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Your data matches 73 different statistics following compositions of up to 3 maps.
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Matching statistic: St000628
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000628: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00109: Permutations —descent word⟶ Binary words
St000628: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => 0 => 0
[2,1] => [1,2] => 0 => 0
[1,2,3] => [1,2,3] => 00 => 0
[1,3,2] => [1,3,2] => 01 => 1
[2,1,3] => [1,3,2] => 01 => 1
[2,3,1] => [1,2,3] => 00 => 0
[3,1,2] => [1,2,3] => 00 => 0
[3,2,1] => [1,2,3] => 00 => 0
[1,2,3,4] => [1,2,3,4] => 000 => 0
[1,2,4,3] => [1,2,4,3] => 001 => 1
[1,3,2,4] => [1,3,2,4] => 010 => 1
[1,3,4,2] => [1,3,4,2] => 001 => 1
[1,4,2,3] => [1,4,2,3] => 010 => 1
[1,4,3,2] => [1,4,2,3] => 010 => 1
[2,1,3,4] => [1,3,4,2] => 001 => 1
[2,1,4,3] => [1,4,2,3] => 010 => 1
[2,3,1,4] => [1,4,2,3] => 010 => 1
[2,3,4,1] => [1,2,3,4] => 000 => 0
[2,4,1,3] => [1,3,2,4] => 010 => 1
[2,4,3,1] => [1,2,4,3] => 001 => 1
[3,1,2,4] => [1,2,4,3] => 001 => 1
[3,1,4,2] => [1,4,2,3] => 010 => 1
[3,2,1,4] => [1,4,2,3] => 010 => 1
[3,2,4,1] => [1,2,4,3] => 001 => 1
[3,4,1,2] => [1,2,3,4] => 000 => 0
[3,4,2,1] => [1,2,3,4] => 000 => 0
[4,1,2,3] => [1,2,3,4] => 000 => 0
[4,1,3,2] => [1,3,2,4] => 010 => 1
[4,2,1,3] => [1,3,2,4] => 010 => 1
[4,2,3,1] => [1,2,3,4] => 000 => 0
[4,3,1,2] => [1,2,3,4] => 000 => 0
[4,3,2,1] => [1,2,3,4] => 000 => 0
[1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,2,3,5,4] => [1,2,3,5,4] => 0001 => 1
[1,2,4,3,5] => [1,2,4,3,5] => 0010 => 1
[1,2,4,5,3] => [1,2,4,5,3] => 0001 => 1
[1,2,5,3,4] => [1,2,5,3,4] => 0010 => 1
[1,2,5,4,3] => [1,2,5,3,4] => 0010 => 1
[1,3,2,4,5] => [1,3,2,4,5] => 0100 => 1
[1,3,2,5,4] => [1,3,2,5,4] => 0101 => 1
[1,3,4,2,5] => [1,3,4,2,5] => 0010 => 1
[1,3,4,5,2] => [1,3,4,5,2] => 0001 => 1
[1,3,5,2,4] => [1,3,5,2,4] => 0010 => 1
[1,3,5,4,2] => [1,3,5,2,4] => 0010 => 1
[1,4,2,3,5] => [1,4,2,3,5] => 0100 => 1
[1,4,2,5,3] => [1,4,2,5,3] => 0101 => 1
[1,4,3,2,5] => [1,4,2,5,3] => 0101 => 1
[1,4,3,5,2] => [1,4,2,3,5] => 0100 => 1
[1,4,5,2,3] => [1,4,5,2,3] => 0010 => 1
[1,4,5,3,2] => [1,4,5,2,3] => 0010 => 1
Description
The balance of a binary word.
The balance of a word is the smallest number $q$ such that the word is $q$-balanced [1].
A binary word $w$ is $q$-balanced if for any two factors $u$, $v$ of $w$ of the same length, the difference between the number of ones in $u$ and $v$ is at most $q$.
Matching statistic: St000485
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000485: Permutations ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 67%
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000485: Permutations ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 67%
Values
[1,2] => [1,2] => [1,2] => [1,2] => 1 = 0 + 1
[2,1] => [1,2] => [1,2] => [1,2] => 1 = 0 + 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,3,2] => [1,3,2] => [3,1,2] => [3,2,1] => 2 = 1 + 1
[2,1,3] => [1,3,2] => [3,1,2] => [3,2,1] => 2 = 1 + 1
[2,3,1] => [1,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[3,1,2] => [1,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [4,1,2,3] => [4,2,3,1] => 2 = 1 + 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[1,3,4,2] => [1,3,4,2] => [3,1,2,4] => [3,2,1,4] => 2 = 1 + 1
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => [1,4,3,2] => 2 = 1 + 1
[1,4,3,2] => [1,4,2,3] => [1,4,2,3] => [1,4,3,2] => 2 = 1 + 1
[2,1,3,4] => [1,3,4,2] => [3,1,2,4] => [3,2,1,4] => 2 = 1 + 1
[2,1,4,3] => [1,4,2,3] => [1,4,2,3] => [1,4,3,2] => 2 = 1 + 1
[2,3,1,4] => [1,4,2,3] => [1,4,2,3] => [1,4,3,2] => 2 = 1 + 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[2,4,1,3] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => [4,2,3,1] => 2 = 1 + 1
[3,1,2,4] => [1,2,4,3] => [4,1,2,3] => [4,2,3,1] => 2 = 1 + 1
[3,1,4,2] => [1,4,2,3] => [1,4,2,3] => [1,4,3,2] => 2 = 1 + 1
[3,2,1,4] => [1,4,2,3] => [1,4,2,3] => [1,4,3,2] => 2 = 1 + 1
[3,2,4,1] => [1,2,4,3] => [4,1,2,3] => [4,2,3,1] => 2 = 1 + 1
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[4,1,3,2] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[4,2,1,3] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => [5,2,3,4,1] => 2 = 1 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,4,2,3,5] => [1,4,3,2,5] => 2 = 1 + 1
[1,2,4,5,3] => [1,2,4,5,3] => [4,1,2,3,5] => [4,2,3,1,5] => 2 = 1 + 1
[1,2,5,3,4] => [1,2,5,3,4] => [1,5,2,3,4] => [1,5,3,4,2] => 2 = 1 + 1
[1,2,5,4,3] => [1,2,5,3,4] => [1,5,2,3,4] => [1,5,3,4,2] => 2 = 1 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,4,2,5] => [1,4,3,2,5] => 2 = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [3,1,5,2,4] => [4,2,5,1,3] => 2 = 1 + 1
[1,3,4,2,5] => [1,3,4,2,5] => [1,3,2,4,5] => [1,3,2,4,5] => 2 = 1 + 1
[1,3,4,5,2] => [1,3,4,5,2] => [3,1,2,4,5] => [3,2,1,4,5] => 2 = 1 + 1
[1,3,5,2,4] => [1,3,5,2,4] => [3,5,1,2,4] => [4,5,3,1,2] => 2 = 1 + 1
[1,3,5,4,2] => [1,3,5,2,4] => [3,5,1,2,4] => [4,5,3,1,2] => 2 = 1 + 1
[1,4,2,3,5] => [1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 2 = 1 + 1
[1,4,2,5,3] => [1,4,2,5,3] => [4,1,5,2,3] => [5,2,4,3,1] => 2 = 1 + 1
[1,4,3,2,5] => [1,4,2,5,3] => [4,1,5,2,3] => [5,2,4,3,1] => 2 = 1 + 1
[1,4,3,5,2] => [1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 2 = 1 + 1
[1,4,5,2,3] => [1,4,5,2,3] => [4,5,1,2,3] => [5,4,3,2,1] => 2 = 1 + 1
[1,4,5,3,2] => [1,4,5,2,3] => [4,5,1,2,3] => [5,4,3,2,1] => 2 = 1 + 1
[1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => [7,2,3,4,5,6,1] => ? = 1 + 1
[1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [1,6,2,3,4,5,7] => [1,6,3,4,5,2,7] => ? = 1 + 1
[1,2,3,4,6,7,5] => [1,2,3,4,6,7,5] => [6,1,2,3,4,5,7] => [6,2,3,4,5,1,7] => ? = 1 + 1
[1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => [1,7,2,3,4,5,6] => [1,7,3,4,5,6,2] => ? = 1 + 1
[1,2,3,4,7,6,5] => [1,2,3,4,7,5,6] => [1,7,2,3,4,5,6] => [1,7,3,4,5,6,2] => ? = 1 + 1
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [5,1,7,2,3,4,6] => [6,2,7,4,5,1,3] => ? = 2 + 1
[1,2,3,5,6,4,7] => [1,2,3,5,6,4,7] => [1,5,2,3,4,6,7] => [1,5,3,4,2,6,7] => ? = 1 + 1
[1,2,3,5,6,7,4] => [1,2,3,5,6,7,4] => [5,1,2,3,4,6,7] => [5,2,3,4,1,6,7] => ? = 1 + 1
[1,2,3,5,7,4,6] => [1,2,3,5,7,4,6] => [5,7,1,2,3,4,6] => [6,7,3,4,5,1,2] => ? = 1 + 1
[1,2,3,5,7,6,4] => [1,2,3,5,7,4,6] => [5,7,1,2,3,4,6] => [6,7,3,4,5,1,2] => ? = 1 + 1
[1,2,3,6,4,7,5] => [1,2,3,6,4,7,5] => [6,1,7,2,3,4,5] => [7,2,6,4,5,3,1] => ? = 2 + 1
[1,2,3,6,5,4,7] => [1,2,3,6,4,7,5] => [6,1,7,2,3,4,5] => [7,2,6,4,5,3,1] => ? = 2 + 1
[1,2,3,6,7,4,5] => [1,2,3,6,7,4,5] => [6,7,1,2,3,4,5] => [7,6,3,4,5,2,1] => ? = 1 + 1
[1,2,3,6,7,5,4] => [1,2,3,6,7,4,5] => [6,7,1,2,3,4,5] => [7,6,3,4,5,2,1] => ? = 1 + 1
[1,2,3,7,4,6,5] => [1,2,3,7,4,6,5] => [7,1,6,2,3,4,5] => [7,2,6,4,5,3,1] => ? = 2 + 1
[1,2,3,7,5,4,6] => [1,2,3,7,4,6,5] => [7,1,6,2,3,4,5] => [7,2,6,4,5,3,1] => ? = 2 + 1
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [4,1,2,7,3,5,6] => [5,2,3,7,1,6,4] => ? = 1 + 1
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [1,4,2,6,3,5,7] => [1,5,3,6,2,4,7] => ? = 1 + 1
[1,2,4,3,6,7,5] => [1,2,4,3,6,7,5] => [4,1,2,6,3,5,7] => [5,2,3,6,1,4,7] => ? = 1 + 1
[1,2,4,3,7,5,6] => [1,2,4,3,7,5,6] => [1,4,2,7,3,5,6] => [1,5,3,7,2,6,4] => ? = 1 + 1
[1,2,4,3,7,6,5] => [1,2,4,3,7,5,6] => [1,4,2,7,3,5,6] => [1,5,3,7,2,6,4] => ? = 1 + 1
[1,2,4,5,3,7,6] => [1,2,4,5,3,7,6] => [4,1,7,2,3,5,6] => [5,2,7,4,1,6,3] => ? = 2 + 1
[1,2,4,5,6,7,3] => [1,2,4,5,6,7,3] => [4,1,2,3,5,6,7] => [4,2,3,1,5,6,7] => ? = 1 + 1
[1,2,4,5,7,3,6] => [1,2,4,5,7,3,6] => [4,7,1,2,3,5,6] => [5,7,3,4,1,6,2] => ? = 1 + 1
[1,2,4,5,7,6,3] => [1,2,4,5,7,3,6] => [4,7,1,2,3,5,6] => [5,7,3,4,1,6,2] => ? = 1 + 1
[1,2,4,6,3,5,7] => [1,2,4,6,3,5,7] => [1,4,6,2,3,5,7] => [1,5,6,4,2,3,7] => ? = 1 + 1
[1,2,4,6,3,7,5] => [1,2,4,6,3,7,5] => [4,1,6,2,3,5,7] => [5,2,6,4,1,3,7] => ? = 2 + 1
[1,2,4,6,5,3,7] => [1,2,4,6,3,7,5] => [4,1,6,2,3,5,7] => [5,2,6,4,1,3,7] => ? = 2 + 1
[1,2,4,6,5,7,3] => [1,2,4,6,3,5,7] => [1,4,6,2,3,5,7] => [1,5,6,4,2,3,7] => ? = 1 + 1
[1,2,4,6,7,3,5] => [1,2,4,6,7,3,5] => [4,6,1,2,3,5,7] => [5,6,3,4,1,2,7] => ? = 1 + 1
