searching the database
Your data matches 195 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000292
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,0,1,0]
=> [1]
=> 10 => 0
[1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> 1010 => 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> 110 => 0
[2,1,3] => [1,1,0,0,1,0]
=> [2]
=> 100 => 0
[2,3,1] => [1,1,0,1,0,0]
=> [1]
=> 10 => 0
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 101010 => 2
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 11010 => 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 100110 => 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 10110 => 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1110 => 0
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1110 => 0
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 10100 => 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 1100 => 0
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> 10010 => 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1010 => 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 110 => 0
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 110 => 0
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 1000 => 0
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2]
=> 100 => 0
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 1000 => 0
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [2]
=> 100 => 0
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> 10 => 0
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> 10 => 0
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 10101010 => 3
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 1101010 => 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 10011010 => 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 1011010 => 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 111010 => 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 111010 => 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 10100110 => 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 1100110 => 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 10010110 => 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 1010110 => 2
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 110110 => 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 110110 => 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 10001110 => 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 1001110 => 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 10001110 => 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 1001110 => 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 101110 => 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 101110 => 1
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 11110 => 0
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 11110 => 0
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 11110 => 0
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 11110 => 0
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 11110 => 0
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 11110 => 0
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 1010100 => 2
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 110100 => 1
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1001100 => 1
Description
The number of ascents of a binary word.
Matching statistic: St000291
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,0,1,0]
=> [1]
=> 10 => 1 = 0 + 1
[1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> 1010 => 2 = 1 + 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> 110 => 1 = 0 + 1
[2,1,3] => [1,1,0,0,1,0]
=> [2]
=> 100 => 1 = 0 + 1
[2,3,1] => [1,1,0,1,0,0]
=> [1]
=> 10 => 1 = 0 + 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 101010 => 3 = 2 + 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 11010 => 2 = 1 + 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 100110 => 2 = 1 + 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 10110 => 2 = 1 + 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1110 => 1 = 0 + 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1110 => 1 = 0 + 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 10100 => 2 = 1 + 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 1100 => 1 = 0 + 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> 10010 => 2 = 1 + 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1010 => 2 = 1 + 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 110 => 1 = 0 + 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 110 => 1 = 0 + 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 1000 => 1 = 0 + 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2]
=> 100 => 1 = 0 + 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 1000 => 1 = 0 + 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [2]
=> 100 => 1 = 0 + 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> 10 => 1 = 0 + 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> 10 => 1 = 0 + 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 10101010 => 4 = 3 + 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 1101010 => 3 = 2 + 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 10011010 => 3 = 2 + 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 1011010 => 3 = 2 + 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 111010 => 2 = 1 + 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 111010 => 2 = 1 + 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 10100110 => 3 = 2 + 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 1100110 => 2 = 1 + 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 10010110 => 3 = 2 + 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 1010110 => 3 = 2 + 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 110110 => 2 = 1 + 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 110110 => 2 = 1 + 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 10001110 => 2 = 1 + 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 1001110 => 2 = 1 + 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 10001110 => 2 = 1 + 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 1001110 => 2 = 1 + 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 101110 => 2 = 1 + 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 101110 => 2 = 1 + 1
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 11110 => 1 = 0 + 1
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 11110 => 1 = 0 + 1
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 11110 => 1 = 0 + 1
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 11110 => 1 = 0 + 1
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 11110 => 1 = 0 + 1
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 11110 => 1 = 0 + 1
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 1010100 => 3 = 2 + 1
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 110100 => 2 = 1 + 1
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1001100 => 2 = 1 + 1
Description
The number of descents of a binary word.
Matching statistic: St000390
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,0,1,0]
=> [1]
=> 10 => 1 = 0 + 1
[1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> 1010 => 2 = 1 + 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> 110 => 1 = 0 + 1
[2,1,3] => [1,1,0,0,1,0]
=> [2]
=> 100 => 1 = 0 + 1
[2,3,1] => [1,1,0,1,0,0]
=> [1]
=> 10 => 1 = 0 + 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 101010 => 3 = 2 + 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 11010 => 2 = 1 + 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 100110 => 2 = 1 + 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 10110 => 2 = 1 + 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1110 => 1 = 0 + 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1110 => 1 = 0 + 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 10100 => 2 = 1 + 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 1100 => 1 = 0 + 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> 10010 => 2 = 1 + 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1010 => 2 = 1 + 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 110 => 1 = 0 + 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 110 => 1 = 0 + 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 1000 => 1 = 0 + 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2]
=> 100 => 1 = 0 + 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 1000 => 1 = 0 + 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [2]
=> 100 => 1 = 0 + 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> 10 => 1 = 0 + 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> 10 => 1 = 0 + 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 10101010 => 4 = 3 + 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 1101010 => 3 = 2 + 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 10011010 => 3 = 2 + 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 1011010 => 3 = 2 + 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 111010 => 2 = 1 + 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 111010 => 2 = 1 + 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 10100110 => 3 = 2 + 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 1100110 => 2 = 1 + 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 10010110 => 3 = 2 + 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 1010110 => 3 = 2 + 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 110110 => 2 = 1 + 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 110110 => 2 = 1 + 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 10001110 => 2 = 1 + 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 1001110 => 2 = 1 + 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 10001110 => 2 = 1 + 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 1001110 => 2 = 1 + 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 101110 => 2 = 1 + 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 101110 => 2 = 1 + 1
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 11110 => 1 = 0 + 1
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 11110 => 1 = 0 + 1
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 11110 => 1 = 0 + 1
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 11110 => 1 = 0 + 1
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 11110 => 1 = 0 + 1
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 11110 => 1 = 0 + 1
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 1010100 => 3 = 2 + 1
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 110100 => 2 = 1 + 1
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1001100 => 2 = 1 + 1
Description
The number of runs of ones in a binary word.
Matching statistic: St000010
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,0,1,0]
=> [1,1] => [1,1]
=> 2 = 0 + 2
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,1] => [1,1,1]
=> 3 = 1 + 2
[1,3,2] => [1,0,1,1,0,0]
=> [1,2] => [2,1]
=> 2 = 0 + 2
[2,1,3] => [1,1,0,0,1,0]
=> [2,1] => [2,1]
=> 2 = 0 + 2
[2,3,1] => [1,1,0,1,0,0]
=> [2,1] => [2,1]
=> 2 = 0 + 2
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1]
=> 4 = 2 + 2
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1]
=> 3 = 1 + 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,1,1]
=> 3 = 1 + 2
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,2,1] => [2,1,1]
=> 3 = 1 + 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,3] => [3,1]
=> 2 = 0 + 2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,3] => [3,1]
=> 2 = 0 + 2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [2,1,1] => [2,1,1]
=> 3 = 1 + 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2] => [2,2]
=> 2 = 0 + 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [2,1,1] => [2,1,1]
=> 3 = 1 + 2
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1,1] => [2,1,1]
=> 3 = 1 + 2
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,2] => [2,2]
=> 2 = 0 + 2
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,2] => [2,2]
=> 2 = 0 + 2
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3,1] => [3,1]
=> 2 = 0 + 2
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [3,1] => [3,1]
=> 2 = 0 + 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3,1] => [3,1]
=> 2 = 0 + 2
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,1] => [3,1]
=> 2 = 0 + 2
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [3,1] => [3,1]
=> 2 = 0 + 2
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [3,1] => [3,1]
=> 2 = 0 + 2
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,1,1,1,1]
=> 5 = 3 + 2
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [2,1,1,1]
=> 4 = 2 + 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1,1]
=> 4 = 2 + 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [2,1,1,1]
=> 4 = 2 + 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1]
=> 3 = 1 + 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1]
=> 3 = 1 + 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [2,1,1,1]
=> 4 = 2 + 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1]
=> 3 = 1 + 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [2,1,1,1]
=> 4 = 2 + 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [2,1,1,1]
=> 4 = 2 + 2
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [2,2,1]
=> 3 = 1 + 2
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [2,2,1]
=> 3 = 1 + 2
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [3,1,1]
=> 3 = 1 + 2
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [3,1,1]
=> 3 = 1 + 2
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [3,1,1]
=> 3 = 1 + 2
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [3,1,1]
=> 3 = 1 + 2
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [3,1,1]
=> 3 = 1 + 2
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [3,1,1]
=> 3 = 1 + 2
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1]
=> 2 = 0 + 2
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1]
=> 2 = 0 + 2
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1]
=> 2 = 0 + 2
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1]
=> 2 = 0 + 2
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1]
=> 2 = 0 + 2
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1]
=> 2 = 0 + 2
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> 4 = 2 + 2
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [2,2,1]
=> 3 = 1 + 2
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,2,1]
=> 3 = 1 + 2
[7,8,3,4,9,10,1,2,5,6] => [1,1,1,1,1,1,1,0,1,0,0,0,1,0,1,0,0,0,0,0]
=> ? => ?
