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Your data matches 18 different statistics following compositions of up to 3 maps.
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Matching statistic: St000628
Mp00069: Permutations —complement⟶ Permutations
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%
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%
Values
[1,2] => [2,1] => [1,2] => 0 => 0
[2,1] => [1,2] => [1,2] => 0 => 0
[1,2,3] => [3,2,1] => [1,2,3] => 00 => 0
[1,3,2] => [3,1,2] => [1,2,3] => 00 => 0
[2,1,3] => [2,3,1] => [1,2,3] => 00 => 0
[2,3,1] => [2,1,3] => [1,3,2] => 01 => 1
[3,1,2] => [1,3,2] => [1,3,2] => 01 => 1
[3,2,1] => [1,2,3] => [1,2,3] => 00 => 0
[1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 000 => 0
[1,2,4,3] => [4,3,1,2] => [1,2,3,4] => 000 => 0
[1,3,2,4] => [4,2,3,1] => [1,2,3,4] => 000 => 0
[1,3,4,2] => [4,2,1,3] => [1,3,2,4] => 010 => 1
[1,4,2,3] => [4,1,3,2] => [1,3,2,4] => 010 => 1
[1,4,3,2] => [4,1,2,3] => [1,2,3,4] => 000 => 0
[2,1,3,4] => [3,4,2,1] => [1,2,3,4] => 000 => 0
[2,1,4,3] => [3,4,1,2] => [1,2,3,4] => 000 => 0
[2,3,1,4] => [3,2,4,1] => [1,2,4,3] => 001 => 1
[2,3,4,1] => [3,2,1,4] => [1,4,2,3] => 010 => 1
[2,4,1,3] => [3,1,4,2] => [1,4,2,3] => 010 => 1
[2,4,3,1] => [3,1,2,4] => [1,2,4,3] => 001 => 1
[3,1,2,4] => [2,4,3,1] => [1,2,4,3] => 001 => 1
[3,1,4,2] => [2,4,1,3] => [1,3,2,4] => 010 => 1
[3,2,1,4] => [2,3,4,1] => [1,2,3,4] => 000 => 0
[3,2,4,1] => [2,3,1,4] => [1,4,2,3] => 010 => 1
[3,4,1,2] => [2,1,4,3] => [1,4,2,3] => 010 => 1
[3,4,2,1] => [2,1,3,4] => [1,3,4,2] => 001 => 1
[4,1,2,3] => [1,4,3,2] => [1,4,2,3] => 010 => 1
[4,1,3,2] => [1,4,2,3] => [1,4,2,3] => 010 => 1
[4,2,1,3] => [1,3,4,2] => [1,3,4,2] => 001 => 1
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => 010 => 1
[4,3,1,2] => [1,2,4,3] => [1,2,4,3] => 001 => 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 000 => 0
[1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => 0000 => 0
[1,2,3,5,4] => [5,4,3,1,2] => [1,2,3,4,5] => 0000 => 0
[1,2,4,3,5] => [5,4,2,3,1] => [1,2,3,4,5] => 0000 => 0
[1,2,4,5,3] => [5,4,2,1,3] => [1,3,2,4,5] => 0100 => 1
[1,2,5,3,4] => [5,4,1,3,2] => [1,3,2,4,5] => 0100 => 1
[1,2,5,4,3] => [5,4,1,2,3] => [1,2,3,4,5] => 0000 => 0
[1,3,2,4,5] => [5,3,4,2,1] => [1,2,3,4,5] => 0000 => 0
[1,3,2,5,4] => [5,3,4,1,2] => [1,2,3,4,5] => 0000 => 0
[1,3,4,2,5] => [5,3,2,4,1] => [1,2,4,3,5] => 0010 => 1
[1,3,4,5,2] => [5,3,2,1,4] => [1,4,2,3,5] => 0100 => 1
[1,3,5,2,4] => [5,3,1,4,2] => [1,4,2,3,5] => 0100 => 1
[1,3,5,4,2] => [5,3,1,2,4] => [1,2,4,3,5] => 0010 => 1
[1,4,2,3,5] => [5,2,4,3,1] => [1,2,4,3,5] => 0010 => 1
[1,4,2,5,3] => [5,2,4,1,3] => [1,3,2,4,5] => 0100 => 1
[1,4,3,2,5] => [5,2,3,4,1] => [1,2,3,4,5] => 0000 => 0
[1,4,3,5,2] => [5,2,3,1,4] => [1,4,2,3,5] => 0100 => 1
[1,4,5,2,3] => [5,2,1,4,3] => [1,4,2,3,5] => 0100 => 1
[1,4,5,3,2] => [5,2,1,3,4] => [1,3,4,2,5] => 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: St001174
(load all 115 compositions to match this statistic)
(load all 115 compositions to match this statistic)
St001174: Permutations ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 67%
Values