[1,2,4,6,7,5,3] => [1,2,4,6,7,3,5] => [4,6,1,2,3,5,7] => [5,6,3,4,1,2,7] => ? = 1 + 1
[1,2,4,7,3,5,6] => [1,2,4,7,3,5,6] => [1,4,7,2,3,5,6] => [1,5,7,4,2,6,3] => ? = 1 + 1
[1,2,4,7,3,6,5] => [1,2,4,7,3,6,5] => [7,4,6,1,2,3,5] => [7,6,5,4,3,2,1] => ? = 2 + 1
[1,2,4,7,5,3,6] => [1,2,4,7,3,6,5] => [7,4,6,1,2,3,5] => [7,6,5,4,3,2,1] => ? = 2 + 1
[1,2,4,7,5,6,3] => [1,2,4,7,3,5,6] => [1,4,7,2,3,5,6] => [1,5,7,4,2,6,3] => ? = 1 + 1
[1,2,4,7,6,3,5] => [1,2,4,7,3,5,6] => [1,4,7,2,3,5,6] => [1,5,7,4,2,6,3] => ? = 1 + 1
[1,2,4,7,6,5,3] => [1,2,4,7,3,5,6] => [1,4,7,2,3,5,6] => [1,5,7,4,2,6,3] => ? = 1 + 1
[1,2,5,3,4,7,6] => [1,2,5,3,4,7,6] => [5,1,2,7,3,4,6] => [6,2,3,7,5,1,4] => ? = 1 + 1
[1,2,5,3,6,4,7] => [1,2,5,3,6,4,7] => [1,5,2,6,3,4,7] => [1,6,3,5,4,2,7] => ? = 1 + 1
[1,2,5,3,6,7,4] => [1,2,5,3,6,7,4] => [5,1,2,6,3,4,7] => [6,2,3,5,4,1,7] => ? = 1 + 1
[1,2,5,3,7,4,6] => [1,2,5,3,7,4,6] => [1,5,2,7,3,4,6] => [1,6,3,7,5,2,4] => ? = 1 + 1
[1,2,5,3,7,6,4] => [1,2,5,3,7,4,6] => [1,5,2,7,3,4,6] => [1,6,3,7,5,2,4] => ? = 1 + 1
[1,2,5,4,3,6,7] => [1,2,5,3,6,7,4] => [5,1,2,6,3,4,7] => [6,2,3,5,4,1,7] => ? = 1 + 1
[1,2,5,4,3,7,6] => [1,2,5,3,7,4,6] => [1,5,2,7,3,4,6] => [1,6,3,7,5,2,4] => ? = 1 + 1
[1,2,5,4,6,3,7] => [1,2,5,3,7,4,6] => [1,5,2,7,3,4,6] => [1,6,3,7,5,2,4] => ? = 1 + 1
[1,2,5,4,7,3,6] => [1,2,5,3,6,4,7] => [1,5,2,6,3,4,7] => [1,6,3,5,4,2,7] => ? = 1 + 1
[1,2,5,4,7,6,3] => [1,2,5,3,4,7,6] => [5,1,2,7,3,4,6] => [6,2,3,7,5,1,4] => ? = 1 + 1
[1,2,5,6,3,4,7] => [1,2,5,6,3,4,7] => [1,5,6,2,3,4,7] => [1,6,5,4,3,2,7] => ? = 1 + 1
[1,2,5,6,3,7,4] => [1,2,5,6,3,7,4] => [5,1,6,2,3,4,7] => [6,2,5,4,3,1,7] => ? = 2 + 1
[1,2,5,6,4,3,7] => [1,2,5,6,3,7,4] => [5,1,6,2,3,4,7] => [6,2,5,4,3,1,7] => ? = 2 + 1
Description
The length of the longest cycle of a permutation.
Matching statistic: St000455
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 33%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 33%
Values
[1,2] => [1,2] => [2] => ([],2)
=> ? = 0 - 1
[2,1] => [1,2] => [2] => ([],2)
=> ? = 0 - 1
[1,2,3] => [1,2,3] => [3] => ([],3)
=> ? = 0 - 1
[1,3,2] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[2,1,3] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[2,3,1] => [1,2,3] => [3] => ([],3)
=> ? = 0 - 1
[3,1,2] => [1,2,3] => [3] => ([],3)
=> ? = 0 - 1
[3,2,1] => [1,2,3] => [3] => ([],3)
=> ? = 0 - 1
[1,2,3,4] => [1,2,3,4] => [4] => ([],4)
=> ? = 0 - 1
[1,2,4,3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,3,2,4] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,3,4,2] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,4,2,3] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,4,3,2] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[2,1,3,4] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[2,1,4,3] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[2,3,1,4] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[2,3,4,1] => [1,2,3,4] => [4] => ([],4)
=> ? = 0 - 1
[2,4,1,3] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[2,4,3,1] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[3,1,2,4] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[3,1,4,2] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[3,2,1,4] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[3,2,4,1] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[3,4,1,2] => [1,2,3,4] => [4] => ([],4)
=> ? = 0 - 1
[3,4,2,1] => [1,2,3,4] => [4] => ([],4)
=> ? = 0 - 1
[4,1,2,3] => [1,2,3,4] => [4] => ([],4)
=> ? = 0 - 1
[4,1,3,2] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[4,2,1,3] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[4,2,3,1] => [1,2,3,4] => [4] => ([],4)
=> ? = 0 - 1
[4,3,1,2] => [1,2,3,4] => [4] => ([],4)
=> ? = 0 - 1
[4,3,2,1] => [1,2,3,4] => [4] => ([],4)
=> ? = 0 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 0 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,2,4,5,3] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,2,5,3,4] => [1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,2,5,4,3] => [1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[1,3,4,2,5] => [1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,3,4,5,2] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,3,5,2,4] => [1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,3,5,4,2] => [1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,4,2,3,5] => [1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,4,2,5,3] => [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[1,4,3,2,5] => [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[1,4,3,5,2] => [1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,4,5,2,3] => [1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,4,5,3,2] => [1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,5,2,3,4] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,5,2,4,3] => [1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[1,5,3,2,4] => [1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[1,5,3,4,2] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,5,4,2,3] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,5,4,3,2] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 0 = 1 - 1
[2,1,3,4,5] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[2,1,3,5,4] => [1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[2,1,4,3,5] => [1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 0 = 1 - 1
[2,1,4,5,3] => [1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[2,1,5,3,4] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 0 = 1 - 1
[2,1,5,4,3] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 0 = 1 - 1
[2,3,1,4,5] => [1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[2,3,1,5,4] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 0 = 1 - 1
[2,3,4,1,5] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 0 = 1 - 1
[2,3,4,5,1] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 0 - 1
[2,3,5,1,4] => [1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 0 = 1 - 1
[2,3,5,4,1] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[2,4,1,3,5] => [1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[2,4,1,5,3] => [1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[2,4,3,1,5] => [1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[2,4,3,5,1] => [1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[2,4,5,1,3] => [1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> 0 = 1 - 1
[2,5,1,4,3] => [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[2,5,3,1,4] => [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[2,5,4,1,3] => [1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[3,1,4,2,5] => [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[3,1,5,2,4] => [1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[3,2,4,1,5] => [1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[3,2,5,1,4] => [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[3,4,5,1,2] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 0 - 1
[3,4,5,2,1] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 0 - 1
[4,1,3,2,5] => [1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[4,2,5,1,3] => [1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[4,5,1,2,3] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 0 - 1
[4,5,2,3,1] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 0 - 1
[4,5,3,1,2] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 0 - 1
[4,5,3,2,1] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 0 - 1
[5,1,2,3,4] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 0 - 1
[5,2,3,4,1] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 0 - 1
[5,3,4,1,2] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 0 - 1
[5,3,4,2,1] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 0 - 1
[5,4,1,2,3] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 0 - 1
[5,4,2,3,1] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 0 - 1
[5,4,3,1,2] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 0 - 1
[5,4,3,2,1] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 0 - 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [6] => ([],6)
=> ? = 0 - 1