=> ? = 2 + 2
[7,8,4,5,9,10,2,3,6,1] => [1,1,1,1,1,1,1,0,1,0,0,0,1,0,1,0,0,0,0,0]
=> ? => ?
=> ? = 2 + 2
[7,8,4,5,9,10,2,6,1,3] => [1,1,1,1,1,1,1,0,1,0,0,0,1,0,1,0,0,0,0,0]
=> ? => ?
=> ? = 2 + 2
[7,8,4,5,9,10,2,1,3,6] => [1,1,1,1,1,1,1,0,1,0,0,0,1,0,1,0,0,0,0,0]
=> ? => ?
=> ? = 2 + 2
[7,8,4,2,5,9,10,1,3,6] => [1,1,1,1,1,1,1,0,1,0,0,0,0,1,0,1,0,0,0,0]
=> ? => ?
=> ? = 2 + 2
[8,11,1,2,3,4,5,6,7,9,10] => [1,1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [8,3] => ?
=> ? = 0 + 2
[2,12,11,10,9,8,7,6,5,4,1,3] => [1,1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> [2,10] => ?
=> ? = 0 + 2
[2,12,1,3,4,5,6,7,8,9,10,11] => [1,1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> [2,10] => ?
=> ? = 0 + 2
[2,12,9,11,4,6,8,10,1,3,5,7] => [1,1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> [2,10] => ?
=> ? = 0 + 2
[2,12,5,11,8,10,7,9,4,6,1,3] => [1,1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> [2,10] => ?
=> ? = 0 + 2
[2,12,7,11,6,10,5,9,4,8,1,3] => [1,1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> [2,10] => ?
=> ? = 0 + 2
[2,12,6,9,5,8,7,11,4,10,1,3] => [1,1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> [2,10] => ?
=> ? = 0 + 2
[2,12,4,5,6,7,8,9,10,11,1,3] => [1,1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> [2,10] => ?
=> ? = 0 + 2
[7,8,1,2,3,9,10,4,5,6] => [1,1,1,1,1,1,1,0,1,0,0,0,0,1,0,1,0,0,0,0]
=> ? => ?
=> ? = 2 + 2
[7,8,1,2,9,10,3,4,5,6] => [1,1,1,1,1,1,1,0,1,0,0,0,1,0,1,0,0,0,0,0]
=> ? => ?
=> ? = 2 + 2
[4,9,3,10,5,7,1,8,2,6] => [1,1,1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> ? => ?
=> ? = 1 + 2
[4,9,3,10,7,1,8,2,5,6] => [1,1,1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> ? => ?
=> ? = 1 + 2
[7,8,2,3,9,10,4,5,1,6] => [1,1,1,1,1,1,1,0,1,0,0,0,1,0,1,0,0,0,0,0]
=> ? => ?
=> ? = 2 + 2
[7,8,2,3,9,10,4,1,5,6] => [1,1,1,1,1,1,1,0,1,0,0,0,1,0,1,0,0,0,0,0]
=> ? => ?
=> ? = 2 + 2
[7,8,2,3,9,10,1,4,5,6] => [1,1,1,1,1,1,1,0,1,0,0,0,1,0,1,0,0,0,0,0]
=> ? => ?
=> ? = 2 + 2
[8,11,1,2,3,4,5,9,10,6,7] => [1,1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [8,3] => ?
=> ? = 0 + 2
Description
The length of the partition.
Matching statistic: St000288
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,0,1,0]
=> [1,1] => 11 => 2 = 0 + 2
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,1] => 111 => 3 = 1 + 2
[1,3,2] => [1,0,1,1,0,0]
=> [1,2] => 110 => 2 = 0 + 2
[2,1,3] => [1,1,0,0,1,0]
=> [2,1] => 101 => 2 = 0 + 2
[2,3,1] => [1,1,0,1,0,0]
=> [2,1] => 101 => 2 = 0 + 2
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1111 => 4 = 2 + 2
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,2] => 1110 => 3 = 1 + 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,2,1] => 1101 => 3 = 1 + 2
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,2,1] => 1101 => 3 = 1 + 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,3] => 1100 => 2 = 0 + 2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,3] => 1100 => 2 = 0 + 2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [2,1,1] => 1011 => 3 = 1 + 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2] => 1010 => 2 = 0 + 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [2,1,1] => 1011 => 3 = 1 + 2
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1,1] => 1011 => 3 = 1 + 2
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,2] => 1010 => 2 = 0 + 2
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,2] => 1010 => 2 = 0 + 2
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3,1] => 1001 => 2 = 0 + 2
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [3,1] => 1001 => 2 = 0 + 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3,1] => 1001 => 2 = 0 + 2
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,1] => 1001 => 2 = 0 + 2
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [3,1] => 1001 => 2 = 0 + 2
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [3,1] => 1001 => 2 = 0 + 2
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 11111 => 5 = 3 + 2
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 11110 => 4 = 2 + 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 11101 => 4 = 2 + 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => 11101 => 4 = 2 + 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 11100 => 3 = 1 + 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 11100 => 3 = 1 + 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 11011 => 4 = 2 + 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 11010 => 3 = 1 + 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => 11011 => 4 = 2 + 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => 11011 => 4 = 2 + 2
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => 11010 => 3 = 1 + 2
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => 11010 => 3 = 1 + 2
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 11001 => 3 = 1 + 2
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => 11001 => 3 = 1 + 2
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 11001 => 3 = 1 + 2
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => 11001 => 3 = 1 + 2
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => 11001 => 3 = 1 + 2
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => 11001 => 3 = 1 + 2
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 11000 => 2 = 0 + 2
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 11000 => 2 = 0 + 2
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 11000 => 2 = 0 + 2
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 11000 => 2 = 0 + 2
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 11000 => 2 = 0 + 2
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 11000 => 2 = 0 + 2
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 10111 => 4 = 2 + 2
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 10110 => 3 = 1 + 2
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 10101 => 3 = 1 + 2
[8,9,5,6,3,4,10,1,2,7] => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0,0,0]
=> [8,1,1] => 1000000011 => ? = 1 + 2
[9,7,6,5,4,3,2,10,1,8] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0,0]
=> [9,1] => 1000000001 => ? = 0 + 2
[4,9,10,1,7,8,5,6,2,3] => [1,1,1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [4,5,1] => 1000100001 => ? = 1 + 2
[6,7,8,9,10,1,2,3,4,5] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [6,1,1,1,1] => 1000001111 => ? = 3 + 2
[9,7,5,10,3,8,2,6,1,4] => [1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0]
=> [9,1] => 1000000001 => ? = 0 + 2
[4,5,10,1,2,9,8,7,6,3] => [1,1,1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [4,1,5] => 1000110000 => ? = 1 + 2
[3,10,1,9,8,7,6,5,4,2] => [1,1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [3,7] => 1001000000 => ? = 0 + 2
[9,10,8,7,6,5,4,3,1,2] => [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [9,1] => 1000000001 => ? = 0 + 2
[8,1,2,3,4,5,6,9,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [8,1] => 100000001 => ? = 0 + 2
[7,1,2,3,4,5,8,9,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,0]
=> [7,1,1] => 100000011 => ? = 1 + 2
[8,1,2,3,4,7,5,9,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [8,1] => 100000001 => ? = 0 + 2
[8,1,2,3,9,4,5,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0]
=> [8,1] => 100000001 => ? = 0 + 2
[7,1,2,8,3,4,5,9,6] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0,0]
=> [7,1,1] => 100000011 => ? = 1 + 2
[8,1,2,5,9,3,4,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0]
=> [8,1] => 100000001 => ? = 0 + 2
[8,1,2,9,6,3,4,5,7] => [1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
=> [8,1] => 100000001 => ? = 0 + 2
[8,1,4,2,3,5,6,9,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [8,1] => 100000001 => ? = 0 + 2
[8,1,7,2,3,4,5,9,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [8,1] => 100000001 => ? = 0 + 2
[3,1,4,9,2,5,6,7,8] => [1,1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,5] => 100110000 => ? = 1 + 2
[3,1,4,9,2,5,8,6,7] => [1,1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,5] => 100110000 => ? = 1 + 2
[7,1,4,5,2,3,8,9,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,0]
=> [7,1,1] => 100000011 => ? = 1 + 2
[7,1,8,9,2,3,4,5,6] => [1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0]
=> [7,1,1] => 100000011 => ? = 1 + 2
[3,1,8,9,7,2,4,5,6] => [1,1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,5,1] => 100100001 => ? = 1 + 2
[8,1,4,5,6,2,3,9,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [8,1] => 100000001 => ? = 0 + 2
[8,1,4,9,7,2,3,5,6] => [1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
=> [8,1] => 100000001 => ? = 0 + 2
[8,1,9,5,6,2,3,4,7] => [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [8,1] => 100000001 => ? = 0 + 2
[8,1,9,7,6,2,3,4,5] => [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [8,1] => 100000001 => ? = 0 + 2
[7,1,4,5,6,2,8,9,3] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,0]
=> [7,1,1] => 100000011 => ? = 1 + 2
[3,1,4,9,8,7,2,5,6] => [1,1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,5] => 100110000 => ? = 1 + 2
[8,1,4,5,6,7,2,9,3] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [8,1] => 100000001 => ? = 0 + 2
[3,1,4,9,6,7,8,2,5] => [1,1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,5] => 100110000 => ? = 1 + 2