[1,2] => 0
[2,1] => 0
[1,2,3] => 0
[1,3,2] => 0
[2,1,3] => 0
[2,3,1] => 1
[3,1,2] => 1
[3,2,1] => 0
[1,2,3,4] => 0
[1,2,4,3] => 0
[1,3,2,4] => 0
[1,3,4,2] => 1
[1,4,2,3] => 1
[1,4,3,2] => 0
[2,1,3,4] => 0
[2,1,4,3] => 0
[2,3,1,4] => 1
[2,3,4,1] => 1
[2,4,1,3] => 1
[2,4,3,1] => 1
[3,1,2,4] => 1
[3,1,4,2] => 1
[3,2,1,4] => 0
[3,2,4,1] => 1
[3,4,1,2] => 1
[3,4,2,1] => 1
[4,1,2,3] => 1
[4,1,3,2] => 1
[4,2,1,3] => 1
[4,2,3,1] => 1
[4,3,1,2] => 1
[4,3,2,1] => 0
[1,2,3,4,5] => 0
[1,2,3,5,4] => 0
[1,2,4,3,5] => 0
[1,2,4,5,3] => 1
[1,2,5,3,4] => 1
[1,2,5,4,3] => 0
[1,3,2,4,5] => 0
[1,3,2,5,4] => 0
[1,3,4,2,5] => 1
[1,3,4,5,2] => 1
[1,3,5,2,4] => 1
[1,3,5,4,2] => 1
[1,4,2,3,5] => 1
[1,4,2,5,3] => 1
[1,4,3,2,5] => 0
[1,4,3,5,2] => 1
[1,4,5,2,3] => 1
[1,4,5,3,2] => 1
[1,2,3,4,5,6,7] => ? = 0
[1,2,3,4,5,7,6] => ? = 0
[1,2,3,4,6,5,7] => ? = 0
[1,2,3,4,6,7,5] => ? = 1
[1,2,3,4,7,5,6] => ? = 1
[1,2,3,4,7,6,5] => ? = 0
[1,2,3,5,4,6,7] => ? = 0
[1,2,3,5,4,7,6] => ? = 0
[1,2,3,5,6,4,7] => ? = 1
[1,2,3,5,6,7,4] => ? = 1
[1,2,3,5,7,4,6] => ? = 1
[1,2,3,5,7,6,4] => ? = 1
[1,2,3,6,4,5,7] => ? = 1
[1,2,3,6,4,7,5] => ? = 1
[1,2,3,6,5,4,7] => ? = 0
[1,2,3,6,5,7,4] => ? = 1
[1,2,3,6,7,4,5] => ? = 1
[1,2,3,6,7,5,4] => ? = 1
[1,2,3,7,4,5,6] => ? = 1
[1,2,3,7,4,6,5] => ? = 1
[1,2,3,7,5,4,6] => ? = 1
[1,2,3,7,5,6,4] => ? = 1
[1,2,3,7,6,4,5] => ? = 1
[1,2,3,7,6,5,4] => ? = 0
[1,2,4,3,5,6,7] => ? = 0
[1,2,4,3,5,7,6] => ? = 0
[1,2,4,3,6,5,7] => ? = 0
[1,2,4,3,6,7,5] => ? = 1
[1,2,4,3,7,5,6] => ? = 1
[1,2,4,3,7,6,5] => ? = 0
[1,2,4,5,3,6,7] => ? = 1
[1,2,4,5,3,7,6] => ? = 1
[1,2,4,5,6,3,7] => ? = 1
[1,2,4,5,6,7,3] => ? = 1
[1,2,4,5,7,3,6] => ? = 1
[1,2,4,5,7,6,3] => ? = 1
[1,2,4,6,3,5,7] => ? = 1
[1,2,4,6,3,7,5] => ? = 1
[1,2,4,6,5,3,7] => ? = 1
[1,2,4,6,5,7,3] => ? = 1
[1,2,4,6,7,3,5] => ? = 1
[1,2,4,6,7,5,3] => ? = 1
[1,2,4,7,3,5,6] => ? = 1
[1,2,4,7,3,6,5] => ? = 1
[1,2,4,7,5,3,6] => ? = 1
[1,2,4,7,5,6,3] => ? = 1
[1,2,4,7,6,3,5] => ? = 1
[1,2,4,7,6,5,3] => ? = 1
[1,2,5,3,4,6,7] => ? = 1
[1,2,5,3,4,7,6] => ? = 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 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St001859: Permutations ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 67%
Mp00223: Permutations —runsort⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St001859: Permutations ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 67%
Values
[1,2] => [2,1] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 0
[1,2,3] => [3,2,1] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [2,3,1] => [1,2,3] => [1,2,3] => 0
[2,1,3] => [3,1,2] => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,3,2] => [1,3,2] => [3,1,2] => 1
[3,1,2] => [2,1,3] => [1,3,2] => [3,1,2] => 1
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [3,4,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,2,4] => [4,2,3,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,4,2] => [2,4,3,1] => [1,2,4,3] => [4,1,2,3] => 1
[1,4,2,3] => [3,2,4,1] => [1,2,4,3] => [4,1,2,3] => 1
[1,4,3,2] => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,3,4] => [4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,4,3] => [3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,1,4] => [4,1,3,2] => [1,3,2,4] => [3,1,2,4] => 1
