3,1,1,1,1,1] => [2,1,6] => [1,1,0,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 5
[1,4,1,1,1,1] => [2,1,1,5] => [1,1,0,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 9
[1,5,1,1,1] => [2,1,1,1,4] => [1,1,0,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 14
Description
The major index north count of a Dyck path.
The descent set $\operatorname{des}(D)$ of a Dyck path $D = D_1 \cdots D_{2n}$ with $D_i \in \{N,E\}$ is given by all indices $i$ such that $D_i = E$ and $D_{i+1} = N$. This is, the positions of the valleys of $D$.
The '''major index''' of a Dyck path is then the sum of the positions of the valleys, $\sum_{i \in \operatorname{des}(D)} i$, see [[St000027]].
The '''major index north count''' is given by $\sum_{i \in \operatorname{des}(D)} \#\{ j \leq i \mid D_j = N\}$.
Matching statistic: St000012
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000012: Dyck paths ⟶ ℤResult quality: 48% ●values known / values provided: 54%●distinct values known / distinct values provided: 48%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000012: Dyck paths ⟶ ℤResult quality: 48% ●values known / values provided: 54%●distinct values known / distinct values provided: 48%
Values
[1] => [1] => [1,0]
=> [1,0]
=> 0
[1,1] => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 0
[2] => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,1,1] => [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[1,2] => [2,1] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[2,1] => [1,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[3] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,3] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 5
[2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 4
[3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 6
[1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
[1,1,2,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[1,1,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 7
[1,2,1,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 6
[1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 5
[1,4] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 9
[2,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 5
[2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 4
[2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 8
[3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 7
[4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 6
[5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 10
[1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 5
[1,1,1,2,1] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 4
[1,1,1,3] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 9
[1,1,2,1,1] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 3
[1,1,2,2] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 8
[1,1,3,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 7
[1,1,4] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 12
[1,2,1,1,1] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 2
[1,2,1,2] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 7
[1,2,2,1] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> 6
[1,2,3] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 11
[1,3,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 5
[1,3,2] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 10
[1,4,1] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 9
[1,5] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 14
[2,1,1,1,1] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[2,1,1,2] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 6
[2,1,2,1] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> 5
[1,1,1,1,1,1,2] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 7
[1,1,1,1,2,2] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 12
[1,1,1,2,1,2] => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 11
[1,1,1,5] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 22
[1,1,2,1,1,2] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 10
[1,1,6] => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 25
[1,2,1,1,1,2] => [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 9
[1,3,1,1,2] => [2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 12
[1,7] => [2,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 27
[2,1,1,1,1,2] => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 8
[2,1,2,1,2] => [1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 12
[2,1,5] => [1,3,1,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> ? = 23
[2,2,1,1,2] => [1,2,4,1] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 11
[2,6] => [1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 26
[3,1,1,1,2] => [1,1,5,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 10
[3,5] => [1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> ? = 25
[4,4] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 24
[5,3] => [1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 23
[6,2] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 22
[8] => [1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 28
[1,1,1,1,1,1,1,1,1] => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,1,1,1,1,1,1,2] => [8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 8
[1,1,1,1,1,1,2,1] => [7,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 7
[1,1,1,1,1,2,1,1] => [6,3] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6
[1,1,1,1,2,1,1,1] => [5,4] => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 5
[1,1,1,1,2,2,1] => [5,2,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 12
[1,1,1,1,3,1,1] => [5,1,3] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 11
[1,1,1,2,1,1,1,1] => [4,5] => [1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 4
[1,1,1,2,1,1,2] => [4,4,1] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 12
[1,1,1,2,2,1,1] => [4,2,3] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 10
[1,1,1,3,1,1,1] => [4,1,4] => [1,1,1,1,0,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 9
[1,1,1,4,1,1] => [4,1,1,3] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 15
[1,1,1,5,1] => [4,1,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 22
[1,1,2,1,1,1,1,1] => [3,6] => [1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 3
[1,1,2,1,1,1,2] => [3,5,1] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 11
[1,1,2,2,1,1,1] => [3,2,4] => [1,1,1,0,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ?