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
[1,2,5,3,6,4] => [1,2,5,3,6,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
[1,2,5,4,3,6] => [1,2,5,3,6,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Matching statistic: St001729
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001729: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 67%
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001729: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 67%
Values
[1,2] => [1,2] => [[1,2]]
=> [1,2] => 0
[2,1] => [1,2] => [[1,2]]
=> [1,2] => 0
[1,2,3] => [1,2,3] => [[1,2,3]]
=> [1,2,3] => 0
[1,3,2] => [1,3,2] => [[1,2],[3]]
=> [3,1,2] => 1
[2,1,3] => [1,3,2] => [[1,2],[3]]
=> [3,1,2] => 1
[2,3,1] => [1,2,3] => [[1,2,3]]
=> [1,2,3] => 0
[3,1,2] => [1,2,3] => [[1,2,3]]
=> [1,2,3] => 0
[3,2,1] => [1,2,3] => [[1,2,3]]
=> [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [[1,2,3],[4]]
=> [4,1,2,3] => 1
[1,3,2,4] => [1,3,2,4] => [[1,2,4],[3]]
=> [3,1,2,4] => 1
[1,3,4,2] => [1,3,4,2] => [[1,2,4],[3]]
=> [3,1,2,4] => 1
[1,4,2,3] => [1,4,2,3] => [[1,2,3],[4]]
=> [4,1,2,3] => 1
[1,4,3,2] => [1,4,2,3] => [[1,2,3],[4]]
=> [4,1,2,3] => 1
[2,1,3,4] => [1,3,4,2] => [[1,2,4],[3]]
=> [3,1,2,4] => 1
[2,1,4,3] => [1,4,2,3] => [[1,2,3],[4]]
=> [4,1,2,3] => 1
[2,3,1,4] => [1,4,2,3] => [[1,2,3],[4]]
=> [4,1,2,3] => 1
[2,3,4,1] => [1,2,3,4] => [[1,2,3,4]]
=> [1,2,3,4] => 0
[2,4,1,3] => [1,3,2,4] => [[1,2,4],[3]]
=> [3,1,2,4] => 1
[2,4,3,1] => [1,2,4,3] => [[1,2,3],[4]]
=> [4,1,2,3] => 1
[3,1,2,4] => [1,2,4,3] => [[1,2,3],[4]]
=> [4,1,2,3] => 1
[3,1,4,2] => [1,4,2,3] => [[1,2,3],[4]]
=> [4,1,2,3] => 1
[3,2,1,4] => [1,4,2,3] => [[1,2,3],[4]]
=> [4,1,2,3] => 1
[3,2,4,1] => [1,2,4,3] => [[1,2,3],[4]]
=> [4,1,2,3] => 1
[3,4,1,2] => [1,2,3,4] => [[1,2,3,4]]
=> [1,2,3,4] => 0
[3,4,2,1] => [1,2,3,4] => [[1,2,3,4]]
=> [1,2,3,4] => 0
[4,1,2,3] => [1,2,3,4] => [[1,2,3,4]]
=> [1,2,3,4] => 0
[4,1,3,2] => [1,3,2,4] => [[1,2,4],[3]]
=> [3,1,2,4] => 1
[4,2,1,3] => [1,3,2,4] => [[1,2,4],[3]]
=> [3,1,2,4] => 1
[4,2,3,1] => [1,2,3,4] => [[1,2,3,4]]
=> [1,2,3,4] => 0
[4,3,1,2] => [1,2,3,4] => [[1,2,3,4]]
=> [1,2,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [[1,2,3,4]]
=> [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1
[1,2,4,3,5] => [1,2,4,3,5] => [[1,2,3,5],[4]]
=> [4,1,2,3,5] => 1
[1,2,4,5,3] => [1,2,4,5,3] => [[1,2,3,5],[4]]
=> [4,1,2,3,5] => 1
[1,2,5,3,4] => [1,2,5,3,4] => [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1
[1,2,5,4,3] => [1,2,5,3,4] => [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1
[1,3,2,4,5] => [1,3,2,4,5] => [[1,2,4,5],[3]]
=> [3,1,2,4,5] => 1
[1,3,2,5,4] => [1,3,2,5,4] => [[1,2,4],[3,5]]
=> [3,5,1,2,4] => 1
[1,3,4,2,5] => [1,3,4,2,5] => [[1,2,4,5],[3]]
=> [3,1,2,4,5] => 1
[1,3,4,5,2] => [1,3,4,5,2] => [[1,2,4,5],[3]]
=> [3,1,2,4,5] => 1
[1,3,5,2,4] => [1,3,5,2,4] => [[1,2,4],[3,5]]
=> [3,5,1,2,4] => 1
[1,3,5,4,2] => [1,3,5,2,4] => [[1,2,4],[3,5]]
=> [3,5,1,2,4] => 1
[1,4,2,3,5] => [1,4,2,3,5] => [[1,2,3,5],[4]]
=> [4,1,2,3,5] => 1
[1,4,2,5,3] => [1,4,2,5,3] => [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 1
[1,4,3,2,5] => [1,4,2,5,3] => [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 1
[1,4,3,5,2] => [1,4,2,3,5] => [[1,2,3,5],[4]]
=> [4,1,2,3,5] => 1
[1,4,5,2,3] => [1,4,5,2,3] => [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 1
[1,4,5,3,2] => [1,4,5,2,3] => [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 1
[1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => ? = 1
[1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [[1,2,3,4,5,7],[6]]
=> [6,1,2,3,4,5,7] => ? = 1
[1,2,3,4,6,7,5] => [1,2,3,4,6,7,5] => [[1,2,3,4,5,7],[6]]
=> [6,1,2,3,4,5,7] => ? = 1
[1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => ? = 1
[1,2,3,4,7,6,5] => [1,2,3,4,7,5,6] => [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => ? = 1
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [[1,2,3,4,6,7],[5]]
=> [5,1,2,3,4,6,7] => ? = 1
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [[1,2,3,4,6],[5,7]]
=> [5,7,1,2,3,4,6] => ? = 2
[1,2,3,5,6,4,7] => [1,2,3,5,6,4,7] => [[1,2,3,4,6,7],[5]]
=> [5,1,2,3,4,6,7] => ? = 1
[1,2,3,5,6,7,4] => [1,2,3,5,6,7,4] => [[1,2,3,4,6,7],[5]]
=> [5,1,2,3,4,6,7] => ? = 1
[1,2,3,5,7,4,6] => [1,2,3,5,7,4,6] => [[1,2,3,4,6],[5,7]]
=> [5,7,1,2,3,4,6] => ? = 1
[1,2,3,5,7,6,4] => [1,2,3,5,7,4,6] => [[1,2,3,4,6],[5,7]]
=> [5,7,1,2,3,4,6] => ? = 1
[1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [[1,2,3,4,5,7],[6]]
=> [6,1,2,3,4,5,7] => ? = 1
[1,2,3,6,4,7,5] => [1,2,3,6,4,7,5] => [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 2
[1,2,3,6,5,4,7] => [1,2,3,6,4,7,5] => [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 2
[1,2,3,6,5,7,4] => [1,2,3,6,4,5,7] => [[1,2,3,4,5,7],[6]]
=> [6,1,2,3,4,5,7] => ? = 1
[1,2,3,6,7,4,5] => [1,2,3,6,7,4,5] => [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 1
[1,2,3,6,7,5,4] => [1,2,3,6,7,4,5] => [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 1
[1,2,3,7,4,5,6] => [1,2,3,7,4,5,6] => [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => ? = 1
[1,2,3,7,4,6,5] => [1,2,3,7,4,6,5] => [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 2
[1,2,3,7,5,4,6] => [1,2,3,7,4,6,5] => [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 2
[1,2,3,7,5,6,4] => [1,2,3,7,4,5,6] => [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => ? = 1
[1,2,3,7,6,4,5] => [1,2,3,7,4,5,6] => [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => ? = 1
[1,2,3,7,6,5,4] => [1,2,3,7,4,5,6] => [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => ? = 1
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [[1,2,3,5,6,7],[4]]
=> [4,1,2,3,5,6,7] => ? = 1
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [[1,2,3,5,6],[4,7]]
=> [4,7,1,2,3,5,6] => ? = 1
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [[1,2,3,5,7],[4,6]]
=> [4,6,1,2,3,5,7] => ? = 1
[1,2,4,3,6,7,5] => [1,2,4,3,6,7,5] => [[1,2,3,5,7],[4,6]]
=> [4,6,1,2,3,5,7] => ? = 1
[1,2,4,3,7,5,6] => [1,2,4,3,7,5,6] => [[1,2,3,5,6],[4,7]]
=> [4,7,1,2,3,5,6] => ? = 1
[1,2,4,3,7,6,5] => [1,2,4,3,7,5,6] => [[1,2,3,5,6],[4,7]]
=> [4,7,1,2,3,5,6] => ? = 1
[1,2,4,5,3,6,7] => [1,2,4,5,3,6,7] => [[1,2,3,5,6,7],[4]]
=> [4,1,2,3,5,6,7] => ? = 1
[1,2,4,5,3,7,6] => [1,2,4,5,3,7,6] => [[1,2,3,5,6],[4,7]]
=> [4,7,1,2,3,5,6] => ? = 2
[1,2,4,5,6,3,7] => [1,2,4,5,6,3,7] => [[1,2,3,5,6,7],[4]]
=> [4,1,2,3,5,6,7] => ? = 1
[1,2,4,5,6,7,3] => [1,2,4,5,6,7,3] => [[1,2,3,5,6,7],[4]]
=> [4,1,2,3,5,6,7] => ? = 1
[1,2,4,5,7,3,6] => [1,2,4,5,7,3,6] => [[1,2,3,5,6],[4,7]]
=> [4,7,1,2,3,5,6] => ? = 1
[1,2,4,5,7,6,3] => [1,2,4,5,7,3,6] => [[1,2,3,5,6],[4,7]]
=> [4,7,1,2,3,5,6] => ? = 1
[1,2,4,6,3,5,7] => [1,2,4,6,3,5,7] => [[1,2,3,5,7],[4,6]]
=> [4,6,1,2,3,5,7] => ? = 1
[1,2,4,6,3,7,5] => [1,2,4,6,3,7,5] => [[1,2,3,5,7],[4,6]]
=> [4,6,1,2,3,5,7] => ? = 2
[1,2,4,6,5,3,7] => [1,2,4,6,3,7,5] => [[1,2,3,5,7],[4,6]]
=> [4,6,1,2,3,5,7] => ? = 2
[1,2,4,6,5,7,3] => [1,2,4,6,3,5,7] => [[1,2,3,5,7],[4,6]]
=> [4,6,1,2,3,5,7] => ? = 1
[1,2,4,6,7,3,5] => [1,2,4,6,7,3,5] => [[1,2,3,5,7],[4,6]]
=> [4,6,1,2,3,5,7] => ? = 1
[1,2,4,6,7,5,3] => [1,2,4,6,7,3,5] => [[1,2,3,5,7],[4,6]]
=> [4,6,1,2,3,5,7] => ? = 1
[1,2,4,7,3,5,6] => [1,2,4,7,3,5,6] => [[1,2,3,5,6],[4,7]]
=> [4,7,1,2,3,5,6] => ? = 1
[1,2,4,7,3,6,5] => [1,2,4,7,3,6,5] => [[1,2,3,5],[4,6],[7]]
=> [7,4,6,1,2,3,5] => ? = 2
[1,2,4,7,5,3,6] => [1,2,4,7,3,6,5] => [[1,2,3,5],[4,6],[7]]
=> [7,4,6,1,2,3,5] => ? = 2
[1,2,4,7,5,6,3] => [1,2,4,7,3,5,6] => [[1,2,3,5,6],[4,7]]
=> [4,7,1,2,3,5,6] => ? = 1
[1,2,4,7,6,3,5] => [1,2,4,7,3,5,6] => [[1,2,3,5,6],[4,7]]
=> [4,7,1,2,3,5,6] => ? = 1
[1,2,4,7,6,5,3] => [1,2,4,7,3,5,6] => [[1,2,3,5,6],[4,7]]
=> [4,7,1,2,3,5,6] => ? = 1
[1,2,5,3,4,6,7] => [1,2,5,3,4,6,7] => [[1,2,3,4,6,7],[5]]
=> [5,1,2,3,4,6,7] => ? = 1
[1,2,5,3,4,7,6] => [1,2,5,3,4,7,6] => [[1,2,3,4,6],[5,7]]
=> [5,7,1,2,3,4,6] => ? = 1
[1,2,5,3,6,4,7] => [1,2,5,3,6,4,7] => [[1,2,3,4,7],[5,6]]
=> [5,6,1,2,3,4,7] => ? = 1
Description
The number of visible descents of a permutation.
A visible descent of a permutation $\pi$ is a position $i$ such that $\pi(i+1) \leq \min(i, \pi(i))$.