[2,3,4,1] => [1,4,3,2] => [1,4,2,3] => [3,4,1,2] => 1
[2,4,1,3] => [3,1,4,2] => [1,4,2,3] => [3,4,1,2] => 1
[2,4,3,1] => [1,3,4,2] => [1,3,4,2] => [2,4,1,3] => 1
[3,1,2,4] => [4,2,1,3] => [1,3,2,4] => [3,1,2,4] => 1
[3,1,4,2] => [2,4,1,3] => [1,3,2,4] => [3,1,2,4] => 1
[3,2,1,4] => [4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 0
[3,2,4,1] => [1,4,2,3] => [1,4,2,3] => [3,4,1,2] => 1
[3,4,1,2] => [2,1,4,3] => [1,4,2,3] => [3,4,1,2] => 1
[3,4,2,1] => [1,2,4,3] => [1,2,4,3] => [4,1,2,3] => 1
[4,1,2,3] => [3,2,1,4] => [1,4,2,3] => [3,4,1,2] => 1
[4,1,3,2] => [2,3,1,4] => [1,4,2,3] => [3,4,1,2] => 1
[4,2,1,3] => [3,1,2,4] => [1,2,4,3] => [4,1,2,3] => 1
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [3,1,2,4] => 1
[4,3,1,2] => [2,1,3,4] => [1,3,4,2] => [2,4,1,3] => 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [4,5,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,3,5] => [5,3,4,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,5,3] => [3,5,4,2,1] => [1,2,3,5,4] => [5,1,2,3,4] => 1
[1,2,5,3,4] => [4,3,5,2,1] => [1,2,3,5,4] => [5,1,2,3,4] => 1
[1,2,5,4,3] => [3,4,5,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,4,5] => [5,4,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,5,4] => [4,5,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,2,5] => [5,2,4,3,1] => [1,2,4,3,5] => [4,1,2,3,5] => 1
[1,3,4,5,2] => [2,5,4,3,1] => [1,2,5,3,4] => [4,5,1,2,3] => 1
[1,3,5,2,4] => [4,2,5,3,1] => [1,2,5,3,4] => [4,5,1,2,3] => 1
[1,3,5,4,2] => [2,4,5,3,1] => [1,2,4,5,3] => [3,5,1,2,4] => 1
[1,4,2,3,5] => [5,3,2,4,1] => [1,2,4,3,5] => [4,1,2,3,5] => 1
[1,4,2,5,3] => [3,5,2,4,1] => [1,2,4,3,5] => [4,1,2,3,5] => 1
[1,4,3,2,5] => [5,2,3,4,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,4,3,5,2] => [2,5,3,4,1] => [1,2,5,3,4] => [4,5,1,2,3] => 1
[1,4,5,2,3] => [3,2,5,4,1] => [1,2,5,3,4] => [4,5,1,2,3] => 1
[1,4,5,3,2] => [2,3,5,4,1] => [1,2,3,5,4] => [5,1,2,3,4] => 1
[1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
[1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
[1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
[1,2,3,4,6,7,5] => [5,7,6,4,3,2,1] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ? = 1
[1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ? = 1
[1,2,3,4,7,6,5] => [5,6,7,4,3,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
[1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
[1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
[1,2,3,5,6,4,7] => [7,4,6,5,3,2,1] => [1,2,3,4,6,5,7] => [6,1,2,3,4,5,7] => ? = 1
[1,2,3,5,6,7,4] => [4,7,6,5,3,2,1] => [1,2,3,4,7,5,6] => [6,7,1,2,3,4,5] => ? = 1
[1,2,3,5,7,4,6] => [6,4,7,5,3,2,1] => [1,2,3,4,7,5,6] => [6,7,1,2,3,4,5] => ? = 1
[1,2,3,5,7,6,4] => [4,6,7,5,3,2,1] => [1,2,3,4,6,7,5] => [5,7,1,2,3,4,6] => ? = 1
[1,2,3,6,4,5,7] => [7,5,4,6,3,2,1] => [1,2,3,4,6,5,7] => [6,1,2,3,4,5,7] => ? = 1
[1,2,3,6,4,7,5] => [5,7,4,6,3,2,1] => [1,2,3,4,6,5,7] => [6,1,2,3,4,5,7] => ? = 1
[1,2,3,6,5,4,7] => [7,4,5,6,3,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
[1,2,3,6,5,7,4] => [4,7,5,6,3,2,1] => [1,2,3,4,7,5,6] => [6,7,1,2,3,4,5] => ? = 1