[2,8,1,3,9,4,5,6,7] => [1,1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [2,6,1] => 101000001 => ? = 1 + 2
[2,8,1,9,7,3,4,5,6] => [1,1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [2,6,1] => 101000001 => ? = 1 + 2
[8,3,1,2,4,5,6,9,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [8,1] => 100000001 => ? = 0 + 2
[7,4,1,2,3,5,8,9,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,0]
=> [7,1,1] => 100000011 => ? = 1 + 2
[5,8,1,2,3,4,6,9,7] => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0,0]
=> [5,3,1] => 100001001 => ? = 1 + 2
[8,6,1,2,3,4,5,9,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [8,1] => 100000001 => ? = 0 + 2
[8,9,1,2,3,4,5,6,7] => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [8,1] => 100000001 => ? = 0 + 2
[8,7,1,2,3,4,5,9,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [8,1] => 100000001 => ? = 0 + 2
[7,6,1,2,3,4,8,9,5] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,0]
=> [7,1,1] => 100000011 => ? = 1 + 2
[8,6,1,2,3,7,4,9,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [8,1] => 100000001 => ? = 0 + 2
[8,4,1,2,9,3,5,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0]
=> [8,1] => 100000001 => ? = 0 + 2
[8,7,1,9,2,3,4,5,6] => [1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
=> [8,1] => 100000001 => ? = 0 + 2
[8,7,1,6,2,3,4,9,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [8,1] => 100000001 => ? = 0 + 2
[8,9,1,5,6,7,2,3,4] => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [8,1] => 100000001 => ? = 0 + 2
[2,3,9,1,4,5,6,7,8] => [1,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,1,6] => 101100000 => ? = 1 + 2
[2,3,9,1,4,5,8,6,7] => [1,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,1,6] => 101100000 => ? = 1 + 2
[2,3,9,1,4,7,8,5,6] => [1,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,1,6] => 101100000 => ? = 1 + 2
[2,3,9,1,8,7,4,5,6] => [1,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,1,6] => 101100000 => ? = 1 + 2
[2,3,9,1,6,7,8,4,5] => [1,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,1,6] => 101100000 => ? = 1 + 2
[2,8,4,1,9,3,5,6,7] => [1,1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [2,6,1] => 101000001 => ? = 1 + 2
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000097
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,0,1,0]
=> [1,1] => ([(0,1)],2)
=> 2 = 0 + 2
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[1,3,2] => [1,0,1,1,0,0]
=> [1,2] => ([(1,2)],3)
=> 2 = 0 + 2
[2,1,3] => [1,1,0,0,1,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[2,3,1] => [1,1,0,1,0,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 2 + 2
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,3] => ([(2,3)],4)
=> 2 = 0 + 2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,3] => ([(2,3)],4)
=> 2 = 0 + 2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 3 + 2
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 2 = 0 + 2
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 2 = 0 + 2
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 2 = 0 + 2
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 2 = 0 + 2
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 2 = 0 + 2
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 2 = 0 + 2
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[8,9,5,6,3,4,10,1,2,7] => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0,0,0]
=> [8,1,1] => ([(0,8),(0,9),(1,8),(1,9),(2,8),(2,9),(3,8),(3,9),(4,8),(4,9),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 1 + 2
[9,7,6,5,4,3,2,10,1,8] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0,0]
=> [9,1] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 0 + 2
[4,9,10,1,7,8,5,6,2,3] => [1,1,1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [4,5,1] => ([(0,9),(1,9),(2,9),(3,9),(4,8),(4,9),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 1 + 2
[6,7,8,9,10,1,2,3,4,5] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [6,1,1,1,1] => ([(0,6),(0,7),(0,8),(0,9),(1,6),(1,7),(1,8),(1,9),(2,6),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 3 + 2
[7,10,5,9,3,8,1,6,4,2] => [1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0]
=> [7,3] => ([(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 0 + 2
[9,7,5,10,3,8,2,6,1,4] => [1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0]
=> [9,1] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 0 + 2
[4,5,10,1,2,9,8,7,6,3] => [1,1,1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [4,1,5] => ([(4,8),(4,9),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 1 + 2
[3,10,1,9,8,7,6,5,4,2] => [1,1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [3,7] => ([(6,9),(7,9),(8,9)],10)
=> ? = 0 + 2
[9,10,8,7,6,5,4,3,1,2] => [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [9,1] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 0 + 2
[8,1,2,3,4,5,6,9,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [8,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 0 + 2
[7,1,2,3,4,5,9,6,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,0]
=> [7,2] => ([(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 0 + 2
[7,1,2,3,4,5,8,9,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,0]
=> [7,1,1] => ([(0,7),(0,8),(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[6,1,2,3,4,9,5,7,8] => [1,1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0,0]
=> [6,3] => ([(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 0 + 2
[8,1,2,3,4,7,5,9,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [8,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 0 + 2
[6,1,2,3,4,7,9,5,8] => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0,0]
=> [6,1,2] => ([(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[6,1,2,3,4,9,8,5,7] => [1,1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0,0]
=> [6,3] => ([(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 0 + 2
[5,1,2,3,9,4,6,7,8] => [1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0]
=> [5,4] => ([(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 0 + 2
[8,1,2,3,9,4,5,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0]
=> [8,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 0 + 2
[5,1,2,3,9,8,4,6,7] => [1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0]
=> [5,4] => ([(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 0 + 2
[4,1,2,9,3,5,8,6,7] => [1,1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [4,5] => ([(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 0 + 2
[4,1,2,9,3,8,5,6,7] => [1,1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [4,5] => ([(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 0 + 2
[7,1,2,8,3,4,5,9,6] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0,0]
=> [7,1,1] => ([(0,7),(0,8),(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[4,1,2,8,9,3,5,6,7] => [1,1,1,1,0,0,0,1,1,1,1,0,1,0,0,0,0,0]
=> [4,4,1] => ([(0,8),(1,8),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[8,1,2,5,9,3,4,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0]
=> [8,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 0 + 2
[8,1,2,9,6,3,4,5,7] => [1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
=> [8,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 0 + 2
[4,1,2,9,6,7,8,3,5] => [1,1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [4,5] => ([(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 0 + 2
[3,1,9,2,4,5,6,7,8] => [1,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,6] => ([(5,8),(6,8),(7,8)],9)
=> ? = 0 + 2
[3,1,9,2,4,5,8,6,7] => [1,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,6] => ([(5,8),(6,8),(7,8)],9)
=> ? = 0 + 2
[8,1,4,2,3,5,6,9,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [8,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 0 + 2
[8,1,7,2,3,4,5,9,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [8,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 0 + 2
[3,1,4,9,2,5,6,7,8] => [1,1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,5] => ([(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[3,1,4,9,2,5,8,6,7] => [1,1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,5] => ([(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[3,1,9,8,2,4,5,6,7] => [1,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,6] => ([(5,8),(6,8),(7,8)],9)
=> ? = 0 + 2
[7,1,4,5,2,3,8,9,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,0]
=> [7,1,1] => ([(0,7),(0,8),(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[7,1,4,9,2,3,5,6,8] => [1,1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0,0]
=> [7,2] => ([(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 0 + 2
[7,1,9,6,2,3,4,5,8] => [1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0]
=> [7,2] => ([(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 0 + 2
[7,1,8,9,2,3,4,5,6] => [1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0]
=> [7,1,1] => ([(0,7),(0,8),(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[3,1,8,9,7,2,4,5,6] => [1,1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,5,1] => ([(0,8),(1,8),(2,8),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[8,1,4,5,6,2,3,9,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [8,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 0 + 2