=> ? = 8
[1,1,3,1,1,1,1] => [3,1,5] => [1,1,1,0,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> ? = 7
[1,1,4,1,1,1] => [3,1,1,4] => [1,1,1,0,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 12
[1,1,5,1,1] => [3,1,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 18
[1,1,6,1] => [3,1,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 25
[1,1,7] => [3,1,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> ? = 33
[1,2,1,1,1,1,1,1] => [2,7] => [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 2
[1,2,1,1,1,1,2] => [2,6,1] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 10
[1,2,1,1,1,2,1] => [2,5,2] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 9
[1,2,1,1,2,1,1] => [2,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 8
[1,2,1,2,1,1,1] => [2,3,4] => [1,1,0,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 7
[1,2,2,1,1,1,1] => [2,2,5] => [1,1,0,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> ? = 6
[1,3,1,1,1,1,1] => [2,1,6] => [1,1,0,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 5
[1,4,1,1,1,1] => [2,1,1,5] => [1,1,0,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 9
[1,5,1,1,1] => [2,1,1,1,4] => [1,1,0,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 14
Description
The area of a Dyck path.
This is the number of complete squares in the integer lattice which are below the path and above the x-axis. The 'half-squares' directly above the axis do not contribute to this statistic.
1. Dyck paths are bijection with '''area sequences''' $(a_1,\ldots,a_n)$ such that $a_1 = 0, a_{k+1} \leq a_k + 1$.
2. The generating function $\mathbf{D}_n(q) = \sum_{D \in \mathfrak{D}_n} q^{\operatorname{area}(D)}$ satisfy the recurrence $$\mathbf{D}_{n+1}(q) = \sum q^k \mathbf{D}_k(q) \mathbf{D}_{n-k}(q).$$
3. The area is equidistributed with [[St000005]] and [[St000006]]. Pairs of these statistics play an important role in the theory of $q,t$-Catalan numbers.
Matching statistic: St000947
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000947: Dyck paths ⟶ ℤResult quality: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 63%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000947: Dyck paths ⟶ ℤResult quality: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 63%
Values
[1] => [1] => [1,0]
=> ? = 0
[1,1] => [2] => [1,1,0,0]
=> 0
[2] => [1,1] => [1,0,1,0]
=> 1
[1,1,1] => [3] => [1,1,1,0,0,0]
=> 0
[1,2] => [2,1] => [1,1,0,0,1,0]
=> 2
[2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[3] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,3] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 5
[2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
[3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3
[4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 6
[1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,1,2,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,1,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 7
[1,2,1,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 6
[1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 5
[1,4] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 9
[2,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 5
[2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4
[2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 8
[3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3
[3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 7
[4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 6
[5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 10
[1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 5
[1,1,1,2,1] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4
[1,1,1,3] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 9
[1,1,2,1,1] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
[1,1,2,2] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 8
[1,1,3,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> 7
[1,1,4] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> 12
[1,2,1,1,1] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
[1,2,1,2] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 7
[1,2,2,1] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 6
[1,2,3] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> 11
[1,3,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 5
[1,3,2] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 10
[1,4,1] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 9
[1,5] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 14
[2,1,1,1,1] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[2,1,1,2] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 6
[2,1,2,1] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 5
[2,1,3] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 10
[1,1,1,1,1,1,1,1] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,1,1,1,1,1,2] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7
[1,1,1,1,1,2,1] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 6
[1,1,1,1,2,1,1] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 5
[1,1,1,1,2,2] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 12
[1,1,1,1,3,1] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 11
[1,1,1,2,1,1,1] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[1,1,1,2,1,2] => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 11
[1,1,1,2,2,1] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 10
[1,1,1,3,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 9
[1,1,1,4,1] => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 15
[1,1,1,5] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 22
[1,1,2,1,1,1,1] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[1,1,2,1,1,2] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 10
[1,1,2,1,2,1] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 9
[1,1,2,2,1,1] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 8
[1,1,3,1,1,1] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 7
[1,1,4,1,1] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 12
[1,1,5,1] => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 18
[1,1,6] => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 25
[1,1,1,1,1,1,1,1,1] => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 0
[1,1,1,1,1,1,1,2] => [8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 8
[1,1,1,1,1,1,2,1] => [7,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> ? = 7
[1,1,1,1,1,2,1,1] => [6,3] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 6
[1,1,1,1,2,1,1,1] => [5,4] => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 5
[1,1,1,1,2,2,1] => [5,2,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 12
[1,1,1,1,3,1,1] => [5,1,3] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 11
[1,1,1,2,1,1,1,1] => [4,5] => [1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 4
[1,1,1,2,1,1,2] => [4,4,1] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 12
[1,1,1,2,2,1,1] => [4,2,3] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 10
[1,1,1,3,1,1,1] => [4,1,4] => [1,1,1,1,0,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 9
[1,1,1,4,1,1] => [4,1,1,3] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 15
[1,1,1,5,1] => [4,1,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 22
[1,1,2,1,1,1,1,1] => [3,6] => [1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3
[1,1,2,1,1,1,2] => [3,5,1] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 11
[1,1,2,2,1,1,1] => [3,2,4] => [1,1,1,0,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 8
[1,1,3,1,1,1,1] => [3,1,5] => [1,1,1,0,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 7
[1,1,4,1,1,1] => [3,1,1,4] => [1,1,1,0,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 12
[1,1,5,1,1] => [3,1,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 18
[1,1,6,1] => [3,1,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 25
[1,1,7] => [3,1,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 33
[1,2,1,1,1,1,1,1] => [2,7] => [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2
[1,2,1,1,1,1,2] => [2,6,1] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 10
[1,2,1,1,1,2,1] => [2,5,2] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 9
[1,2,1,1,2,1,1] => [2,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 8
[1,2,1,2,1,1,1] => [2,3,4] => [1,1,0,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 7
[1,2,2,1,1,1,1] => [2,2,5] => [1,1,0,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 6
[1,3,1,1,1,1,1] => [2,1,6] => [1,1,0,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 5
[1,4,1,1,1,1] => [2,1,1,5] => [1,1,0,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 9
Description
The major index east count of a Dyck path.