[1,2,3,6,7,4,5] => [5,4,7,6,3,2,1] => [1,2,3,4,7,5,6] => [6,7,1,2,3,4,5] => ? = 1
[1,2,3,6,7,5,4] => [4,5,7,6,3,2,1] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ? = 1
[1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => [1,2,3,4,7,5,6] => [6,7,1,2,3,4,5] => ? = 1
[1,2,3,7,4,6,5] => [5,6,4,7,3,2,1] => [1,2,3,4,7,5,6] => [6,7,1,2,3,4,5] => ? = 1
[1,2,3,7,5,4,6] => [6,4,5,7,3,2,1] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ? = 1
[1,2,3,7,5,6,4] => [4,6,5,7,3,2,1] => [1,2,3,4,6,5,7] => [6,1,2,3,4,5,7] => ? = 1
[1,2,3,7,6,4,5] => [5,4,6,7,3,2,1] => [1,2,3,4,6,7,5] => [5,7,1,2,3,4,6] => ? = 1
[1,2,3,7,6,5,4] => [4,5,6,7,3,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
[1,2,4,3,5,6,7] => [7,6,5,3,4,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
[1,2,4,3,5,7,6] => [6,7,5,3,4,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
[1,2,4,3,6,5,7] => [7,5,6,3,4,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
[1,2,4,3,6,7,5] => [5,7,6,3,4,2,1] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ? = 1
[1,2,4,3,7,5,6] => [6,5,7,3,4,2,1] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ? = 1
[1,2,4,3,7,6,5] => [5,6,7,3,4,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
[1,2,4,5,3,6,7] => [7,6,3,5,4,2,1] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => ? = 1
[1,2,4,5,3,7,6] => [6,7,3,5,4,2,1] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => ? = 1
[1,2,4,5,6,3,7] => [7,3,6,5,4,2,1] => [1,2,3,6,4,5,7] => [5,6,1,2,3,4,7] => ? = 1
[1,2,4,5,6,7,3] => [3,7,6,5,4,2,1] => [1,2,3,7,4,5,6] => [5,6,7,1,2,3,4] => ? = 1
[1,2,4,5,7,3,6] => [6,3,7,5,4,2,1] => [1,2,3,7,4,5,6] => [5,6,7,1,2,3,4] => ? = 1
[1,2,4,5,7,6,3] => [3,6,7,5,4,2,1] => [1,2,3,6,7,4,5] => [6,4,7,1,2,3,5] => ? = 1
[1,2,4,6,3,5,7] => [7,5,3,6,4,2,1] => [1,2,3,6,4,5,7] => [5,6,1,2,3,4,7] => ? = 1
[1,2,4,6,3,7,5] => [5,7,3,6,4,2,1] => [1,2,3,6,4,5,7] => [5,6,1,2,3,4,7] => ? = 1
[1,2,4,6,5,3,7] => [7,3,5,6,4,2,1] => [1,2,3,5,6,4,7] => [4,6,1,2,3,5,7] => ? = 1
[1,2,4,6,5,7,3] => [3,7,5,6,4,2,1] => [1,2,3,7,4,5,6] => [5,6,7,1,2,3,4] => ? = 1
[1,2,4,6,7,3,5] => [5,3,7,6,4,2,1] => [1,2,3,7,4,5,6] => [5,6,7,1,2,3,4] => ? = 1
[1,2,4,6,7,5,3] => [3,5,7,6,4,2,1] => [1,2,3,5,7,4,6] => [4,6,7,1,2,3,5] => ? = 1
[1,2,4,7,3,5,6] => [6,5,3,7,4,2,1] => [1,2,3,7,4,5,6] => [5,6,7,1,2,3,4] => ? = 1
[1,2,4,7,3,6,5] => [5,6,3,7,4,2,1] => [1,2,3,7,4,5,6] => [5,6,7,1,2,3,4] => ? = 1
[1,2,4,7,5,3,6] => [6,3,5,7,4,2,1] => [1,2,3,5,7,4,6] => [4,6,7,1,2,3,5] => ? = 1
[1,2,4,7,5,6,3] => [3,6,5,7,4,2,1] => [1,2,3,6,4,5,7] => [5,6,1,2,3,4,7] => ? = 1
[1,2,4,7,6,3,5] => [5,3,6,7,4,2,1] => [1,2,3,6,7,4,5] => [6,4,7,1,2,3,5] => ? = 1
[1,2,4,7,6,5,3] => [3,5,6,7,4,2,1] => [1,2,3,5,6,7,4] => [4,5,7,1,2,3,6] => ? = 1
[1,2,5,3,4,6,7] => [7,6,4,3,5,2,1] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => ? = 1
[1,2,5,3,4,7,6] => [6,7,4,3,5,2,1] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => ? = 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 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
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%
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%