[8,1,4,9,7,2,3,5,6] => [1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
=> [8,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 0 + 2
[8,1,9,5,6,2,3,4,7] => [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [8,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 0 + 2
[8,1,9,7,6,2,3,4,5] => [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [8,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 0 + 2
[7,1,4,5,6,2,8,9,3] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,0]
=> [7,1,1] => ([(0,7),(0,8),(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[3,1,4,9,8,7,2,5,6] => [1,1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,5] => ([(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[3,1,9,8,6,7,2,4,5] => [1,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,6] => ([(5,8),(6,8),(7,8)],9)
=> ? = 0 + 2
[8,1,4,5,6,7,2,9,3] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [8,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 0 + 2
[3,1,4,5,6,7,9,2,8] => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [3,1,1,1,1,2] => ([(1,4),(1,5),(1,6),(1,7),(1,8),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 4 + 2
[3,1,4,9,6,7,8,2,5] => [1,1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,5] => ([(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[3,1,9,5,6,7,8,2,4] => [1,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,6] => ([(5,8),(6,8),(7,8)],9)
=> ? = 0 + 2
[2,9,1,3,4,5,6,7,8] => [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,7] => ([(6,8),(7,8)],9)
=> ? = 0 + 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: St001581
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001581: Graphs ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001581: Graphs ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,0,1,0]
=> [1,1] => ([(0,1)],2)
=> 2 = 0 + 2
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[1,3,2] => [1,0,1,1,0,0]
=> [1,2] => ([(1,2)],3)
=> 2 = 0 + 2
[2,1,3] => [1,1,0,0,1,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[2,3,1] => [1,1,0,1,0,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 2 + 2
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,3] => ([(2,3)],4)
=> 2 = 0 + 2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,3] => ([(2,3)],4)
=> 2 = 0 + 2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 3 + 2
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 2 = 0 + 2
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 2 = 0 + 2
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 2 = 0 + 2
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 2 = 0 + 2
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 2 = 0 + 2
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 2 = 0 + 2
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[4,5,6,7,8,3,2,1] => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 2
[4,5,6,7,3,8,2,1] => [1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 2
[4,3,5,6,7,8,2,1] => [1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
=> [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 2
[3,4,5,6,7,8,2,1] => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [3,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 + 2
[4,5,6,7,8,2,3,1] => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 2
[3,4,5,6,7,2,8,1] => [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [3,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 + 2
[3,4,5,6,2,7,8,1] => [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> [3,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 + 2
[3,4,5,2,6,7,8,1] => [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [3,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 + 2
[3,4,2,5,6,7,8,1] => [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 + 2
[2,3,4,5,6,7,8,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 + 2
[4,5,6,7,8,3,1,2] => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 2
[4,5,6,7,3,8,1,2] => [1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 2
[4,5,6,3,7,8,1,2] => [1,1,1,1,0,1,0,1,0,0,1,0,1,0,0,0]
=> [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 2
[4,5,3,6,7,8,1,2] => [1,1,1,1,0,1,0,0,1,0,1,0,1,0,0,0]
=> [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 2
[4,3,5,6,7,8,1,2] => [1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
=> [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 2
[3,4,5,6,7,8,1,2] => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [3,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 + 2
[4,5,6,7,8,2,1,3] => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 2
[4,5,6,7,8,1,2,3] => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 2
[4,5,3,6,2,7,1,8] => [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 2
[4,3,5,6,2,7,1,8] => [1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
=> [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 2
[4,5,2,3,6,7,1,8] => [1,1,1,1,0,1,0,0,0,1,0,1,0,0,1,0]
=> [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 2
[4,3,5,6,7,1,2,8] => [1,1,1,1,0,0,1,0,1,0,1,0,0,0,1,0]
=> [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 2
[4,3,5,2,1,6,7,8] => [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
=> [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 2
[4,3,5,1,2,6,7,8] => [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
=> [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 2
[3,5,6,7,8,4,2,1] => [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> [3,2,1,1,1] => ([(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 2
[3,4,6,7,8,5,2,1] => [1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,0]
=> [3,1,2,1,1] => ([(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 2
[3,4,5,7,8,6,2,1] => [1,1,1,0,1,0,1,0,1,1,0,1,0,0,0,0]
=> [3,1,1,2,1] => ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 2
[3,4,5,6,8,7,2,1] => [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [3,1,1,1,2] => ([(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 2
[3,5,6,7,4,8,2,1] => [1,1,1,0,1,1,0,1,0,1,0,0,1,0,0,0]
=> [3,2,1,1,1] => ([(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 2
[3,4,6,7,5,8,2,1] => [1,1,1,0,1,0,1,1,0,1,0,0,1,0,0,0]
=> [3,1,2,1,1] => ([(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 2
[3,4,5,7,6,8,2,1] => [1,1,1,0,1,0,1,0,1,1,0,0,1,0,0,0]
=> [3,1,1,2,1] => ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 2
[3,5,6,4,7,8,2,1] => [1,1,1,0,1,1,0,1,0,0,1,0,1,0,0,0]
=> [3,2,1,1,1] => ([(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 2
[3,4,6,5,7,8,2,1] => [1,1,1,0,1,0,1,1,0,0,1,0,1,0,0,0]
=> [3,1,2,1,1] => ([(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 2
[3,5,4,6,7,8,2,1] => [1,1,1,0,1,1,0,0,1,0,1,0,1,0,0,0]
=> [3,2,1,1,1] => ([(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 2
[3,2,5,6,7,8,4,1] => [1,1,1,0,0,1,1,0,1,0,1,0,1,0,0,0]
=> [3,2,1,1,1] => ([(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 2
[3,4,5,2,7,8,6,1] => [1,1,1,0,1,0,1,0,0,1,1,0,1,0,0,0]
=> [3,1,1,2,1] => ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 2
[3,4,2,5,7,8,6,1] => [1,1,1,0,1,0,0,1,0,1,1,0,1,0,0,0]
=> [3,1,1,2,1] => ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 2
[3,2,4,5,7,8,6,1] => [1,1,1,0,0,1,0,1,0,1,1,0,1,0,0,0]
=> [3,1,1,2,1] => ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 2
[3,4,2,5,6,8,7,1] => [1,1,1,0,1,0,0,1,0,1,0,1,1,0,0,0]
=> [3,1,1,1,2] => ([(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 2
[3,2,4,5,6,8,7,1] => [1,1,1,0,0,1,0,1,0,1,0,1,1,0,0,0]
=> [3,1,1,1,2] => ([(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 2
[2,1,4,5,6,7,8,3] => [1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,2,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 + 2
[2,1,3,5,6,7,8,4] => [1,1,0,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,1,2,1,1,1] => ([(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 + 2
[2,1,3,4,6,7,8,5] => [1,1,0,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,1,1,2,1,1] => ([(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 + 2
[2,3,1,4,5,7,8,6] => [1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [2,1,1,1,2,1] => ([(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 + 2
[2,1,3,4,5,7,8,6] => [1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,1,1,1,2,1] => ([(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 + 2
[2,3,4,5,6,1,8,7] => [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [2,1,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 + 2
[2,3,4,5,1,6,8,7] => [1,1,0,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [2,1,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 + 2
[2,3,1,4,5,6,8,7] => [1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 + 2
[2,1,3,4,5,6,8,7] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 + 2
[3,1,4,6,7,8,2,5] => [1,1,1,0,0,1,0,1,1,0,1,0,1,0,0,0]
=> [3,1,2,1,1] => ([(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 2
Description
The achromatic number of a graph.
This is the maximal number of colours of a proper colouring, such that for any pair of colours there are two adjacent vertices with these colours.