Matching statistic: St001737
Mp00223: Permutations —runsort⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001737: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 67%
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001737: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 67%
Values
[1,2] => [1,2] => [[1,2]]
=> [1,2] => 0
[2,1] => [1,2] => [[1,2]]
=> [1,2] => 0
[1,2,3] => [1,2,3] => [[1,2,3]]
=> [1,2,3] => 0
[1,3,2] => [1,3,2] => [[1,2],[3]]
=> [3,1,2] => 1
[2,1,3] => [1,3,2] => [[1,2],[3]]
=> [3,1,2] => 1
[2,3,1] => [1,2,3] => [[1,2,3]]
=> [1,2,3] => 0
[3,1,2] => [1,2,3] => [[1,2,3]]
=> [1,2,3] => 0
[3,2,1] => [1,2,3] => [[1,2,3]]
=> [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [[1,2,3],[4]]
=> [4,1,2,3] => 1
[1,3,2,4] => [1,3,2,4] => [[1,2,4],[3]]
=> [3,1,2,4] => 1
[1,3,4,2] => [1,3,4,2] => [[1,2,3],[4]]
=> [4,1,2,3] => 1
[1,4,2,3] => [1,4,2,3] => [[1,2,4],[3]]
=> [3,1,2,4] => 1
[1,4,3,2] => [1,4,2,3] => [[1,2,4],[3]]
=> [3,1,2,4] => 1
[2,1,3,4] => [1,3,4,2] => [[1,2,3],[4]]
=> [4,1,2,3] => 1
[2,1,4,3] => [1,4,2,3] => [[1,2,4],[3]]
=> [3,1,2,4] => 1
[2,3,1,4] => [1,4,2,3] => [[1,2,4],[3]]
=> [3,1,2,4] => 1
[2,3,4,1] => [1,2,3,4] => [[1,2,3,4]]
=> [1,2,3,4] => 0
[2,4,1,3] => [1,3,2,4] => [[1,2,4],[3]]
=> [3,1,2,4] => 1
[2,4,3,1] => [1,2,4,3] => [[1,2,3],[4]]
=> [4,1,2,3] => 1
[3,1,2,4] => [1,2,4,3] => [[1,2,3],[4]]
=> [4,1,2,3] => 1
[3,1,4,2] => [1,4,2,3] => [[1,2,4],[3]]
=> [3,1,2,4] => 1
[3,2,1,4] => [1,4,2,3] => [[1,2,4],[3]]
=> [3,1,2,4] => 1
[3,2,4,1] => [1,2,4,3] => [[1,2,3],[4]]
=> [4,1,2,3] => 1
[3,4,1,2] => [1,2,3,4] => [[1,2,3,4]]
=> [1,2,3,4] => 0
[3,4,2,1] => [1,2,3,4] => [[1,2,3,4]]
=> [1,2,3,4] => 0
[4,1,2,3] => [1,2,3,4] => [[1,2,3,4]]
=> [1,2,3,4] => 0
[4,1,3,2] => [1,3,2,4] => [[1,2,4],[3]]
=> [3,1,2,4] => 1
[4,2,1,3] => [1,3,2,4] => [[1,2,4],[3]]
=> [3,1,2,4] => 1
[4,2,3,1] => [1,2,3,4] => [[1,2,3,4]]
=> [1,2,3,4] => 0
[4,3,1,2] => [1,2,3,4] => [[1,2,3,4]]
=> [1,2,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [[1,2,3,4]]
=> [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1
[1,2,4,3,5] => [1,2,4,3,5] => [[1,2,3,5],[4]]
=> [4,1,2,3,5] => 1
[1,2,4,5,3] => [1,2,4,5,3] => [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1
[1,2,5,3,4] => [1,2,5,3,4] => [[1,2,3,5],[4]]
=> [4,1,2,3,5] => 1
[1,2,5,4,3] => [1,2,5,3,4] => [[1,2,3,5],[4]]
=> [4,1,2,3,5] => 1
[1,3,2,4,5] => [1,3,2,4,5] => [[1,2,4,5],[3]]
=> [3,1,2,4,5] => 1
[1,3,2,5,4] => [1,3,2,5,4] => [[1,2,4],[3,5]]
=> [3,5,1,2,4] => 1
[1,3,4,2,5] => [1,3,4,2,5] => [[1,2,3,5],[4]]
=> [4,1,2,3,5] => 1
[1,3,4,5,2] => [1,3,4,5,2] => [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1
[1,3,5,2,4] => [1,3,5,2,4] => [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 1
[1,3,5,4,2] => [1,3,5,2,4] => [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 1
[1,4,2,3,5] => [1,4,2,3,5] => [[1,2,4,5],[3]]
=> [3,1,2,4,5] => 1
[1,4,2,5,3] => [1,4,2,5,3] => [[1,2,4],[3,5]]
=> [3,5,1,2,4] => 1
[1,4,3,2,5] => [1,4,2,5,3] => [[1,2,4],[3,5]]
=> [3,5,1,2,4] => 1
[1,4,3,5,2] => [1,4,2,3,5] => [[1,2,4,5],[3]]
=> [3,1,2,4,5] => 1
[1,4,5,2,3] => [1,4,5,2,3] => [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 1
[1,4,5,3,2] => [1,4,5,2,3] => [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 1
[1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => ? = 1
[1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [[1,2,3,4,5,7],[6]]
=> [6,1,2,3,4,5,7] => ? = 1
[1,2,3,4,6,7,5] => [1,2,3,4,6,7,5] => [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => ? = 1
[1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => [[1,2,3,4,5,7],[6]]
=> [6,1,2,3,4,5,7] => ? = 1
[1,2,3,4,7,6,5] => [1,2,3,4,7,5,6] => [[1,2,3,4,5,7],[6]]
=> [6,1,2,3,4,5,7] => ? = 1
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [[1,2,3,4,6,7],[5]]
=> [5,1,2,3,4,6,7] => ? = 1
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [[1,2,3,4,6],[5,7]]
=> [5,7,1,2,3,4,6] => ? = 2
[1,2,3,5,6,4,7] => [1,2,3,5,6,4,7] => [[1,2,3,4,5,7],[6]]
=> [6,1,2,3,4,5,7] => ? = 1
[1,2,3,5,6,7,4] => [1,2,3,5,6,7,4] => [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => ? = 1
[1,2,3,5,7,4,6] => [1,2,3,5,7,4,6] => [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 1
[1,2,3,5,7,6,4] => [1,2,3,5,7,4,6] => [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 1
[1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [[1,2,3,4,6,7],[5]]
=> [5,1,2,3,4,6,7] => ? = 1
[1,2,3,6,4,7,5] => [1,2,3,6,4,7,5] => [[1,2,3,4,6],[5,7]]
=> [5,7,1,2,3,4,6] => ? = 2
[1,2,3,6,5,4,7] => [1,2,3,6,4,7,5] => [[1,2,3,4,6],[5,7]]
=> [5,7,1,2,3,4,6] => ? = 2
[1,2,3,6,5,7,4] => [1,2,3,6,4,5,7] => [[1,2,3,4,6,7],[5]]
=> [5,1,2,3,4,6,7] => ? = 1
[1,2,3,6,7,4,5] => [1,2,3,6,7,4,5] => [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 1
[1,2,3,6,7,5,4] => [1,2,3,6,7,4,5] => [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 1
[1,2,3,7,4,5,6] => [1,2,3,7,4,5,6] => [[1,2,3,4,6,7],[5]]
=> [5,1,2,3,4,6,7] => ? = 1
[1,2,3,7,4,6,5] => [1,2,3,7,4,6,5] => [[1,2,3,4,6],[5],[7]]
=> [7,5,1,2,3,4,6] => ? = 2
[1,2,3,7,5,4,6] => [1,2,3,7,4,6,5] => [[1,2,3,4,6],[5],[7]]
=> [7,5,1,2,3,4,6] => ? = 2
[1,2,3,7,5,6,4] => [1,2,3,7,4,5,6] => [[1,2,3,4,6,7],[5]]
=> [5,1,2,3,4,6,7] => ? = 1
[1,2,3,7,6,4,5] => [1,2,3,7,4,5,6] => [[1,2,3,4,6,7],[5]]
=> [5,1,2,3,4,6,7] => ? = 1
[1,2,3,7,6,5,4] => [1,2,3,7,4,5,6] => [[1,2,3,4,6,7],[5]]
=> [5,1,2,3,4,6,7] => ? = 1
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [[1,2,3,5,6,7],[4]]
=> [4,1,2,3,5,6,7] => ? = 1
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [[1,2,3,5,6],[4,7]]
=> [4,7,1,2,3,5,6] => ? = 1
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [[1,2,3,5,7],[4,6]]
=> [4,6,1,2,3,5,7] => ? = 1
[1,2,4,3,6,7,5] => [1,2,4,3,6,7,5] => [[1,2,3,5,6],[4,7]]
=> [4,7,1,2,3,5,6] => ? = 1
[1,2,4,3,7,5,6] => [1,2,4,3,7,5,6] => [[1,2,3,5,7],[4,6]]
=> [4,6,1,2,3,5,7] => ? = 1
[1,2,4,3,7,6,5] => [1,2,4,3,7,5,6] => [[1,2,3,5,7],[4,6]]
=> [4,6,1,2,3,5,7] => ? = 1
[1,2,4,5,3,6,7] => [1,2,4,5,3,6,7] => [[1,2,3,4,6,7],[5]]
=> [5,1,2,3,4,6,7] => ? = 1
[1,2,4,5,3,7,6] => [1,2,4,5,3,7,6] => [[1,2,3,4,6],[5,7]]
=> [5,7,1,2,3,4,6] => ? = 2
[1,2,4,5,6,3,7] => [1,2,4,5,6,3,7] => [[1,2,3,4,5,7],[6]]
=> [6,1,2,3,4,5,7] => ? = 1
[1,2,4,5,6,7,3] => [1,2,4,5,6,7,3] => [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => ? = 1
[1,2,4,5,7,3,6] => [1,2,4,5,7,3,6] => [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 1
[1,2,4,5,7,6,3] => [1,2,4,5,7,3,6] => [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 1
[1,2,4,6,3,5,7] => [1,2,4,6,3,5,7] => [[1,2,3,4,7],[5,6]]
=> [5,6,1,2,3,4,7] => ? = 1
[1,2,4,6,3,7,5] => [1,2,4,6,3,7,5] => [[1,2,3,4,6],[5,7]]
=> [5,7,1,2,3,4,6] => ? = 2
[1,2,4,6,5,3,7] => [1,2,4,6,3,7,5] => [[1,2,3,4,6],[5,7]]
=> [5,7,1,2,3,4,6] => ? = 2
[1,2,4,6,5,7,3] => [1,2,4,6,3,5,7] => [[1,2,3,4,7],[5,6]]
=> [5,6,1,2,3,4,7] => ? = 1
[1,2,4,6,7,3,5] => [1,2,4,6,7,3,5] => [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 1
[1,2,4,6,7,5,3] => [1,2,4,6,7,3,5] => [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 1
[1,2,4,7,3,5,6] => [1,2,4,7,3,5,6] => [[1,2,3,4,7],[5,6]]
=> [5,6,1,2,3,4,7] => ? = 1
[1,2,4,7,3,6,5] => [1,2,4,7,3,6,5] => [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 2
[1,2,4,7,5,3,6] => [1,2,4,7,3,6,5] => [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 2
[1,2,4,7,5,6,3] => [1,2,4,7,3,5,6] => [[1,2,3,4,7],[5,6]]
=> [5,6,1,2,3,4,7] => ? = 1
[1,2,4,7,6,3,5] => [1,2,4,7,3,5,6] => [[1,2,3,4,7],[5,6]]
=> [5,6,1,2,3,4,7] => ? = 1
[1,2,4,7,6,5,3] => [1,2,4,7,3,5,6] => [[1,2,3,4,7],[5,6]]
=> [5,6,1,2,3,4,7] => ? = 1
[1,2,5,3,4,6,7] => [1,2,5,3,4,6,7] => [[1,2,3,5,6,7],[4]]
=> [4,1,2,3,5,6,7] => ? = 1
[1,2,5,3,4,7,6] => [1,2,5,3,4,7,6] => [[1,2,3,5,6],[4,7]]
=> [4,7,1,2,3,5,6] => ? = 1
[1,2,5,3,6,4,7] => [1,2,5,3,6,4,7] => [[1,2,3,5,7],[4,6]]
=> [4,6,1,2,3,5,7] => ? = 1
Description
The number of descents of type 2 in a permutation.
A position $i\in[1,n-1]$ is a descent of type 2 of a permutation $\pi$ of $n$ letters, if it is a descent and if $\pi(j) < \pi(i)$ for all $j < i$.