Values
[1,2] => [2,1] => [1,2] => [2,1] => 1 = 0 + 1
[2,1] => [1,2] => [1,2] => [2,1] => 1 = 0 + 1
[1,2,3] => [3,2,1] => [1,2,3] => [3,2,1] => 1 = 0 + 1
[1,3,2] => [2,3,1] => [1,2,3] => [3,2,1] => 1 = 0 + 1
[2,1,3] => [3,1,2] => [1,2,3] => [3,2,1] => 1 = 0 + 1
[2,3,1] => [1,3,2] => [1,3,2] => [2,3,1] => 2 = 1 + 1
[3,1,2] => [2,1,3] => [1,3,2] => [2,3,1] => 2 = 1 + 1
[3,2,1] => [1,2,3] => [1,2,3] => [3,2,1] => 1 = 0 + 1
[1,2,3,4] => [4,3,2,1] => [1,2,3,4] => [4,3,2,1] => 1 = 0 + 1
[1,2,4,3] => [3,4,2,1] => [1,2,3,4] => [4,3,2,1] => 1 = 0 + 1
[1,3,2,4] => [4,2,3,1] => [1,2,3,4] => [4,3,2,1] => 1 = 0 + 1
[1,3,4,2] => [2,4,3,1] => [1,2,4,3] => [3,4,2,1] => 2 = 1 + 1
[1,4,2,3] => [3,2,4,1] => [1,2,4,3] => [3,4,2,1] => 2 = 1 + 1
[1,4,3,2] => [2,3,4,1] => [1,2,3,4] => [4,3,2,1] => 1 = 0 + 1
[2,1,3,4] => [4,3,1,2] => [1,2,3,4] => [4,3,2,1] => 1 = 0 + 1
[2,1,4,3] => [3,4,1,2] => [1,2,3,4] => [4,3,2,1] => 1 = 0 + 1
[2,3,1,4] => [4,1,3,2] => [1,3,2,4] => [4,2,3,1] => 2 = 1 + 1
[2,3,4,1] => [1,4,3,2] => [1,4,2,3] => [3,2,4,1] => 2 = 1 + 1
[2,4,1,3] => [3,1,4,2] => [1,4,2,3] => [3,2,4,1] => 2 = 1 + 1
[2,4,3,1] => [1,3,4,2] => [1,3,4,2] => [2,4,3,1] => 2 = 1 + 1
[3,1,2,4] => [4,2,1,3] => [1,3,2,4] => [4,2,3,1] => 2 = 1 + 1
[3,1,4,2] => [2,4,1,3] => [1,3,2,4] => [4,2,3,1] => 2 = 1 + 1
[3,2,1,4] => [4,1,2,3] => [1,2,3,4] => [4,3,2,1] => 1 = 0 + 1
[3,2,4,1] => [1,4,2,3] => [1,4,2,3] => [3,2,4,1] => 2 = 1 + 1
[3,4,1,2] => [2,1,4,3] => [1,4,2,3] => [3,2,4,1] => 2 = 1 + 1
[3,4,2,1] => [1,2,4,3] => [1,2,4,3] => [3,4,2,1] => 2 = 1 + 1
[4,1,2,3] => [3,2,1,4] => [1,4,2,3] => [3,2,4,1] => 2 = 1 + 1
[4,1,3,2] => [2,3,1,4] => [1,4,2,3] => [3,2,4,1] => 2 = 1 + 1
[4,2,1,3] => [3,1,2,4] => [1,2,4,3] => [3,4,2,1] => 2 = 1 + 1
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 2 = 1 + 1
[4,3,1,2] => [2,1,3,4] => [1,3,4,2] => [2,4,3,1] => 2 = 1 + 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 1 = 0 + 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => 1 = 0 + 1
[1,2,3,5,4] => [4,5,3,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => 1 = 0 + 1
[1,2,4,3,5] => [5,3,4,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => 1 = 0 + 1
[1,2,4,5,3] => [3,5,4,2,1] => [1,2,3,5,4] => [4,5,3,2,1] => 2 = 1 + 1
[1,2,5,3,4] => [4,3,5,2,1] => [1,2,3,5,4] => [4,5,3,2,1] => 2 = 1 + 1
[1,2,5,4,3] => [3,4,5,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => 1 = 0 + 1
[1,3,2,4,5] => [5,4,2,3,1] => [1,2,3,4,5] => [5,4,3,2,1] => 1 = 0 + 1
[1,3,2,5,4] => [4,5,2,3,1] => [1,2,3,4,5] => [5,4,3,2,1] => 1 = 0 + 1
[1,3,4,2,5] => [5,2,4,3,1] => [1,2,4,3,5] => [5,3,4,2,1] => 2 = 1 + 1
[1,3,4,5,2] => [2,5,4,3,1] => [1,2,5,3,4] => [4,3,5,2,1] => 2 = 1 + 1
[1,3,5,2,4] => [4,2,5,3,1] => [1,2,5,3,4] => [4,3,5,2,1] => 2 = 1 + 1
[1,3,5,4,2] => [2,4,5,3,1] => [1,2,4,5,3] => [3,5,4,2,1] => 2 = 1 + 1
[1,4,2,3,5] => [5,3,2,4,1] => [1,2,4,3,5] => [5,3,4,2,1] => 2 = 1 + 1
[1,4,2,5,3] => [3,5,2,4,1] => [1,2,4,3,5] => [5,3,4,2,1] => 2 = 1 + 1
[1,4,3,2,5] => [5,2,3,4,1] => [1,2,3,4,5] => [5,4,3,2,1] => 1 = 0 + 1
[1,4,3,5,2] => [2,5,3,4,1] => [1,2,5,3,4] => [4,3,5,2,1] => 2 = 1 + 1
[1,4,5,2,3] => [3,2,5,4,1] => [1,2,5,3,4] => [4,3,5,2,1] => 2 = 1 + 1