Matching statistic: St000159
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000159: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 91%●distinct values known / distinct values provided: 67%
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000159: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 91%●distinct values known / distinct values provided: 67%
Values
[1,2] => [1,0,1,0]
=> [1]
=> 1 = 0 + 1
[1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> 2 = 1 + 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> 1 = 0 + 1
[2,1,3] => [1,1,0,0,1,0]
=> [2]
=> 1 = 0 + 1
[2,3,1] => [1,1,0,1,0,0]
=> [1]
=> 1 = 0 + 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 3 = 2 + 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 2 = 1 + 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 2 = 1 + 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 2 = 1 + 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1 = 0 + 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1 = 0 + 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 2 = 1 + 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 1 = 0 + 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> 2 = 1 + 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> 2 = 1 + 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 1 = 0 + 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 1 = 0 + 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 1 = 0 + 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2]
=> 1 = 0 + 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 1 = 0 + 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [2]
=> 1 = 0 + 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> 1 = 0 + 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> 1 = 0 + 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 4 = 3 + 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 3 = 2 + 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 3 = 2 + 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 3 = 2 + 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 2 = 1 + 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 2 = 1 + 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 3 = 2 + 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 2 = 1 + 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 3 = 2 + 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 3 = 2 + 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 2 = 1 + 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 2 = 1 + 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 2 = 1 + 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 2 = 1 + 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 2 = 1 + 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 2 = 1 + 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 2 = 1 + 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 2 = 1 + 1
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 1 = 0 + 1
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 1 = 0 + 1
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 1 = 0 + 1
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 1 = 0 + 1
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 1 = 0 + 1
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 1 = 0 + 1
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 3 = 2 + 1
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 2 = 1 + 1
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 2 = 1 + 1
[1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> ? = 4 + 1
[1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> ? = 3 + 1
[1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> ? = 3 + 1
[1,2,3,5,6,4] => [1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> ? = 3 + 1
[1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> ? = 3 + 1
[1,2,4,3,6,5] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> ? = 2 + 1
[1,2,4,5,3,6] => [1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> ? = 3 + 1
[1,3,2,4,5,6] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> ? = 3 + 1
[1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> ? = 2 + 1
[1,3,2,5,4,6] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> ? = 2 + 1
[1,3,4,2,5,6] => [1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> ? = 3 + 1
[2,1,3,4,5,6] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> ? = 3 + 1
[2,1,3,4,6,5] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> ? = 2 + 1
[2,1,3,5,4,6] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> ? = 2 + 1
[2,1,4,3,5,6] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> ? = 2 + 1
[2,3,1,4,5,6] => [1,1,0,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1]
=> ? = 3 + 1
[1,2,3,4,5,6,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1]
=> ? = 5 + 1
[1,2,3,7,4,5,6] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,2,1]
=> ? = 2 + 1
[1,2,3,7,4,6,5] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,2,1]
=> ? = 2 + 1
[1,2,3,7,5,4,6] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,2,1]
=> ? = 2 + 1
[1,2,3,7,5,6,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,2,1]
=> ? = 2 + 1
[1,2,3,7,6,4,5] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,2,1]
=> ? = 2 + 1
[1,2,3,7,6,5,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,2,1]
=> ? = 2 + 1
[1,2,4,7,3,5,6] => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,2,1]
=> ? = 2 + 1
[1,2,4,7,3,6,5] => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,2,1]
=> ? = 2 + 1
[1,2,4,7,5,3,6] => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,2,1]
=> ? = 2 + 1
[1,2,4,7,5,6,3] => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,2,1]
=> ? = 2 + 1
[1,2,4,7,6,3,5] => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,2,1]
=> ? = 2 + 1
[1,2,4,7,6,5,3] => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,2,1]
=> ? = 2 + 1
[1,2,5,7,3,4,6] => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,3,2,2,2,1]
=> ? = 2 + 1
[1,2,5,7,3,6,4] => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,3,2,2,2,1]
=> ? = 2 + 1
[1,2,5,7,4,3,6] => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,3,2,2,2,1]
=> ? = 2 + 1
[1,2,5,7,4,6,3] => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,3,2,2,2,1]
=> ? = 2 + 1
[1,2,5,7,6,3,4] => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,3,2,2,2,1]
=> ? = 2 + 1
[1,2,5,7,6,4,3] => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,3,2,2,2,1]
=> ? = 2 + 1
[1,3,2,7,4,5,6] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,1,1]
=> ? = 1 + 1
[1,3,2,7,4,6,5] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,1,1]
=> ? = 1 + 1
[1,3,2,7,5,4,6] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,1,1]
=> ? = 1 + 1
[1,3,2,7,5,6,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,1,1]
=> ? = 1 + 1
[1,3,2,7,6,4,5] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,1,1]
=> ? = 1 + 1
[1,3,2,7,6,5,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,1,1]
=> ? = 1 + 1
[1,3,4,7,2,5,6] => [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,1,1]
=> ? = 2 + 1
[1,3,4,7,2,6,5] => [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,1,1]
=> ? = 2 + 1
[1,3,4,7,5,2,6] => [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,1,1]
=> ? = 2 + 1
[1,3,4,7,5,6,2] => [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,1,1]
=> ? = 2 + 1
[1,3,4,7,6,2,5] => [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,1,1]
=> ? = 2 + 1
[1,3,4,7,6,5,2] => [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,1,1]
=> ? = 2 + 1
[1,4,2,5,7,3,6] => [1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [4,4,3,1,1,1]
=> ? = 2 + 1
[1,4,2,5,7,6,3] => [1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [4,4,3,1,1,1]
=> ? = 2 + 1
[1,4,3,5,7,2,6] => [1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [4,4,3,1,1,1]
=> ? = 2 + 1
Description
The number of distinct parts of the integer partition.
This statistic is also the number of removeable cells of the partition, and the number of valleys of the Dyck path tracing the shape of the partition.
Matching statistic: St000147
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,0,1,0]
=> [1,2] => [2]
=> 2 = 0 + 2
[1,2,3] => [1,0,1,0,1,0]
=> [1,2,3] => [3]
=> 3 = 1 + 2
[1,3,2] => [1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> 2 = 0 + 2
[2,1,3] => [1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> 2 = 0 + 2
[2,3,1] => [1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> 2 = 0 + 2
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4]
=> 4 = 2 + 2
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [3,1]
=> 3 = 1 + 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,1]
=> 3 = 1 + 2
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,1]
=> 3 = 1 + 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [2,1,1]
=> 2 = 0 + 2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [2,1,1]
=> 2 = 0 + 2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [3,1]
=> 3 = 1 + 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> 2 = 0 + 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1]
=> 3 = 1 + 2
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [3,1]
=> 3 = 1 + 2
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,1,1]
=> 2 = 0 + 2
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,1,1]
=> 2 = 0 + 2
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,1,1]
=> 2 = 0 + 2
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,1,1]
=> 2 = 0 + 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,1,1]
=> 2 = 0 + 2
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,1,1]
=> 2 = 0 + 2
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,1,1]
=> 2 = 0 + 2
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,1,1]
=> 2 = 0 + 2
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [5]
=> 5 = 3 + 2
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [4,1]
=> 4 = 2 + 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [4,1]
=> 4 = 2 + 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [4,1]
=> 4 = 2 + 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,1,1]
=> 3 = 1 + 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,1,1]
=> 3 = 1 + 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [4,1]
=> 4 = 2 + 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [3,2]
=> 3 = 1 + 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [4,1]
=> 4 = 2 + 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [4,1]
=> 4 = 2 + 2
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [3,1,1]
=> 3 = 1 + 2
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [3,1,1]
=> 3 = 1 + 2
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [3,1,1]
=> 3 = 1 + 2
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [3,1,1]
=> 3 = 1 + 2
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [3,1,1]
=> 3 = 1 + 2
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [3,1,1]
=> 3 = 1 + 2
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [3,1,1]
=> 3 = 1 + 2
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [3,1,1]
=> 3 = 1 + 2
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,1,1,1]
=> 2 = 0 + 2
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,1,1,1]
=> 2 = 0 + 2
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,1,1,1]
=> 2 = 0 + 2
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,1,1,1]
=> 2 = 0 + 2
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,1,1,1]
=> 2 = 0 + 2
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,1,1,1]
=> 2 = 0 + 2
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [4,1]
=> 4 = 2 + 2
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [3,2]
=> 3 = 1 + 2
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [3,2]
=> 3 = 1 + 2
[6,7,5,8,4,3,2,1] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [8,7,4,5,3,6,2,1] => ?