Matching statistic: St001741
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001741: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 67%
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001741: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 67%
Values
[1,2] => [1,2] => [[1,2]]
=> [1,2] => 1 = 0 + 1
[2,1] => [1,2] => [[1,2]]
=> [1,2] => 1 = 0 + 1
[1,2,3] => [1,2,3] => [[1,2,3]]
=> [1,2,3] => 1 = 0 + 1
[1,3,2] => [1,3,2] => [[1,2],[3]]
=> [3,1,2] => 2 = 1 + 1
[2,1,3] => [1,3,2] => [[1,2],[3]]
=> [3,1,2] => 2 = 1 + 1
[2,3,1] => [1,2,3] => [[1,2,3]]
=> [1,2,3] => 1 = 0 + 1
[3,1,2] => [1,2,3] => [[1,2,3]]
=> [1,2,3] => 1 = 0 + 1
[3,2,1] => [1,2,3] => [[1,2,3]]
=> [1,2,3] => 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> [1,2,3,4] => 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [[1,2,3],[4]]
=> [4,1,2,3] => 2 = 1 + 1
[1,3,2,4] => [1,3,2,4] => [[1,2,4],[3]]
=> [3,1,2,4] => 2 = 1 + 1
[1,3,4,2] => [1,3,4,2] => [[1,2,3],[4]]
=> [4,1,2,3] => 2 = 1 + 1
[1,4,2,3] => [1,4,2,3] => [[1,2,4],[3]]
=> [3,1,2,4] => 2 = 1 + 1
[1,4,3,2] => [1,4,2,3] => [[1,2,4],[3]]
=> [3,1,2,4] => 2 = 1 + 1
[2,1,3,4] => [1,3,4,2] => [[1,2,3],[4]]
=> [4,1,2,3] => 2 = 1 + 1
[2,1,4,3] => [1,4,2,3] => [[1,2,4],[3]]
=> [3,1,2,4] => 2 = 1 + 1
[2,3,1,4] => [1,4,2,3] => [[1,2,4],[3]]
=> [3,1,2,4] => 2 = 1 + 1
[2,3,4,1] => [1,2,3,4] => [[1,2,3,4]]
=> [1,2,3,4] => 1 = 0 + 1
[2,4,1,3] => [1,3,2,4] => [[1,2,4],[3]]
=> [3,1,2,4] => 2 = 1 + 1
[2,4,3,1] => [1,2,4,3] => [[1,2,3],[4]]
=> [4,1,2,3] => 2 = 1 + 1
[3,1,2,4] => [1,2,4,3] => [[1,2,3],[4]]
=> [4,1,2,3] => 2 = 1 + 1
[3,1,4,2] => [1,4,2,3] => [[1,2,4],[3]]
=> [3,1,2,4] => 2 = 1 + 1
[3,2,1,4] => [1,4,2,3] => [[1,2,4],[3]]
=> [3,1,2,4] => 2 = 1 + 1
[3,2,4,1] => [1,2,4,3] => [[1,2,3],[4]]
=> [4,1,2,3] => 2 = 1 + 1
[3,4,1,2] => [1,2,3,4] => [[1,2,3,4]]
=> [1,2,3,4] => 1 = 0 + 1
[3,4,2,1] => [1,2,3,4] => [[1,2,3,4]]
=> [1,2,3,4] => 1 = 0 + 1
[4,1,2,3] => [1,2,3,4] => [[1,2,3,4]]
=> [1,2,3,4] => 1 = 0 + 1
[4,1,3,2] => [1,3,2,4] => [[1,2,4],[3]]
=> [3,1,2,4] => 2 = 1 + 1
[4,2,1,3] => [1,3,2,4] => [[1,2,4],[3]]
=> [3,1,2,4] => 2 = 1 + 1
[4,2,3,1] => [1,2,3,4] => [[1,2,3,4]]
=> [1,2,3,4] => 1 = 0 + 1
[4,3,1,2] => [1,2,3,4] => [[1,2,3,4]]
=> [1,2,3,4] => 1 = 0 + 1
[4,3,2,1] => [1,2,3,4] => [[1,2,3,4]]
=> [1,2,3,4] => 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> [1,2,3,4,5] => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 2 = 1 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [[1,2,3,5],[4]]
=> [4,1,2,3,5] => 2 = 1 + 1
[1,2,4,5,3] => [1,2,4,5,3] => [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 2 = 1 + 1
[1,2,5,3,4] => [1,2,5,3,4] => [[1,2,3,5],[4]]
=> [4,1,2,3,5] => 2 = 1 + 1
[1,2,5,4,3] => [1,2,5,3,4] => [[1,2,3,5],[4]]
=> [4,1,2,3,5] => 2 = 1 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [[1,2,4,5],[3]]
=> [3,1,2,4,5] => 2 = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [[1,2,4],[3,5]]
=> [3,5,1,2,4] => 2 = 1 + 1
[1,3,4,2,5] => [1,3,4,2,5] => [[1,2,3,5],[4]]
=> [4,1,2,3,5] => 2 = 1 + 1
[1,3,4,5,2] => [1,3,4,5,2] => [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 2 = 1 + 1
[1,3,5,2,4] => [1,3,5,2,4] => [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 2 = 1 + 1
[1,3,5,4,2] => [1,3,5,2,4] => [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 2 = 1 + 1
[1,4,2,3,5] => [1,4,2,3,5] => [[1,2,4,5],[3]]
=> [3,1,2,4,5] => 2 = 1 + 1
[1,4,2,5,3] => [1,4,2,5,3] => [[1,2,4],[3,5]]
=> [3,5,1,2,4] => 2 = 1 + 1
[1,4,3,2,5] => [1,4,2,5,3] => [[1,2,4],[3,5]]
=> [3,5,1,2,4] => 2 = 1 + 1
[1,4,3,5,2] => [1,4,2,3,5] => [[1,2,4,5],[3]]
=> [3,1,2,4,5] => 2 = 1 + 1
[1,4,5,2,3] => [1,4,5,2,3] => [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 2 = 1 + 1
[1,4,5,3,2] => [1,4,5,2,3] => [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 2 = 1 + 1
[1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => ? = 1 + 1
[1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [[1,2,3,4,5,7],[6]]
=> [6,1,2,3,4,5,7] => ? = 1 + 1
[1,2,3,4,6,7,5] => [1,2,3,4,6,7,5] => [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => ? = 1 + 1
[1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => [[1,2,3,4,5,7],[6]]
=> [6,1,2,3,4,5,7] => ? = 1 + 1
[1,2,3,4,7,6,5] => [1,2,3,4,7,5,6] => [[1,2,3,4,5,7],[6]]
=> [6,1,2,3,4,5,7] => ? = 1 + 1
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [[1,2,3,4,6,7],[5]]
=> [5,1,2,3,4,6,7] => ? = 1 + 1
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [[1,2,3,4,6],[5,7]]
=> [5,7,1,2,3,4,6] => ? = 2 + 1
[1,2,3,5,6,4,7] => [1,2,3,5,6,4,7] => [[1,2,3,4,5,7],[6]]
=> [6,1,2,3,4,5,7] => ? = 1 + 1
[1,2,3,5,6,7,4] => [1,2,3,5,6,7,4] => [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => ? = 1 + 1
[1,2,3,5,7,4,6] => [1,2,3,5,7,4,6] => [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 1 + 1
[1,2,3,5,7,6,4] => [1,2,3,5,7,4,6] => [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 1 + 1
[1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [[1,2,3,4,6,7],[5]]
=> [5,1,2,3,4,6,7] => ? = 1 + 1
[1,2,3,6,4,7,5] => [1,2,3,6,4,7,5] => [[1,2,3,4,6],[5,7]]
=> [5,7,1,2,3,4,6] => ? = 2 + 1
[1,2,3,6,5,4,7] => [1,2,3,6,4,7,5] => [[1,2,3,4,6],[5,7]]
=> [5,7,1,2,3,4,6] => ? = 2 + 1
[1,2,3,6,5,7,4] => [1,2,3,6,4,5,7] => [[1,2,3,4,6,7],[5]]
=> [5,1,2,3,4,6,7] => ? = 1 + 1
[1,2,3,6,7,4,5] => [1,2,3,6,7,4,5] => [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 1 + 1
[1,2,3,6,7,5,4] => [1,2,3,6,7,4,5] => [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 1 + 1
[1,2,3,7,4,5,6] => [1,2,3,7,4,5,6] => [[1,2,3,4,6,7],[5]]
=> [5,1,2,3,4,6,7] => ? = 1 + 1
[1,2,3,7,4,6,5] => [1,2,3,7,4,6,5] => [[1,2,3,4,6],[5],[7]]
=> [7,5,1,2,3,4,6] => ? = 2 + 1
[1,2,3,7,5,4,6] => [1,2,3,7,4,6,5] => [[1,2,3,4,6],[5],[7]]
=> [7,5,1,2,3,4,6] => ? = 2 + 1
[1,2,3,7,5,6,4] => [1,2,3,7,4,5,6] => [[1,2,3,4,6,7],[5]]
=> [5,1,2,3,4,6,7] => ? = 1 + 1
[1,2,3,7,6,4,5] => [1,2,3,7,4,5,6] => [[1,2,3,4,6,7],[5]]
=> [5,1,2,3,4,6,7] => ? = 1 + 1
[1,2,3,7,6,5,4] => [1,2,3,7,4,5,6] => [[1,2,3,4,6,7],[5]]
=> [5,1,2,3,4,6,7] => ? = 1 + 1
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [[1,2,3,5,6,7],[4]]
=> [4,1,2,3,5,6,7] => ? = 1 + 1
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [[1,2,3,5,6],[4,7]]
=> [4,7,1,2,3,5,6] => ? = 1 + 1
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [[1,2,3,5,7],[4,6]]
=> [4,6,1,2,3,5,7] => ? = 1 + 1
[1,2,4,3,6,7,5] => [1,2,4,3,6,7,5] => [[1,2,3,5,6],[4,7]]
=> [4,7,1,2,3,5,6] => ? = 1 + 1
[1,2,4,3,7,5,6] => [1,2,4,3,7,5,6] => [[1,2,3,5,7],[4,6]]
=> [4,6,1,2,3,5,7] => ? = 1 + 1
[1,2,4,3,7,6,5] => [1,2,4,3,7,5,6] => [[1,2,3,5,7],[4,6]]
=> [4,6,1,2,3,5,7] => ? = 1 + 1
[1,2,4,5,3,6,7] => [1,2,4,5,3,6,7] => [[1,2,3,4,6,7],[5]]
=> [5,1,2,3,4,6,7] => ? = 1 + 1
[1,2,4,5,3,7,6] => [1,2,4,5,3,7,6] => [[1,2,3,4,6],[5,7]]
=> [5,7,1,2,3,4,6] => ? = 2 + 1
[1,2,4,5,6,3,7] => [1,2,4,5,6,3,7] => [[1,2,3,4,5,7],[6]]
=> [6,1,2,3,4,5,7] => ? = 1 + 1
[1,2,4,5,6,7,3] => [1,2,4,5,6,7,3] => [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => ? = 1 + 1
[1,2,4,5,7,3,6] => [1,2,4,5,7,3,6] => [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 1 + 1
[1,2,4,5,7,6,3] => [1,2,4,5,7,3,6] => [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 1 + 1
[1,2,4,6,3,5,7] => [1,2,4,6,3,5,7] => [[1,2,3,4,7],[5,6]]
=> [5,6,1,2,3,4,7] => ? = 1 + 1
[1,2,4,6,3,7,5] => [1,2,4,6,3,7,5] => [[1,2,3,4,6],[5,7]]
=> [5,7,1,2,3,4,6] => ? = 2 + 1
[1,2,4,6,5,3,7] => [1,2,4,6,3,7,5] => [[1,2,3,4,6],[5,7]]
=> [5,7,1,2,3,4,6] => ? = 2 + 1
[1,2,4,6,5,7,3] => [1,2,4,6,3,5,7] => [[1,2,3,4,7],[5,6]]
=> [5,6,1,2,3,4,7] => ? = 1 + 1
[1,2,4,6,7,3,5] => [1,2,4,6,7,3,5] => [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 1 + 1
[1,2,4,6,7,5,3] => [1,2,4,6,7,3,5] => [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 1 + 1
[1,2,4,7,3,5,6] => [1,2,4,7,3,5,6] => [[1,2,3,4,7],[5,6]]
=> [5,6,1,2,3,4,7] => ? = 1 + 1
[1,2,4,7,3,6,5] => [1,2,4,7,3,6,5] => [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 2 + 1
[1,2,4,7,5,3,6] => [1,2,4,7,3,6,5] => [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 2 + 1
[1,2,4,7,5,6,3] => [1,2,4,7,3,5,6] => [[1,2,3,4,7],[5,6]]
=> [5,6,1,2,3,4,7] => ? = 1 + 1
[1,2,4,7,6,3,5] => [1,2,4,7,3,5,6] => [[1,2,3,4,7],[5,6]]
=> [5,6,1,2,3,4,7] => ? = 1 + 1
[1,2,4,7,6,5,3] => [1,2,4,7,3,5,6] => [[1,2,3,4,7],[5,6]]
=> [5,6,1,2,3,4,7] => ? = 1 + 1
[1,2,5,3,4,6,7] => [1,2,5,3,4,6,7] => [[1,2,3,5,6,7],[4]]
=> [4,1,2,3,5,6,7] => ? = 1 + 1
[1,2,5,3,4,7,6] => [1,2,5,3,4,7,6] => [[1,2,3,5,6],[4,7]]
=> [4,7,1,2,3,5,6] => ? = 1 + 1
[1,2,5,3,6,4,7] => [1,2,5,3,6,4,7] => [[1,2,3,5,7],[4,6]]
=> [4,6,1,2,3,5,7] => ? = 1 + 1
Description
The largest integer such that all patterns of this size are contained in the permutation.