[1,4,5,3,2] => [2,3,5,4,1] => [1,2,3,5,4] => [4,5,3,2,1] => 2 = 1 + 1
[1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 0 + 1
[1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 0 + 1
[1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 0 + 1
[1,2,3,4,6,7,5] => [5,7,6,4,3,2,1] => [1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => ? = 1 + 1
[1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => [1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => ? = 1 + 1
[1,2,3,4,7,6,5] => [5,6,7,4,3,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 0 + 1
[1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 0 + 1
[1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 0 + 1
[1,2,3,5,6,4,7] => [7,4,6,5,3,2,1] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => ? = 1 + 1
[1,2,3,5,6,7,4] => [4,7,6,5,3,2,1] => [1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => ? = 1 + 1
[1,2,3,5,7,4,6] => [6,4,7,5,3,2,1] => [1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => ? = 1 + 1
[1,2,3,5,7,6,4] => [4,6,7,5,3,2,1] => [1,2,3,4,6,7,5] => [5,7,6,4,3,2,1] => ? = 1 + 1
[1,2,3,6,4,5,7] => [7,5,4,6,3,2,1] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => ? = 1 + 1
[1,2,3,6,4,7,5] => [5,7,4,6,3,2,1] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => ? = 1 + 1
[1,2,3,6,5,4,7] => [7,4,5,6,3,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 0 + 1
[1,2,3,6,5,7,4] => [4,7,5,6,3,2,1] => [1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => ? = 1 + 1
[1,2,3,6,7,4,5] => [5,4,7,6,3,2,1] => [1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => ? = 1 + 1
[1,2,3,6,7,5,4] => [4,5,7,6,3,2,1] => [1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => ? = 1 + 1
[1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => [1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => ? = 1 + 1
[1,2,3,7,4,6,5] => [5,6,4,7,3,2,1] => [1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => ? = 1 + 1
[1,2,3,7,5,4,6] => [6,4,5,7,3,2,1] => [1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => ? = 1 + 1
[1,2,3,7,5,6,4] => [4,6,5,7,3,2,1] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => ? = 1 + 1
[1,2,3,7,6,4,5] => [5,4,6,7,3,2,1] => [1,2,3,4,6,7,5] => [5,7,6,4,3,2,1] => ? = 1 + 1
[1,2,3,7,6,5,4] => [4,5,6,7,3,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 0 + 1
[1,2,4,3,5,6,7] => [7,6,5,3,4,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 0 + 1
[1,2,4,3,5,7,6] => [6,7,5,3,4,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 0 + 1
[1,2,4,3,6,5,7] => [7,5,6,3,4,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 0 + 1
[1,2,4,3,6,7,5] => [5,7,6,3,4,2,1] => [1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => ? = 1 + 1
[1,2,4,3,7,5,6] => [6,5,7,3,4,2,1] => [1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => ? = 1 + 1
[1,2,4,3,7,6,5] => [5,6,7,3,4,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 0 + 1
[1,2,4,5,3,6,7] => [7,6,3,5,4,2,1] => [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => ? = 1 + 1
[1,2,4,5,3,7,6] => [6,7,3,5,4,2,1] => [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => ? = 1 + 1