=> ? = 1 + 2
[6,7,5,4,8,3,2,1] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> [8,6,4,5,3,7,2,1] => ?
=> ? = 1 + 2
[5,6,7,4,8,3,2,1] => [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
=> [8,6,3,4,5,7,2,1] => ?
=> ? = 2 + 2
[6,7,4,5,8,3,2,1] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> [8,6,4,5,3,7,2,1] => ?
=> ? = 1 + 2
[6,5,4,7,8,3,2,1] => [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [8,5,4,3,6,7,2,1] => ?
=> ? = 1 + 2
[6,4,5,7,8,3,2,1] => [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [8,5,4,3,6,7,2,1] => ?
=> ? = 1 + 2
[6,7,5,8,3,4,2,1] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [8,7,4,5,3,6,2,1] => ?
=> ? = 1 + 2
[6,7,5,4,3,8,2,1] => [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0]
=> [8,6,4,5,3,2,7,1] => ?
=> ? = 1 + 2
[5,6,7,4,3,8,2,1] => [1,1,1,1,1,0,1,0,1,0,0,0,1,0,0,0]
=> [8,6,3,4,5,2,7,1] => ?
=> ? = 2 + 2
[6,7,4,5,3,8,2,1] => [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0]
=> [8,6,4,5,3,2,7,1] => ?
=> ? = 1 + 2
[6,5,4,7,3,8,2,1] => [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0]
=> [8,5,4,3,6,2,7,1] => ?
=> ? = 1 + 2
[6,4,5,7,3,8,2,1] => [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0]
=> [8,5,4,3,6,2,7,1] => ?
=> ? = 1 + 2
[4,5,6,7,3,8,2,1] => [1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> [8,3,4,5,6,2,7,1] => ?
=> ? = 3 + 2
[6,7,5,8,4,2,3,1] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [8,7,4,5,3,6,2,1] => ?
=> ? = 1 + 2
[6,7,4,5,8,2,3,1] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> [8,6,4,5,3,7,2,1] => ?
=> ? = 1 + 2
[6,7,5,8,3,2,4,1] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [8,7,4,5,3,6,2,1] => ?
=> ? = 1 + 2
[6,7,5,8,2,3,4,1] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [8,7,4,5,3,6,2,1] => ?
=> ? = 1 + 2
[6,5,4,3,7,2,8,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> [7,5,4,3,2,6,8,1] => ?
=> ? = 1 + 2
[6,4,5,3,7,2,8,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> [7,5,4,3,2,6,8,1] => ?
=> ? = 1 + 2
[6,5,3,4,7,2,8,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> [7,5,4,3,2,6,8,1] => ?
=> ? = 1 + 2
[5,6,3,4,7,2,8,1] => [1,1,1,1,1,0,1,0,0,0,1,0,0,1,0,0]
=> [7,5,3,4,2,6,8,1] => ?
=> ? = 2 + 2
[6,4,3,5,7,2,8,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> [7,5,4,3,2,6,8,1] => ?
=> ? = 1 + 2
[6,3,4,5,7,2,8,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> [7,5,4,3,2,6,8,1] => ?
=> ? = 1 + 2
[5,4,3,6,7,2,8,1] => [1,1,1,1,1,0,0,0,1,0,1,0,0,1,0,0]
=> [7,4,3,2,5,6,8,1] => ?
=> ? = 2 + 2
[5,4,6,7,2,3,8,1] => [1,1,1,1,1,0,0,1,0,1,0,0,0,1,0,0]
=> [7,4,3,5,6,2,8,1] => ?
=> ? = 2 + 2
[5,4,6,3,2,7,8,1] => [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> [6,4,3,5,2,7,8,1] => ?
=> ? = 2 + 2
[5,4,3,6,2,7,8,1] => [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
=> [6,4,3,2,5,7,8,1] => ?
=> ? = 2 + 2
[5,4,6,2,3,7,8,1] => [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> [6,4,3,5,2,7,8,1] => ?
=> ? = 2 + 2
[6,7,5,8,4,3,1,2] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [8,7,4,5,3,6,2,1] => ?
=> ? = 1 + 2
[6,7,4,5,8,3,1,2] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> [8,6,4,5,3,7,2,1] => ?
=> ? = 1 + 2
[6,5,4,7,8,3,1,2] => [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [8,5,4,3,6,7,2,1] => ?
=> ? = 1 + 2
[6,7,5,8,3,4,1,2] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [8,7,4,5,3,6,2,1] => ?
=> ? = 1 + 2
[5,6,7,4,3,8,1,2] => [1,1,1,1,1,0,1,0,1,0,0,0,1,0,0,0]
=> [8,6,3,4,5,2,7,1] => ?
=> ? = 2 + 2
[6,7,4,5,3,8,1,2] => [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0]
=> [8,6,4,5,3,2,7,1] => ?
=> ? = 1 + 2
[6,5,4,7,3,8,1,2] => [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0]
=> [8,5,4,3,6,2,7,1] => ?
=> ? = 1 + 2
[5,6,4,7,3,8,1,2] => [1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0]
=> [8,5,3,4,6,2,7,1] => ?
=> ? = 2 + 2
[6,4,5,7,3,8,1,2] => [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0]
=> [8,5,4,3,6,2,7,1] => ?
=> ? = 1 + 2
[4,5,6,7,3,8,1,2] => [1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> [8,3,4,5,6,2,7,1] => ?
=> ? = 3 + 2
[5,4,6,3,7,8,1,2] => [1,1,1,1,1,0,0,1,0,0,1,0,1,0,0,0]
=> [8,4,3,5,2,6,7,1] => ?
=> ? = 2 + 2
[6,7,5,8,4,2,1,3] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [8,7,4,5,3,6,2,1] => ?
=> ? = 1 + 2
[6,7,5,4,8,2,1,3] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> [8,6,4,5,3,7,2,1] => ?
=> ? = 1 + 2
[5,6,7,4,8,2,1,3] => [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
=> [8,6,3,4,5,7,2,1] => ?
=> ? = 2 + 2
[6,7,5,8,4,1,2,3] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [8,7,4,5,3,6,2,1] => ?
=> ? = 1 + 2
[6,7,5,4,8,1,2,3] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> [8,6,4,5,3,7,2,1] => ?
=> ? = 1 + 2
[5,6,7,4,8,1,2,3] => [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
=> [8,6,3,4,5,7,2,1] => ?
=> ? = 2 + 2
[6,7,4,5,8,1,2,3] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> [8,6,4,5,3,7,2,1] => ?
=> ? = 1 + 2
[6,5,4,7,8,1,2,3] => [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [8,5,4,3,6,7,2,1] => ?
=> ? = 1 + 2
[6,4,5,7,8,1,2,3] => [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [8,5,4,3,6,7,2,1] => ?
=> ? = 1 + 2
[6,7,5,8,3,2,1,4] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [8,7,4,5,3,6,2,1] => ?
=> ? = 1 + 2
[6,7,5,8,2,3,1,4] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [8,7,4,5,3,6,2,1] => ?
=> ? = 1 + 2
Description
The largest part of an integer partition.