Matching statistic: St001195
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 67%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 67%
Values
[1,2] => [1,2] => [2]
=> [1,0,1,0]
=> ? = 0
[2,1] => [1,2] => [2]
=> [1,0,1,0]
=> ? = 0
[1,2,3] => [1,2,3] => [3]
=> [1,0,1,0,1,0]
=> 0
[1,3,2] => [1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,1,3] => [1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,3,1] => [1,2,3] => [3]
=> [1,0,1,0,1,0]
=> 0
[3,1,2] => [1,2,3] => [3]
=> [1,0,1,0,1,0]
=> 0
[3,2,1] => [1,2,3] => [3]
=> [1,0,1,0,1,0]
=> 0
[1,2,3,4] => [1,2,3,4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3] => [1,2,4,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,3,2,4] => [1,3,2,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,3,4,2] => [1,3,4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,4,2,3] => [1,4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,4,3,2] => [1,4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,1,3,4] => [1,3,4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,1,4,3] => [1,4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,3,1,4] => [1,4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,3,4,1] => [1,2,3,4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[2,4,1,3] => [1,3,2,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,4,3,1] => [1,2,4,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,1,2,4] => [1,2,4,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,1,4,2] => [1,4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,2,1,4] => [1,4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,2,4,1] => [1,2,4,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,4,1,2] => [1,2,3,4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[3,4,2,1] => [1,2,3,4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[4,1,2,3] => [1,2,3,4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[4,1,3,2] => [1,3,2,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[4,2,1,3] => [1,3,2,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[4,2,3,1] => [1,2,3,4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[4,3,1,2] => [1,2,3,4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[4,3,2,1] => [1,2,3,4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,2,3,4,5] => [1,2,3,4,5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,4,5,3] => [1,2,4,5,3] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,5,3,4] => [1,2,5,3,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,5,4,3] => [1,2,5,3,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,3,4,2,5] => [1,3,4,2,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,3,4,5,2] => [1,3,4,5,2] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,3,5,2,4] => [1,3,5,2,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,3,5,4,2] => [1,3,5,2,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,4,2,3,5] => [1,4,2,3,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,4,2,5,3] => [1,4,2,5,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,4,3,2,5] => [1,4,2,5,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,4,3,5,2] => [1,4,2,3,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,4,5,2,3] => [1,4,5,2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,4,5,3,2] => [1,4,5,2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,5,2,3,4] => [1,5,2,3,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,5,2,4,3] => [1,5,2,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,2,3,4,6,5] => [1,2,3,4,6,5] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,2,3,5,6,4] => [1,2,3,5,6,4] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,2,3,6,4,5] => [1,2,3,6,4,5] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,2,3,6,5,4] => [1,2,3,6,4,5] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,2,4,3,5,6] => [1,2,4,3,5,6] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,2,4,5,3,6] => [1,2,4,5,3,6] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,2,4,5,6,3] => [1,2,4,5,6,3] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,2,5,3,4,6] => [1,2,5,3,4,6] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,2,5,4,6,3] => [1,2,5,3,4,6] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,2,6,3,4,5] => [1,2,6,3,4,5] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,2,6,3,5,4] => [1,2,6,3,5,4] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
[1,2,6,4,3,5] => [1,2,6,3,5,4] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
[1,2,6,4,5,3] => [1,2,6,3,4,5] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,2,6,5,3,4] => [1,2,6,3,4,5] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,2,6,5,4,3] => [1,2,6,3,4,5] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,3,2,4,5,6] => [1,3,2,4,5,6] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,3,4,2,5,6] => [1,3,4,2,5,6] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,3,4,5,2,6] => [1,3,4,5,2,6] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,3,4,5,6,2] => [1,3,4,5,6,2] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,4,2,3,5,6] => [1,4,2,3,5,6] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,4,3,5,6,2] => [1,4,2,3,5,6] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,5,2,3,4,6] => [1,5,2,3,4,6] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,5,2,4,3,6] => [1,5,2,4,3,6] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
[1,5,2,4,6,3] => [1,5,2,4,6,3] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
[1,5,3,2,4,6] => [1,5,2,4,6,3] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
[1,5,3,4,6,2] => [1,5,2,3,4,6] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,5,3,6,2,4] => [1,5,2,4,3,6] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
[1,5,4,6,2,3] => [1,5,2,3,4,6] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,5,4,6,3,2] => [1,5,2,3,4,6] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,6,2,3,4,5] => [1,6,2,3,4,5] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,6,2,3,5,4] => [1,6,2,3,5,4] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
[1,6,2,4,3,5] => [1,6,2,4,3,5] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
[1,6,2,4,5,3] => [1,6,2,4,5,3] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
[1,6,2,5,3,4] => [1,6,2,5,3,4] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
[1,6,2,5,4,3] => [1,6,2,5,3,4] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
[1,6,3,2,4,5] => [1,6,2,4,5,3] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
[1,6,3,2,5,4] => [1,6,2,5,3,4] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
[1,6,3,4,2,5] => [1,6,2,5,3,4] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
[1,6,3,4,5,2] => [1,6,2,3,4,5] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,6,3,5,2,4] => [1,6,2,4,3,5] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
[1,6,3,5,4,2] => [1,6,2,3,5,4] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
[1,6,4,2,3,5] => [1,6,2,3,5,4] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
[1,6,4,2,5,3] => [1,6,2,5,3,4] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
[1,6,4,3,2,5] => [1,6,2,5,3,4] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
[1,6,4,3,5,2] => [1,6,2,3,5,4] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
[1,6,4,5,2,3] => [1,6,2,3,4,5] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
Description
The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$.
Matching statistic: St001174
(load all 145 compositions to match this statistic)
(load all 145 compositions to match this statistic)
Mp00069: Permutations —complement⟶ Permutations
St001174: Permutations ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 67%
St001174: Permutations ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 67%
Values
[1,2] => [2,1] => 0
[2,1] => [1,2] => 0
[1,2,3] => [3,2,1] => 0
[1,3,2] => [3,1,2] => 1
[2,1,3] => [2,3,1] => 1
[2,3,1] => [2,1,3] => 0
[3,1,2] => [1,3,2] => 0
[3,2,1] => [1,2,3] => 0
[1,2,3,4] => [4,3,2,1] => 0
[1,2,4,3] => [4,3,1,2] => 1
[1,3,2,4] => [4,2,3,1] => 1
[1,3,4,2] => [4,2,1,3] => 1
[1,4,2,3] => [4,1,3,2] => 1
[1,4,3,2] => [4,1,2,3] => 1
[2,1,3,4] => [3,4,2,1] => 1
[2,1,4,3] => [3,4,1,2] => 1
[2,3,1,4] => [3,2,4,1] => 1
[2,3,4,1] => [3,2,1,4] => 0
[2,4,1,3] => [3,1,4,2] => 1
[2,4,3,1] => [3,1,2,4] => 1
[3,1,2,4] => [2,4,3,1] => 1
[3,1,4,2] => [2,4,1,3] => 1
[3,2,1,4] => [2,3,4,1] => 1
[3,2,4,1] => [2,3,1,4] => 1
[3,4,1,2] => [2,1,4,3] => 0
[3,4,2,1] => [2,1,3,4] => 0
[4,1,2,3] => [1,4,3,2] => 0
[4,1,3,2] => [1,4,2,3] => 1
[4,2,1,3] => [1,3,4,2] => 1
[4,2,3,1] => [1,3,2,4] => 0
[4,3,1,2] => [1,2,4,3] => 0