[1,2,4,5,6,3,7] => [7,3,6,5,4,2,1] => [1,2,3,6,4,5,7] => [7,5,4,6,3,2,1] => ? = 1 + 1
[1,2,4,5,6,7,3] => [3,7,6,5,4,2,1] => [1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => ? = 1 + 1
[1,2,4,5,7,3,6] => [6,3,7,5,4,2,1] => [1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => ? = 1 + 1
[1,2,4,5,7,6,3] => [3,6,7,5,4,2,1] => [1,2,3,6,7,4,5] => [5,4,7,6,3,2,1] => ? = 1 + 1
[1,2,4,6,3,5,7] => [7,5,3,6,4,2,1] => [1,2,3,6,4,5,7] => [7,5,4,6,3,2,1] => ? = 1 + 1
[1,2,4,6,3,7,5] => [5,7,3,6,4,2,1] => [1,2,3,6,4,5,7] => [7,5,4,6,3,2,1] => ? = 1 + 1
[1,2,4,6,5,3,7] => [7,3,5,6,4,2,1] => [1,2,3,5,6,4,7] => [7,4,6,5,3,2,1] => ? = 1 + 1
[1,2,4,6,5,7,3] => [3,7,5,6,4,2,1] => [1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => ? = 1 + 1
[1,2,4,6,7,3,5] => [5,3,7,6,4,2,1] => [1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => ? = 1 + 1
[1,2,4,6,7,5,3] => [3,5,7,6,4,2,1] => [1,2,3,5,7,4,6] => [6,4,7,5,3,2,1] => ? = 1 + 1
[1,2,4,7,3,5,6] => [6,5,3,7,4,2,1] => [1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => ? = 1 + 1
[1,2,4,7,3,6,5] => [5,6,3,7,4,2,1] => [1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => ? = 1 + 1
[1,2,4,7,5,3,6] => [6,3,5,7,4,2,1] => [1,2,3,5,7,4,6] => [6,4,7,5,3,2,1] => ? = 1 + 1
[1,2,4,7,5,6,3] => [3,6,5,7,4,2,1] => [1,2,3,6,4,5,7] => [7,5,4,6,3,2,1] => ? = 1 + 1
[1,2,4,7,6,3,5] => [5,3,6,7,4,2,1] => [1,2,3,6,7,4,5] => [5,4,7,6,3,2,1] => ? = 1 + 1
[1,2,4,7,6,5,3] => [3,5,6,7,4,2,1] => [1,2,3,5,6,7,4] => [4,7,6,5,3,2,1] => ? = 1 + 1
[1,2,5,3,4,6,7] => [7,6,4,3,5,2,1] => [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => ? = 1 + 1
[1,2,5,3,4,7,6] => [6,7,4,3,5,2,1] => [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => ? = 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.
Matching statistic: St001235
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St001235: Integer compositions ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 67%
Mp00223: Permutations —runsort⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St001235: Integer compositions ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 67%
Values
[1,2] => [2,1] => [1,2] => [2] => 1 = 0 + 1
[2,1] => [1,2] => [1,2] => [2] => 1 = 0 + 1
[1,2,3] => [3,2,1] => [1,2,3] => [3] => 1 = 0 + 1
[1,3,2] => [2,3,1] => [1,2,3] => [3] => 1 = 0 + 1
[2,1,3] => [3,1,2] => [1,2,3] => [3] => 1 = 0 + 1
[2,3,1] => [1,3,2] => [1,3,2] => [2,1] => 2 = 1 + 1
[3,1,2] => [2,1,3] => [1,3,2] => [2,1] => 2 = 1 + 1
[3,2,1] => [1,2,3] => [1,2,3] => [3] => 1 = 0 + 1
[1,2,3,4] => [4,3,2,1] => [1,2,3,4] => [4] => 1 = 0 + 1
[1,2,4,3] => [3,4,2,1] => [1,2,3,4] => [4] => 1 = 0 + 1
[1,3,2,4] => [4,2,3,1] => [1,2,3,4] => [4] => 1 = 0 + 1
[1,3,4,2] => [2,4,3,1] => [1,2,4,3] => [3,1] => 2 = 1 + 1
[1,4,2,3] => [3,2,4,1] => [1,2,4,3] => [3,1] => 2 = 1 + 1
[1,4,3,2] => [2,3,4,1] => [1,2,3,4] => [4] => 1 = 0 + 1
[2,1,3,4] => [4,3,1,2] => [1,2,3,4] => [4] => 1 = 0 + 1
[2,1,4,3] => [3,4,1,2] => [1,2,3,4] => [4] => 1 = 0 + 1
[2,3,1,4] => [4,1,3,2] => [1,3,2,4] => [2,2] => 2 = 1 + 1
[2,3,4,1] => [1,4,3,2] => [1,4,2,3] => [2,2] => 2 = 1 + 1
[2,4,1,3] => [3,1,4,2] => [1,4,2,3] => [2,2] => 2 = 1 + 1
[2,4,3,1] => [1,3,4,2] => [1,3,4,2] => [3,1] => 2 = 1 + 1