The descent set $\operatorname{des}(D)$ of a Dyck path $D = D_1 \cdots D_{2n}$ with $D_i \in \{N,E\}$ is given by all indices $i$ such that $D_i = E$ and $D_{i+1} = N$. This is, the positions of the valleys of $D$.
The '''major index''' of a Dyck path is then the sum of the positions of the valleys, $\sum_{i \in \operatorname{des}(D)} i$, see [[St000027]].
The '''major index east count''' is given by $\sum_{i \in \operatorname{des}(D)} \#\{ j \leq i \mid D_j = E\}$.
Matching statistic: St000246
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000246: Permutations ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 80%
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000246: Permutations ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 80%
Values
[1] => [1,0]
=> [1,0]
=> [1] => 0
[1,1] => [1,0,1,0]
=> [1,0,1,0]
=> [2,1] => 0
[2] => [1,1,0,0]
=> [1,1,0,0]
=> [1,2] => 1
[1,1,1] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [3,2,1] => 0
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,1,3] => 2
[2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => 1
[3] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 3
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 0
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 3
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 2
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 5
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 4
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 6
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => 4
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => 3
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => 7
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => 6
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 5
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => 9
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => 5
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => 4
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => 8
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 3
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => 7
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 6
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 10
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1,6] => 5
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,1,5] => 4
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [4,3,2,1,5,6] => 9
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1,4] => 3
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [5,3,2,1,4,6] => 8
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,4,5] => 7
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [3,2,1,4,5,6] => 12
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1,3] => 2
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [5,4,2,1,3,6] => 7
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [6,4,2,1,3,5] => 6
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,3,5,6] => 11
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [6,5,2,1,3,4] => 5
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [5,2,1,3,4,6] => 10
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => 9
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => 14
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,2] => 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [5,4,3,1,2,6] => 6
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [6,4,3,1,2,5] => 5
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,2,1,6] => ? = 5
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,2,1,5] => ? = 4
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [6,4,3,2,1,5,7] => ? = 10
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [7,4,3,2,1,5,6] => ? = 9
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,2,1,4] => ? = 3
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [6,5,3,2,1,4,7] => ? = 9
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [7,5,3,2,1,4,6] => ? = 8
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [7,6,3,2,1,4,5] => ? = 7
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,2,1,3] => ? = 2
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [6,5,4,2,1,3,7] => ? = 8
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [7,5,4,2,1,3,6] => ? = 7
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [5,4,2,1,3,6,7] => ? = 13
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [7,6,4,2,1,3,5] => ? = 6
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [6,4,2,1,3,5,7] => ? = 12
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [7,6,5,2,1,3,4] => ? = 5
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [6,5,2,1,3,4,7] => ? = 11
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [7,5,2,1,3,4,6] => ? = 10
[1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [7,6,2,1,3,4,5] => ? = 9
[2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,1,2,7] => ? = 7
[2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,1,2,6] => ? = 6
[2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [5,4,3,1,2,6,7] => ? = 12
[2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,1,2,5] => ? = 5
[2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [6,4,3,1,2,5,7] => ? = 11
[2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [7,4,3,1,2,5,6] => ? = 10
[2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [4,3,1,2,5,6,7] => ? = 16
[2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,1,2,4] => ? = 4
[2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> [6,5,3,1,2,4,7] => ? = 10
[2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [7,5,3,1,2,4,6] => ? = 9
[2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [7,6,3,1,2,4,5] => ? = 8
[3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [6,5,4,1,2,3,7] => ? = 9
[3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [7,5,4,1,2,3,6] => ? = 8
[3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [5,4,1,2,3,6,7] => ? = 14
[3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [7,6,4,1,2,3,5] => ? = 7