Matching statistic: St001918
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001918: Integer partitions ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001918: Integer partitions ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,0,1,0]
=> [2,1] => [2]
=> 1 = 0 + 1
[1,2,3] => [1,0,1,0,1,0]
=> [3,2,1] => [3]
=> 2 = 1 + 1
[1,3,2] => [1,0,1,1,0,0]
=> [2,3,1] => [2,1]
=> 1 = 0 + 1
[2,1,3] => [1,1,0,0,1,0]
=> [3,1,2] => [2,1]
=> 1 = 0 + 1
[2,3,1] => [1,1,0,1,0,0]
=> [2,1,3] => [2,1]
=> 1 = 0 + 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4]
=> 3 = 2 + 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,1]
=> 2 = 1 + 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [3,1]
=> 2 = 1 + 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [3,1]
=> 2 = 1 + 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,1,1]
=> 1 = 0 + 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,1,1]
=> 1 = 0 + 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [3,1]
=> 2 = 1 + 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> 1 = 0 + 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [3,1]
=> 2 = 1 + 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,1]
=> 2 = 1 + 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [2,1,1]
=> 1 = 0 + 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [2,1,1]
=> 1 = 0 + 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,1,1]
=> 1 = 0 + 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,1,1]
=> 1 = 0 + 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,1,1]
=> 1 = 0 + 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,1,1]
=> 1 = 0 + 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,1]
=> 1 = 0 + 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,1]
=> 1 = 0 + 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5]
=> 4 = 3 + 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [4,1]
=> 3 = 2 + 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [4,1]
=> 3 = 2 + 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [4,1]
=> 3 = 2 + 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,1,1]
=> 2 = 1 + 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,1,1]
=> 2 = 1 + 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [4,1]
=> 3 = 2 + 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [3,1,1]
=> 2 = 1 + 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [4,1]
=> 3 = 2 + 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [4,1]
=> 3 = 2 + 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,1,1]
=> 2 = 1 + 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,1,1]
=> 2 = 1 + 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [3,1,1]
=> 2 = 1 + 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [3,1,1]
=> 2 = 1 + 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [3,1,1]
=> 2 = 1 + 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [3,1,1]
=> 2 = 1 + 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,1,1]
=> 2 = 1 + 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,1,1]
=> 2 = 1 + 1
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 1 = 0 + 1
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 1 = 0 + 1
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 1 = 0 + 1
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 1 = 0 + 1
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 1 = 0 + 1
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 1 = 0 + 1
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [4,1]
=> 3 = 2 + 1
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [3,1,1]
=> 2 = 1 + 1
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [3,1,1]
=> 2 = 1 + 1
[6,5,4,3,7,8,2,1] => [1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0]
=> [6,5,1,2,3,4,7,8] => ?
=> ? = 1 + 1
[6,5,3,4,7,8,2,1] => [1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0]
=> [6,5,1,2,3,4,7,8] => ?
=> ? = 1 + 1
[6,3,4,5,7,8,2,1] => [1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0]
=> [6,5,1,2,3,4,7,8] => ?
=> ? = 1 + 1
[5,6,3,4,7,2,8,1] => [1,1,1,1,1,0,1,0,0,0,1,0,0,1,0,0]
=> [7,5,2,1,3,4,6,8] => ?
=> ? = 2 + 1
[5,4,3,6,7,2,8,1] => [1,1,1,1,1,0,0,0,1,0,1,0,0,1,0,0]
=> [7,5,4,1,2,3,6,8] => ?
=> ? = 2 + 1
[5,6,4,7,2,3,8,1] => [1,1,1,1,1,0,1,0,0,1,0,0,0,1,0,0]
=> [7,4,2,1,3,5,6,8] => ?
=> ? = 2 + 1
[5,4,6,3,2,7,8,1] => [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> [7,6,3,1,2,4,5,8] => ?
=> ? = 2 + 1
[5,4,3,6,2,7,8,1] => [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
=> [7,6,4,1,2,3,5,8] => ?
=> ? = 2 + 1
[5,6,4,2,3,7,8,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,1,0,0]
=> [7,6,2,1,3,4,5,8] => ?
=> ? = 2 + 1
[5,4,6,2,3,7,8,1] => [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> [7,6,3,1,2,4,5,8] => ?
=> ? = 2 + 1
[6,5,4,3,7,8,1,2] => [1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0]
=> [6,5,1,2,3,4,7,8] => ?
=> ? = 1 + 1
[5,6,4,3,7,8,1,2] => [1,1,1,1,1,0,1,0,0,0,1,0,1,0,0,0]
=> [6,5,2,1,3,4,7,8] => ?
=> ? = 2 + 1
[5,4,6,3,7,8,1,2] => [1,1,1,1,1,0,0,1,0,0,1,0,1,0,0,0]
=> [6,5,3,1,2,4,7,8] => ?
=> ? = 2 + 1
[5,6,3,4,7,8,1,2] => [1,1,1,1,1,0,1,0,0,0,1,0,1,0,0,0]
=> [6,5,2,1,3,4,7,8] => ?
=> ? = 2 + 1
[6,4,3,5,7,8,1,2] => [1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0]
=> [6,5,1,2,3,4,7,8] => ?
=> ? = 1 + 1
[6,3,4,5,7,8,1,2] => [1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0]
=> [6,5,1,2,3,4,7,8] => ?
=> ? = 1 + 1
[4,3,5,6,7,1,2,8] => [1,1,1,1,0,0,1,0,1,0,1,0,0,0,1,0]
=> [8,5,4,3,1,2,6,7] => ?
=> ? = 3 + 1
[3,5,7,6,8,4,2,1] => [1,1,1,0,1,1,0,1,1,0,0,1,0,0,0,0]
=> [5,3,4,2,6,1,7,8] => ?
=> ? = 2 + 1
[3,4,8,7,6,5,2,1] => [1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0]
=> [3,4,5,6,2,1,7,8] => ?
=> ? = 1 + 1
[5,4,3,7,8,6,2,1] => [1,1,1,1,1,0,0,0,1,1,0,1,0,0,0,0]
=> [5,4,6,1,2,3,7,8] => ?
=> ? = 1 + 1
[4,5,3,7,8,6,2,1] => [1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0]
=> [5,4,6,2,1,3,7,8] => ?
=> ? = 2 + 1
[4,3,5,7,8,6,2,1] => [1,1,1,1,0,0,1,0,1,1,0,1,0,0,0,0]
=> [5,4,6,3,1,2,7,8] => ?
=> ? = 2 + 1
[6,5,4,3,8,7,2,1] => [1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
=> [5,6,1,2,3,4,7,8] => ?
=> ? = 0 + 1
[5,4,3,6,8,7,2,1] => [1,1,1,1,1,0,0,0,1,0,1,1,0,0,0,0]
=> [5,6,4,1,2,3,7,8] => ?
=> ? = 1 + 1
[4,3,5,6,8,7,2,1] => [1,1,1,1,0,0,1,0,1,0,1,1,0,0,0,0]
=> [5,6,4,3,1,2,7,8] => ?
=> ? = 2 + 1
[4,7,6,5,3,8,2,1] => [1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0]
=> [6,2,3,4,1,5,7,8] => ?
=> ? = 1 + 1
[5,4,7,6,3,8,2,1] => [1,1,1,1,1,0,0,1,1,0,0,0,1,0,0,0]
=> [6,3,4,1,2,5,7,8] => ?
=> ? = 1 + 1
[3,5,7,6,4,8,2,1] => [1,1,1,0,1,1,0,1,1,0,0,0,1,0,0,0]
=> [6,3,4,2,5,1,7,8] => ?
=> ? = 2 + 1
[4,3,7,6,5,8,2,1] => [1,1,1,1,0,0,1,1,1,0,0,0,1,0,0,0]
=> [6,3,4,5,1,2,7,8] => ?
=> ? = 1 + 1
[3,4,7,6,5,8,2,1] => [1,1,1,0,1,0,1,1,1,0,0,0,1,0,0,0]
=> [6,3,4,5,2,1,7,8] => ?
=> ? = 2 + 1
[3,4,6,7,5,8,2,1] => [1,1,1,0,1,0,1,1,0,1,0,0,1,0,0,0]
=> [6,4,3,5,2,1,7,8] => ?
=> ? = 3 + 1
[3,5,4,7,6,8,2,1] => [1,1,1,0,1,1,0,0,1,1,0,0,1,0,0,0]
=> [6,4,5,2,3,1,7,8] => ?
=> ? = 2 + 1
[3,4,5,7,6,8,2,1] => [1,1,1,0,1,0,1,0,1,1,0,0,1,0,0,0]
=> [6,4,5,3,2,1,7,8] => ?
=> ? = 3 + 1
[3,5,6,4,7,8,2,1] => [1,1,1,0,1,1,0,1,0,0,1,0,1,0,0,0]
=> [6,5,3,2,4,1,7,8] => ?
=> ? = 3 + 1
[4,3,6,5,7,8,2,1] => [1,1,1,1,0,0,1,1,0,0,1,0,1,0,0,0]
=> [6,5,3,4,1,2,7,8] => ?
=> ? = 2 + 1
[3,5,4,6,7,8,2,1] => [1,1,1,0,1,1,0,0,1,0,1,0,1,0,0,0]
=> [6,5,4,2,3,1,7,8] => ?
=> ? = 3 + 1
[2,5,8,7,6,4,3,1] => [1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [3,4,5,2,6,7,1,8] => ?