[4,3,2,1] => [1,2,3,4] => 0
[1,2,3,4,5] => [5,4,3,2,1] => 0
[1,2,3,5,4] => [5,4,3,1,2] => 1
[1,2,4,3,5] => [5,4,2,3,1] => 1
[1,2,4,5,3] => [5,4,2,1,3] => 1
[1,2,5,3,4] => [5,4,1,3,2] => 1
[1,2,5,4,3] => [5,4,1,2,3] => 1
[1,3,2,4,5] => [5,3,4,2,1] => 1
[1,3,2,5,4] => [5,3,4,1,2] => 1
[1,3,4,2,5] => [5,3,2,4,1] => 1
[1,3,4,5,2] => [5,3,2,1,4] => 1
[1,3,5,2,4] => [5,3,1,4,2] => 1
[1,3,5,4,2] => [5,3,1,2,4] => 1
[1,4,2,3,5] => [5,2,4,3,1] => 1
[1,4,2,5,3] => [5,2,4,1,3] => 1
[1,4,3,2,5] => [5,2,3,4,1] => 1
[1,4,3,5,2] => [5,2,3,1,4] => 1
[1,4,5,2,3] => [5,2,1,4,3] => 1
[1,4,5,3,2] => [5,2,1,3,4] => 1
[1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 0
[1,2,3,4,5,7,6] => [7,6,5,4,3,1,2] => ? = 1
[1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => ? = 1
[1,2,3,4,6,7,5] => [7,6,5,4,2,1,3] => ? = 1
[1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => ? = 1
[1,2,3,4,7,6,5] => [7,6,5,4,1,2,3] => ? = 1
[1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => ? = 1
[1,2,3,5,4,7,6] => [7,6,5,3,4,1,2] => ? = 2
[1,2,3,5,6,4,7] => [7,6,5,3,2,4,1] => ? = 1
[1,2,3,5,6,7,4] => [7,6,5,3,2,1,4] => ? = 1
[1,2,3,5,7,4,6] => [7,6,5,3,1,4,2] => ? = 1
[1,2,3,5,7,6,4] => [7,6,5,3,1,2,4] => ? = 1
[1,2,3,6,4,5,7] => [7,6,5,2,4,3,1] => ? = 1
[1,2,3,6,4,7,5] => [7,6,5,2,4,1,3] => ? = 2
[1,2,3,6,5,4,7] => [7,6,5,2,3,4,1] => ? = 2
[1,2,3,6,5,7,4] => [7,6,5,2,3,1,4] => ? = 1
[1,2,3,6,7,4,5] => [7,6,5,2,1,4,3] => ? = 1
[1,2,3,6,7,5,4] => [7,6,5,2,1,3,4] => ? = 1
[1,2,3,7,4,5,6] => [7,6,5,1,4,3,2] => ? = 1
[1,2,3,7,4,6,5] => [7,6,5,1,4,2,3] => ? = 2
[1,2,3,7,5,4,6] => [7,6,5,1,3,4,2] => ? = 2
[1,2,3,7,5,6,4] => [7,6,5,1,3,2,4] => ? = 1
[1,2,3,7,6,4,5] => [7,6,5,1,2,4,3] => ? = 1
[1,2,3,7,6,5,4] => [7,6,5,1,2,3,4] => ? = 1
[1,2,4,3,5,6,7] => [7,6,4,5,3,2,1] => ? = 1
[1,2,4,3,5,7,6] => [7,6,4,5,3,1,2] => ? = 1
[1,2,4,3,6,5,7] => [7,6,4,5,2,3,1] => ? = 1
[1,2,4,3,6,7,5] => [7,6,4,5,2,1,3] => ? = 1
[1,2,4,3,7,5,6] => [7,6,4,5,1,3,2] => ? = 1
[1,2,4,3,7,6,5] => [7,6,4,5,1,2,3] => ? = 1
[1,2,4,5,3,6,7] => [7,6,4,3,5,2,1] => ? = 1
[1,2,4,5,3,7,6] => [7,6,4,3,5,1,2] => ? = 2
[1,2,4,5,6,3,7] => [7,6,4,3,2,5,1] => ? = 1
[1,2,4,5,6,7,3] => [7,6,4,3,2,1,5] => ? = 1
[1,2,4,5,7,3,6] => [7,6,4,3,1,5,2] => ? = 1
[1,2,4,5,7,6,3] => [7,6,4,3,1,2,5] => ? = 1
[1,2,4,6,3,5,7] => [7,6,4,2,5,3,1] => ? = 1
[1,2,4,6,3,7,5] => [7,6,4,2,5,1,3] => ? = 2
[1,2,4,6,5,3,7] => [7,6,4,2,3,5,1] => ? = 2
[1,2,4,6,5,7,3] => [7,6,4,2,3,1,5] => ? = 1
[1,2,4,6,7,3,5] => [7,6,4,2,1,5,3] => ? = 1
[1,2,4,6,7,5,3] => [7,6,4,2,1,3,5] => ? = 1
[1,2,4,7,3,5,6] => [7,6,4,1,5,3,2] => ? = 1
[1,2,4,7,3,6,5] => [7,6,4,1,5,2,3] => ? = 2
[1,2,4,7,5,3,6] => [7,6,4,1,3,5,2] => ? = 2
[1,2,4,7,5,6,3] => [7,6,4,1,3,2,5] => ? = 1
[1,2,4,7,6,3,5] => [7,6,4,1,2,5,3] => ? = 1
[1,2,4,7,6,5,3] => [7,6,4,1,2,3,5] => ? = 1
[1,2,5,3,4,6,7] => [7,6,3,5,4,2,1] => ? = 1
[1,2,5,3,4,7,6] => [7,6,3,5,4,1,2] => ? = 1
Description
The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
Matching statistic: St001859
(load all 45 compositions to match this statistic)
(load all 45 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St001859: Permutations ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 67%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St001859: Permutations ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 67%
Values
[1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [3,1,2] => 1
[2,1,3] => [1,3,2] => [3,1,2] => 1
[2,3,1] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,2,3] => [1,2,3] => 0
[3,2,1] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [4,1,2,3] => 1
[1,3,2,4] => [1,3,2,4] => [3,1,2,4] => 1
[1,3,4,2] => [1,3,4,2] => [3,4,1,2] => 1
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 1
[1,4,3,2] => [1,4,2,3] => [1,4,2,3] => 1
[2,1,3,4] => [1,3,4,2] => [3,4,1,2] => 1
[2,1,4,3] => [1,4,2,3] => [1,4,2,3] => 1
[2,3,1,4] => [1,4,2,3] => [1,4,2,3] => 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,3,2,4] => [3,1,2,4] => 1
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => 1
[3,1,2,4] => [1,2,4,3] => [4,1,2,3] => 1
[3,1,4,2] => [1,4,2,3] => [1,4,2,3] => 1
[3,2,1,4] => [1,4,2,3] => [1,4,2,3] => 1
[3,2,4,1] => [1,2,4,3] => [4,1,2,3] => 1
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,3,2] => [1,3,2,4] => [3,1,2,4] => 1
[4,2,1,3] => [1,3,2,4] => [3,1,2,4] => 1
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => 1
[1,2,4,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => 1
[1,2,4,5,3] => [1,2,4,5,3] => [4,5,1,2,3] => 1
[1,2,5,3,4] => [1,2,5,3,4] => [1,5,2,3,4] => 1
[1,2,5,4,3] => [1,2,5,3,4] => [1,5,2,3,4] => 1
[1,3,2,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => 1
[1,3,2,5,4] => [1,3,2,5,4] => [3,5,1,2,4] => 1
[1,3,4,2,5] => [1,3,4,2,5] => [3,4,1,2,5] => 1
[1,3,4,5,2] => [1,3,4,5,2] => [3,4,5,1,2] => 1
[1,3,5,2,4] => [1,3,5,2,4] => [5,3,1,2,4] => 1
[1,3,5,4,2] => [1,3,5,2,4] => [5,3,1,2,4] => 1
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => 1
[1,4,2,5,3] => [1,4,2,5,3] => [1,4,5,2,3] => 1
[1,4,3,2,5] => [1,4,2,5,3] => [1,4,5,2,3] => 1
[1,4,3,5,2] => [1,4,2,3,5] => [1,4,2,3,5] => 1
[1,4,5,2,3] => [1,4,5,2,3] => [4,1,5,2,3] => 1
[1,4,5,3,2] => [1,4,5,2,3] => [4,1,5,2,3] => 1
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
[1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ? = 1
[1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [6,1,2,3,4,5,7] => ? = 1
[1,2,3,4,6,7,5] => [1,2,3,4,6,7,5] => [6,7,1,2,3,4,5] => ? = 1
[1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => [1,7,2,3,4,5,6] => ? = 1
[1,2,3,4,7,6,5] => [1,2,3,4,7,5,6] => [1,7,2,3,4,5,6] => ? = 1
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => ? = 1
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [5,7,1,2,3,4,6] => ? = 2
[1,2,3,5,6,4,7] => [1,2,3,5,6,4,7] => [5,6,1,2,3,4,7] => ? = 1
[1,2,3,5,6,7,4] => [1,2,3,5,6,7,4] => [5,6,7,1,2,3,4] => ? = 1
[1,2,3,5,7,4,6] => [1,2,3,5,7,4,6] => [7,5,1,2,3,4,6] => ? = 1
[1,2,3,5,7,6,4] => [1,2,3,5,7,4,6] => [7,5,1,2,3,4,6] => ? = 1
[1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => ? = 1
[1,2,3,6,4,7,5] => [1,2,3,6,4,7,5] => [1,6,7,2,3,4,5] => ? = 2
[1,2,3,6,5,4,7] => [1,2,3,6,4,7,5] => [1,6,7,2,3,4,5] => ? = 2
[1,2,3,6,5,7,4] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => ? = 1
[1,2,3,6,7,4,5] => [1,2,3,6,7,4,5] => [6,1,7,2,3,4,5] => ? = 1
[1,2,3,6,7,5,4] => [1,2,3,6,7,4,5] => [6,1,7,2,3,4,5] => ? = 1
[1,2,3,7,4,5,6] => [1,2,3,7,4,5,6] => [1,2,7,3,4,5,6] => ? = 1
[1,2,3,7,4,6,5] => [1,2,3,7,4,6,5] => [7,1,6,2,3,4,5] => ? = 2
[1,2,3,7,5,4,6] => [1,2,3,7,4,6,5] => [7,1,6,2,3,4,5] => ? = 2
[1,2,3,7,5,6,4] => [1,2,3,7,4,5,6] => [1,2,7,3,4,5,6] => ? = 1
[1,2,3,7,6,4,5] => [1,2,3,7,4,5,6] => [1,2,7,3,4,5,6] => ? = 1
[1,2,3,7,6,5,4] => [1,2,3,7,4,5,6] => [1,2,7,3,4,5,6] => ? = 1
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [4,1,2,3,5,6,7] => ? = 1
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [4,7,1,2,3,5,6] => ? = 1
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [4,6,1,2,3,5,7] => ? = 1
[1,2,4,3,6,7,5] => [1,2,4,3,6,7,5] => [4,6,7,1,2,3,5] => ? = 1
[1,2,4,3,7,5,6] => [1,2,4,3,7,5,6] => [7,4,1,2,3,5,6] => ? = 1
[1,2,4,3,7,6,5] => [1,2,4,3,7,5,6] => [7,4,1,2,3,5,6] => ? = 1
[1,2,4,5,3,6,7] => [1,2,4,5,3,6,7] => [4,5,1,2,3,6,7] => ? = 1
[1,2,4,5,3,7,6] => [1,2,4,5,3,7,6] => [4,5,7,1,2,3,6] => ? = 2
[1,2,4,5,6,3,7] => [1,2,4,5,6,3,7] => [4,5,6,1,2,3,7] => ? = 1
[1,2,4,5,6,7,3] => [1,2,4,5,6,7,3] => [4,5,6,7,1,2,3] => ? = 1
[1,2,4,5,7,3,6] => [1,2,4,5,7,3,6] => [7,4,5,1,2,3,6] => ? = 1
[1,2,4,5,7,6,3] => [1,2,4,5,7,3,6] => [7,4,5,1,2,3,6] => ? = 1
[1,2,4,6,3,5,7] => [1,2,4,6,3,5,7] => [6,4,1,2,3,5,7] => ? = 1
[1,2,4,6,3,7,5] => [1,2,4,6,3,7,5] => [6,4,7,1,2,3,5] => ? = 2
[1,2,4,6,5,3,7] => [1,2,4,6,3,7,5] => [6,4,7,1,2,3,5] => ? = 2
[1,2,4,6,5,7,3] => [1,2,4,6,3,5,7] => [6,4,1,2,3,5,7] => ? = 1
[1,2,4,6,7,3,5] => [1,2,4,6,7,3,5] => [6,7,4,1,2,3,5] => ? = 1
[1,2,4,6,7,5,3] => [1,2,4,6,7,3,5] => [6,7,4,1,2,3,5] => ? = 1
[1,2,4,7,3,5,6] => [1,2,4,7,3,5,6] => [7,1,4,2,3,5,6] => ? = 1
[1,2,4,7,3,6,5] => [1,2,4,7,3,6,5] => [4,7,6,1,2,3,5] => ? = 2
[1,2,4,7,5,3,6] => [1,2,4,7,3,6,5] => [4,7,6,1,2,3,5] => ? = 2
[1,2,4,7,5,6,3] => [1,2,4,7,3,5,6] => [7,1,4,2,3,5,6] => ? = 1
[1,2,4,7,6,3,5] => [1,2,4,7,3,5,6] => [7,1,4,2,3,5,6] => ? = 1
[1,2,4,7,6,5,3] => [1,2,4,7,3,5,6] => [7,1,4,2,3,5,6] => ? = 1
[1,2,5,3,4,6,7] => [1,2,5,3,4,6,7] => [1,5,2,3,4,6,7] => ? = 1
[1,2,5,3,4,7,6] => [1,2,5,3,4,7,6] => [1,5,7,2,3,4,6] => ? = 1
Description
The number of factors of the Stanley symmetric function associated with a permutation.