[4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [6,5,1,2,3,4,7] => ? = 12
[1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [8,7,5,4,3,2,1,6] => ? = 5
[1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [7,5,4,3,2,1,6,8] => ? = 12
[1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> [8,5,4,3,2,1,6,7] => ? = 11
[1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> [7,6,4,3,2,1,5,8] => ? = 11
[1,1,1,3,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> [8,7,4,3,2,1,5,6] => ? = 9
[1,1,1,4,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> [8,4,3,2,1,5,6,7] => ? = 15
[1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [8,7,6,5,3,2,1,4] => ? = 3
[1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [7,6,5,3,2,1,4,8] => ? = 10
[1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0,1,0]
=> [8,6,5,3,2,1,4,7] => ? = 9
[1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0,1,0]
=> [8,7,5,3,2,1,4,6] => ? = 8
[1,1,3,1,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
=> [8,7,6,3,2,1,4,5] => ? = 7
[1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
=> [8,7,3,2,1,4,5,6] => ? = 12
[1,2,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [7,6,5,4,2,1,3,8] => ? = 9
[1,2,1,2,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
=> [8,7,5,4,2,1,3,6] => ? = 7
[1,2,1,3,1] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0]
=> [8,5,4,2,1,3,6,7] => ? = 13
[1,2,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0]
=> [8,7,4,2,1,3,5,6] => ? = 11
Description
The number of non-inversions of a permutation.
For a permutation of $\{1,\ldots,n\}$, this is given by $\operatorname{noninv}(\pi) = \binom{n}{2}-\operatorname{inv}(\pi)$.
The following 84 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000018The number of inversions of a permutation. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000578The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton. St000081The number of edges of a graph. St000492The rob statistic of a set partition. St000499The rcb statistic of a set partition. St000579The number of occurrences of the pattern {{1},{2}} such that 2 is a maximal element. St000493The los statistic of a set partition. St000498The lcs statistic of a set partition. St000041The number of nestings of a perfect matching. St000992The alternating sum of the parts of an integer partition. St001814The number of partitions interlacing the given partition. St001759The Rajchgot index of a permutation. St000147The largest part of an integer partition. St000459The hook length of the base cell of a partition. St001400The total number of Littlewood-Richardson tableaux of given shape. St001360The number of covering relations in Young's lattice below a partition. St001389The number of partitions of the same length below the given integer partition. St001527The cyclic permutation representation number of an integer partition. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000384The maximal part of the shifted composition of an integer partition. St000784The maximum of the length and the largest part of the integer partition. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St001055The Grundy value for the game of removing cells of a row in an integer partition. St000063The number of linear extensions of a certain poset defined for an integer partition. St000532The total number of rook placements on a Ferrers board. St000161The sum of the sizes of the right subtrees of a binary tree. St000460The hook length of the last cell along the main diagonal of an integer partition. St000667The greatest common divisor of the parts of the partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000446The disorder of a permutation. St000833The comajor index of a permutation. St001671Haglund's hag of a permutation. St000477The weight of a partition according to Alladi. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000681The Grundy value of Chomp on Ferrers diagrams. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001571The Cartan determinant of the integer partition. St000108The number of partitions contained in the given partition. St000794The mak of a permutation. St000797The stat`` of a permutation. St000798The makl of a permutation. St000472The sum of the ascent bottoms of a permutation. St000067The inversion number of the alternating sign matrix. St000057The Shynar inversion number of a standard tableau. St000076The rank of the alternating sign matrix in the alternating sign matrix poset. St000145The Dyson rank of a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St000004The major index of a permutation. St000005The bounce statistic of a Dyck path. St000154The sum of the descent bottoms of a permutation. St000305The inverse major index of a permutation. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St000006The dinv of a Dyck path. St000133The "bounce" of a permutation. St000156The Denert index of a permutation. St000304The load of a permutation. St000796The stat' of a permutation. St000448The number of pairs of vertices of a graph with distance 2. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001341The number of edges in the center of a graph. St001311The cyclomatic 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. St001746The coalition number of a graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001120The length of a longest path in a graph. St001725The harmonious chromatic number of a graph. St001645The pebbling number of a connected graph. St000719The number of alignments in a perfect matching. St001622The number of join-irreducible elements of a lattice. St000360The number of occurrences of the pattern 32-1. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000450The number of edges minus the number of vertices plus 2 of a graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001330The hat guessing number of a graph. St001621The number of atoms of a lattice.
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