=> ? = 1 + 1
[2,6,7,5,8,4,3,1] => [1,1,0,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [5,3,2,4,6,7,1,8] => ?
=> ? = 2 + 1
[2,5,4,8,7,6,3,1] => [1,1,0,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [4,5,6,2,3,7,1,8] => ?
=> ? = 1 + 1
[3,2,6,8,7,5,4,1] => [1,1,1,0,0,1,1,1,0,1,1,0,0,0,0,0]
=> [4,5,3,6,7,1,2,8] => ?
=> ? = 1 + 1
[3,2,5,7,8,6,4,1] => [1,1,1,0,0,1,1,0,1,1,0,1,0,0,0,0]
=> [5,4,6,3,7,1,2,8] => ?
=> ? = 2 + 1
[2,3,5,8,7,6,4,1] => [1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [4,5,6,3,7,2,1,8] => ?
=> ? = 2 + 1
[4,3,2,6,8,7,5,1] => [1,1,1,1,0,0,0,1,1,0,1,1,0,0,0,0]
=> [5,6,4,7,1,2,3,8] => ?
=> ? = 1 + 1
[4,3,5,2,8,7,6,1] => [1,1,1,1,0,0,1,0,0,1,1,1,0,0,0,0]
=> [5,6,7,3,1,2,4,8] => ?
=> ? = 1 + 1
[3,4,5,2,7,8,6,1] => [1,1,1,0,1,0,1,0,0,1,1,0,1,0,0,0]
=> [6,5,7,3,2,1,4,8] => ?
=> ? = 3 + 1
[3,2,5,4,7,8,6,1] => [1,1,1,0,0,1,1,0,0,1,1,0,1,0,0,0]
=> [6,5,7,3,4,1,2,8] => ?
=> ? = 2 + 1
[4,3,2,5,7,8,6,1] => [1,1,1,1,0,0,0,1,0,1,1,0,1,0,0,0]
=> [6,5,7,4,1,2,3,8] => ?
=> ? = 2 + 1
[3,4,2,5,8,7,6,1] => [1,1,1,0,1,0,0,1,0,1,1,1,0,0,0,0]
=> [5,6,7,4,2,1,3,8] => ?
=> ? = 2 + 1
[3,2,4,5,7,8,6,1] => [1,1,1,0,0,1,0,1,0,1,1,0,1,0,0,0]
=> [6,5,7,4,3,1,2,8] => ?
=> ? = 3 + 1
[5,4,3,6,2,8,7,1] => [1,1,1,1,1,0,0,0,1,0,0,1,1,0,0,0]
=> [6,7,4,1,2,3,5,8] => ?
=> ? = 1 + 1
Description
The degree of the cyclic sieving polynomial corresponding to an integer partition.
Let $\lambda$ be an integer partition of $n$ and let $N$ be the least common multiple of the parts of $\lambda$. Fix an arbitrary permutation $\pi$ of cycle type $\lambda$. Then $\pi$ induces a cyclic action of order $N$ on $\{1,\dots,n\}$.
The corresponding character can be identified with the cyclic sieving polynomial $C_\lambda(q)$ of this action, modulo $q^N-1$. Explicitly, it is
$$
\sum_{p\in\lambda} [p]_{q^{N/p}},
$$
where $[p]_q = 1+\dots+q^{p-1}$ is the $q$-integer.
This statistic records the degree of $C_\lambda(q)$. Equivalently, it equals
$$
\left(1 - \frac{1}{\lambda_1}\right) N,
$$
where $\lambda_1$ is the largest part of $\lambda$.
The statistic is undefined for the empty partition.
The following 185 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St001279The sum of the parts of an integer partition that are at least two. St001389The number of partitions of the same length below the given integer partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000157The number of descents of a standard tableau. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St000507The number of ascents of a standard tableau. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000053The number of valleys of the Dyck path. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000340The number of non-final maximal constant sub-paths of length greater than one. St000093The cardinality of a maximal independent set of vertices of a graph. St000378The diagonal inversion number of an integer partition. St000733The row containing the largest entry of a standard tableau. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St000015The number of peaks of a Dyck path. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000675The number of centered multitunnels of a Dyck path. St000098The chromatic number of a graph. St000996The number of exclusive left-to-right maxima of a permutation. St000678The number of up steps after the last double rise of a Dyck path. St000925The number of topologically connected components of a set partition. St000011The number of touch points (or returns) of a Dyck path. St000662The staircase size of the code of a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St000245The number of ascents of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000153The number of adjacent cycles of a permutation. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St000676The number of odd rises of a Dyck path. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000439The position of the first down step of a Dyck path. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000141The maximum drop size of a permutation. St000052The number of valleys of a Dyck path not on the x-axis. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000912The number of maximal antichains in a poset. St000071The number of maximal chains in a poset. St000007The number of saliances of the permutation. St000167The number of leaves of an ordered tree. St000068The number of minimal elements in a poset. St000528The height of a poset. St000069The number of maximal elements of a poset. St000024The number of double up and double down steps of a Dyck path. St000105The number of blocks in the set partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000211The rank of the set partition. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St000025The number of initial rises of a Dyck path. St000172The Grundy number of a graph. St001029The size of the core of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St000234The number of global ascents of a permutation. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001963The tree-depth of a graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St001330The hat guessing number of a graph. St000527The width of the poset. St000632The jump number of the poset. St000164The number of short pairs. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000470The number of runs in a permutation. St001489The maximum of the number of descents and the number of inverse descents. St000237The number of small exceedances. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St000546The number of global descents of a permutation. St000989The number of final rises of a permutation. St000054The first entry of the permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St001461The number of topologically connected components of the chord diagram of a permutation. St000809The reduced reflection length of the permutation. St000702The number of weak deficiencies of a permutation. St000740The last entry of a permutation. St000308The height of the tree associated to a permutation. St001427The number of descents of a signed permutation. St000654The first descent of a permutation. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St000843The decomposition number of a perfect matching. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000332The positive inversions of an alternating sign matrix. St000065The number of entries equal to -1 in an alternating sign matrix. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001298The number of repeated entries in the Lehmer code of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000314The number of left-to-right-maxima 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. St000542The number of left-to-right-minima of a permutation. St000991The number of right-to-left minima of a permutation. St000021The number of descents of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000155The number of exceedances (also excedences) of a permutation. St000061The number of nodes on the left branch of a binary tree. St000062The length of the longest increasing subsequence of the permutation. St000084The number of subtrees. St000239The number of small weak excedances. St000325The width of the tree associated to a permutation. St000443The number of long tunnels of a Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001180Number of indecomposable injective modules with projective dimension at most 1. St000080The rank of the poset. St000120The number of left tunnels of a Dyck path. St000133The "bounce" of a permutation. St000168The number of internal nodes of an ordered tree. St000216The absolute length of a permutation. St000238The number of indices that are not small weak excedances. St000316The number of non-left-to-right-maxima of a permutation. St000834The number of right outer peaks of a permutation. St000871The number of very big ascents of a permutation. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001083The number of boxed occurrences of 132 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000035The number of left outer peaks of a permutation. St000056The decomposition (or block) number of a permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000236The number of cyclical small weak excedances. St000286The number of connected components of the complement of a graph. St000389The number of runs of ones of odd length in a binary word. St000742The number of big ascents of a permutation after prepending zero. St000822The Hadwiger number of the graph. St000990The first ascent of a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St000203The number of external nodes of a binary tree. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001812The biclique partition number of a graph. St000711The number of big exceedences of a permutation. St001651The Frankl number of a lattice. St001621The number of atoms of a lattice. St001960The number of descents of a permutation minus one if its first entry is not one. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001152The number of pairs with even minimum in a perfect matching. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000181The number of connected components of the Hasse diagram for the poset. St000619The number of cyclic descents of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001845The number of join irreducibles minus the rank of a lattice. St001864The number of excedances of a signed permutation. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001645The pebbling number of a connected graph. St001626The number of maximal proper sublattices of a lattice. St001875The number of simple modules with projective dimension at most 1. St000942The number of critical left to right maxima of the parking functions. St001712The number of natural descents of a standard Young tableau. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001896The number of right descents of a signed permutations. St001935The number of ascents in a parking function. St001946The number of descents in a parking function. St000710The number of big deficiencies of a permutation. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001462The number of factors of a standard tableaux under concatenation. St001773The number of minimal elements in Bruhat order not less than the signed permutation.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!