For example, the Stanley symmetric function of $\pi=321645$ equals
$20 m_{1,1,1,1,1} + 11 m_{2,1,1,1} + 6 m_{2,2,1} + 4 m_{3,1,1} + 2 m_{3,2} + m_{4,1} = (m_{1,1} + m_{2})(2 m_{1,1,1} + m_{2,1}).$
Matching statistic: St000308
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000308: Permutations ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 67%
Mp00064: Permutations —reverse⟶ Permutations
St000308: Permutations ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 67%
Values
[1,2] => [1,2] => [2,1] => 1 = 0 + 1
[2,1] => [1,2] => [2,1] => 1 = 0 + 1
[1,2,3] => [1,2,3] => [3,2,1] => 1 = 0 + 1
[1,3,2] => [1,3,2] => [2,3,1] => 2 = 1 + 1
[2,1,3] => [1,3,2] => [2,3,1] => 2 = 1 + 1
[2,3,1] => [1,2,3] => [3,2,1] => 1 = 0 + 1
[3,1,2] => [1,2,3] => [3,2,1] => 1 = 0 + 1
[3,2,1] => [1,2,3] => [3,2,1] => 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [3,4,2,1] => 2 = 1 + 1
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 2 = 1 + 1
[1,3,4,2] => [1,3,4,2] => [2,4,3,1] => 2 = 1 + 1
[1,4,2,3] => [1,4,2,3] => [3,2,4,1] => 2 = 1 + 1
[1,4,3,2] => [1,4,2,3] => [3,2,4,1] => 2 = 1 + 1
[2,1,3,4] => [1,3,4,2] => [2,4,3,1] => 2 = 1 + 1
[2,1,4,3] => [1,4,2,3] => [3,2,4,1] => 2 = 1 + 1
[2,3,1,4] => [1,4,2,3] => [3,2,4,1] => 2 = 1 + 1
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => 1 = 0 + 1
[2,4,1,3] => [1,3,2,4] => [4,2,3,1] => 2 = 1 + 1
[2,4,3,1] => [1,2,4,3] => [3,4,2,1] => 2 = 1 + 1
[3,1,2,4] => [1,2,4,3] => [3,4,2,1] => 2 = 1 + 1
[3,1,4,2] => [1,4,2,3] => [3,2,4,1] => 2 = 1 + 1
[3,2,1,4] => [1,4,2,3] => [3,2,4,1] => 2 = 1 + 1
[3,2,4,1] => [1,2,4,3] => [3,4,2,1] => 2 = 1 + 1
[3,4,1,2] => [1,2,3,4] => [4,3,2,1] => 1 = 0 + 1
[3,4,2,1] => [1,2,3,4] => [4,3,2,1] => 1 = 0 + 1
[4,1,2,3] => [1,2,3,4] => [4,3,2,1] => 1 = 0 + 1
[4,1,3,2] => [1,3,2,4] => [4,2,3,1] => 2 = 1 + 1
[4,2,1,3] => [1,3,2,4] => [4,2,3,1] => 2 = 1 + 1
[4,2,3,1] => [1,2,3,4] => [4,3,2,1] => 1 = 0 + 1
[4,3,1,2] => [1,2,3,4] => [4,3,2,1] => 1 = 0 + 1
[4,3,2,1] => [1,2,3,4] => [4,3,2,1] => 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [4,5,3,2,1] => 2 = 1 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => 2 = 1 + 1
[1,2,4,5,3] => [1,2,4,5,3] => [3,5,4,2,1] => 2 = 1 + 1
[1,2,5,3,4] => [1,2,5,3,4] => [4,3,5,2,1] => 2 = 1 + 1
[1,2,5,4,3] => [1,2,5,3,4] => [4,3,5,2,1] => 2 = 1 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => 2 = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [4,5,2,3,1] => 2 = 1 + 1
[1,3,4,2,5] => [1,3,4,2,5] => [5,2,4,3,1] => 2 = 1 + 1
[1,3,4,5,2] => [1,3,4,5,2] => [2,5,4,3,1] => 2 = 1 + 1
[1,3,5,2,4] => [1,3,5,2,4] => [4,2,5,3,1] => 2 = 1 + 1
[1,3,5,4,2] => [1,3,5,2,4] => [4,2,5,3,1] => 2 = 1 + 1
[1,4,2,3,5] => [1,4,2,3,5] => [5,3,2,4,1] => 2 = 1 + 1
[1,4,2,5,3] => [1,4,2,5,3] => [3,5,2,4,1] => 2 = 1 + 1
[1,4,3,2,5] => [1,4,2,5,3] => [3,5,2,4,1] => 2 = 1 + 1
[1,4,3,5,2] => [1,4,2,3,5] => [5,3,2,4,1] => 2 = 1 + 1
[1,4,5,2,3] => [1,4,5,2,3] => [3,2,5,4,1] => 2 = 1 + 1
[1,4,5,3,2] => [1,4,5,2,3] => [3,2,5,4,1] => 2 = 1 + 1
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 0 + 1
[1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => ? = 1 + 1
[1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => ? = 1 + 1
[1,2,3,4,6,7,5] => [1,2,3,4,6,7,5] => [5,7,6,4,3,2,1] => ? = 1 + 1
[1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => ? = 1 + 1
[1,2,3,4,7,6,5] => [1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => ? = 1 + 1
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => ? = 1 + 1
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => ? = 2 + 1
[1,2,3,5,6,4,7] => [1,2,3,5,6,4,7] => [7,4,6,5,3,2,1] => ? = 1 + 1
[1,2,3,5,6,7,4] => [1,2,3,5,6,7,4] => [4,7,6,5,3,2,1] => ? = 1 + 1
[1,2,3,5,7,4,6] => [1,2,3,5,7,4,6] => [6,4,7,5,3,2,1] => ? = 1 + 1
[1,2,3,5,7,6,4] => [1,2,3,5,7,4,6] => [6,4,7,5,3,2,1] => ? = 1 + 1
[1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [7,5,4,6,3,2,1] => ? = 1 + 1
[1,2,3,6,4,7,5] => [1,2,3,6,4,7,5] => [5,7,4,6,3,2,1] => ? = 2 + 1
[1,2,3,6,5,4,7] => [1,2,3,6,4,7,5] => [5,7,4,6,3,2,1] => ? = 2 + 1
[1,2,3,6,5,7,4] => [1,2,3,6,4,5,7] => [7,5,4,6,3,2,1] => ? = 1 + 1
[1,2,3,6,7,4,5] => [1,2,3,6,7,4,5] => [5,4,7,6,3,2,1] => ? = 1 + 1
[1,2,3,6,7,5,4] => [1,2,3,6,7,4,5] => [5,4,7,6,3,2,1] => ? = 1 + 1
[1,2,3,7,4,5,6] => [1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => ? = 1 + 1
[1,2,3,7,4,6,5] => [1,2,3,7,4,6,5] => [5,6,4,7,3,2,1] => ? = 2 + 1
[1,2,3,7,5,4,6] => [1,2,3,7,4,6,5] => [5,6,4,7,3,2,1] => ? = 2 + 1
[1,2,3,7,5,6,4] => [1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => ? = 1 + 1
[1,2,3,7,6,4,5] => [1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => ? = 1 + 1
[1,2,3,7,6,5,4] => [1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => ? = 1 + 1
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [7,6,5,3,4,2,1] => ? = 1 + 1
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [6,7,5,3,4,2,1] => ? = 1 + 1
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [7,5,6,3,4,2,1] => ? = 1 + 1
[1,2,4,3,6,7,5] => [1,2,4,3,6,7,5] => [5,7,6,3,4,2,1] => ? = 1 + 1
[1,2,4,3,7,5,6] => [1,2,4,3,7,5,6] => [6,5,7,3,4,2,1] => ? = 1 + 1
[1,2,4,3,7,6,5] => [1,2,4,3,7,5,6] => [6,5,7,3,4,2,1] => ? = 1 + 1
[1,2,4,5,3,6,7] => [1,2,4,5,3,6,7] => [7,6,3,5,4,2,1] => ? = 1 + 1
[1,2,4,5,3,7,6] => [1,2,4,5,3,7,6] => [6,7,3,5,4,2,1] => ? = 2 + 1
[1,2,4,5,6,3,7] => [1,2,4,5,6,3,7] => [7,3,6,5,4,2,1] => ? = 1 + 1
[1,2,4,5,6,7,3] => [1,2,4,5,6,7,3] => [3,7,6,5,4,2,1] => ? = 1 + 1
[1,2,4,5,7,3,6] => [1,2,4,5,7,3,6] => [6,3,7,5,4,2,1] => ? = 1 + 1
[1,2,4,5,7,6,3] => [1,2,4,5,7,3,6] => [6,3,7,5,4,2,1] => ? = 1 + 1
[1,2,4,6,3,5,7] => [1,2,4,6,3,5,7] => [7,5,3,6,4,2,1] => ? = 1 + 1
[1,2,4,6,3,7,5] => [1,2,4,6,3,7,5] => [5,7,3,6,4,2,1] => ? = 2 + 1
[1,2,4,6,5,3,7] => [1,2,4,6,3,7,5] => [5,7,3,6,4,2,1] => ? = 2 + 1
[1,2,4,6,5,7,3] => [1,2,4,6,3,5,7] => [7,5,3,6,4,2,1] => ? = 1 + 1
[1,2,4,6,7,3,5] => [1,2,4,6,7,3,5] => [5,3,7,6,4,2,1] => ? = 1 + 1
[1,2,4,6,7,5,3] => [1,2,4,6,7,3,5] => [5,3,7,6,4,2,1] => ? = 1 + 1
[1,2,4,7,3,5,6] => [1,2,4,7,3,5,6] => [6,5,3,7,4,2,1] => ? = 1 + 1
[1,2,4,7,3,6,5] => [1,2,4,7,3,6,5] => [5,6,3,7,4,2,1] => ? = 2 + 1
[1,2,4,7,5,3,6] => [1,2,4,7,3,6,5] => [5,6,3,7,4,2,1] => ? = 2 + 1
[1,2,4,7,5,6,3] => [1,2,4,7,3,5,6] => [6,5,3,7,4,2,1] => ? = 1 + 1
[1,2,4,7,6,3,5] => [1,2,4,7,3,5,6] => [6,5,3,7,4,2,1] => ? = 1 + 1
[1,2,4,7,6,5,3] => [1,2,4,7,3,5,6] => [6,5,3,7,4,2,1] => ? = 1 + 1
[1,2,5,3,4,6,7] => [1,2,5,3,4,6,7] => [7,6,4,3,5,2,1] => ? = 1 + 1
[1,2,5,3,4,7,6] => [1,2,5,3,4,7,6] => [6,7,4,3,5,2,1] => ? = 1 + 1
Description
The height of the tree associated to a permutation.
A permutation can be mapped to a rooted tree with vertices $\{0,1,2,\ldots,n\}$ and root $0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1].
The statistic is given by the height of this tree.
See also [[St000325]] for the width of this tree.
The following 63 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001235The global dimension of the corresponding Comp-Nakayama algebra. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000387The matching number of a graph. St000486The number of cycles of length at least 3 of a permutation. St000535The rank-width of a graph. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001071The beta invariant of the graph. St001271The competition number of a graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001393The induced matching number of a graph. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001512The minimum rank of a graph. St001728The number of invisible descents of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000166The depth minus 1 of an ordered tree. St000258The burning number of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000328The maximum number of child nodes in a tree. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000918The 2-limited packing number of a graph. St001093The detour number of a graph. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001261The Castelnuovo-Mumford regularity of a graph. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001530The depth of a Dyck path. St001674The number of vertices of the largest induced star graph in the graph. St001734The lettericity of a graph. St000397The Strahler number of a rooted tree. St001330The hat guessing number of a graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000454The largest eigenvalue of a graph if it is integral. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001555The order of a signed permutation. St001488The number of corners of a skew partition. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001570The minimal number of edges to add to make a graph Hamiltonian. St000264The girth of a graph, which is not a tree. St001060The distinguishing index of a graph. St001896The number of right descents of a signed permutations. St001624The breadth of a lattice. St000699The toughness times the least common multiple of 1,. St000640The rank of the largest boolean interval in a poset. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001823The Stasinski-Voll length of a signed permutation. St001946The number of descents in a parking function. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001935The number of ascents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one. St001569The maximal modular displacement of a permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation.
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