[3,1,2,4] => [4,2,1,3] => [1,3,2,4] => [2,2] => 2 = 1 + 1
[3,1,4,2] => [2,4,1,3] => [1,3,2,4] => [2,2] => 2 = 1 + 1
[3,2,1,4] => [4,1,2,3] => [1,2,3,4] => [4] => 1 = 0 + 1
[3,2,4,1] => [1,4,2,3] => [1,4,2,3] => [2,2] => 2 = 1 + 1
[3,4,1,2] => [2,1,4,3] => [1,4,2,3] => [2,2] => 2 = 1 + 1
[3,4,2,1] => [1,2,4,3] => [1,2,4,3] => [3,1] => 2 = 1 + 1
[4,1,2,3] => [3,2,1,4] => [1,4,2,3] => [2,2] => 2 = 1 + 1
[4,1,3,2] => [2,3,1,4] => [1,4,2,3] => [2,2] => 2 = 1 + 1
[4,2,1,3] => [3,1,2,4] => [1,2,4,3] => [3,1] => 2 = 1 + 1
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [2,2] => 2 = 1 + 1
[4,3,1,2] => [2,1,3,4] => [1,3,4,2] => [3,1] => 2 = 1 + 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [4] => 1 = 0 + 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => [5] => 1 = 0 + 1
[1,2,3,5,4] => [4,5,3,2,1] => [1,2,3,4,5] => [5] => 1 = 0 + 1
[1,2,4,3,5] => [5,3,4,2,1] => [1,2,3,4,5] => [5] => 1 = 0 + 1
[1,2,4,5,3] => [3,5,4,2,1] => [1,2,3,5,4] => [4,1] => 2 = 1 + 1
[1,2,5,3,4] => [4,3,5,2,1] => [1,2,3,5,4] => [4,1] => 2 = 1 + 1
[1,2,5,4,3] => [3,4,5,2,1] => [1,2,3,4,5] => [5] => 1 = 0 + 1
[1,3,2,4,5] => [5,4,2,3,1] => [1,2,3,4,5] => [5] => 1 = 0 + 1
[1,3,2,5,4] => [4,5,2,3,1] => [1,2,3,4,5] => [5] => 1 = 0 + 1
[1,3,4,2,5] => [5,2,4,3,1] => [1,2,4,3,5] => [3,2] => 2 = 1 + 1
[1,3,4,5,2] => [2,5,4,3,1] => [1,2,5,3,4] => [3,2] => 2 = 1 + 1
[1,3,5,2,4] => [4,2,5,3,1] => [1,2,5,3,4] => [3,2] => 2 = 1 + 1
[1,3,5,4,2] => [2,4,5,3,1] => [1,2,4,5,3] => [4,1] => 2 = 1 + 1
[1,4,2,3,5] => [5,3,2,4,1] => [1,2,4,3,5] => [3,2] => 2 = 1 + 1
[1,4,2,5,3] => [3,5,2,4,1] => [1,2,4,3,5] => [3,2] => 2 = 1 + 1
[1,4,3,2,5] => [5,2,3,4,1] => [1,2,3,4,5] => [5] => 1 = 0 + 1
[1,4,3,5,2] => [2,5,3,4,1] => [1,2,5,3,4] => [3,2] => 2 = 1 + 1
[1,4,5,2,3] => [3,2,5,4,1] => [1,2,5,3,4] => [3,2] => 2 = 1 + 1
[1,4,5,3,2] => [2,3,5,4,1] => [1,2,3,5,4] => [4,1] => 2 = 1 + 1
[1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => [7] => ? = 0 + 1
[1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => [1,2,3,4,5,6,7] => [7] => ? = 0 + 1
[1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [1,2,3,4,5,6,7] => [7] => ? = 0 + 1
[1,2,3,4,6,7,5] => [5,7,6,4,3,2,1] => [1,2,3,4,5,7,6] => [6,1] => ? = 1 + 1
[1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => [1,2,3,4,5,7,6] => [6,1] => ? = 1 + 1
[1,2,3,4,7,6,5] => [5,6,7,4,3,2,1] => [1,2,3,4,5,6,7] => [7] => ? = 0 + 1
[1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => [1,2,3,4,5,6,7] => [7] => ? = 0 + 1
[1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => [1,2,3,4,5,6,7] => [7] => ? = 0 + 1
[1,2,3,5,6,4,7] => [7,4,6,5,3,2,1] => [1,2,3,4,6,5,7] => [5,2] => ? = 1 + 1
[1,2,3,5,6,7,4] => [4,7,6,5,3,2,1] => [1,2,3,4,7,5,6] => [5,2] => ? = 1 + 1
[1,2,3,5,7,4,6] => [6,4,7,5,3,2,1] => [1,2,3,4,7,5,6] => [5,2] => ? = 1 + 1
[1,2,3,5,7,6,4] => [4,6,7,5,3,2,1] => [1,2,3,4,6,7,5] => [6,1] => ? = 1 + 1
[1,2,3,6,4,5,7] => [7,5,4,6,3,2,1] => [1,2,3,4,6,5,7] => [5,2] => ? = 1 + 1
[1,2,3,6,4,7,5] => [5,7,4,6,3,2,1] => [1,2,3,4,6,5,7] => [5,2] => ? = 1 + 1
[1,2,3,6,5,4,7] => [7,4,5,6,3,2,1] => [1,2,3,4,5,6,7] => [7] => ? = 0 + 1
[1,2,3,6,5,7,4] => [4,7,5,6,3,2,1] => [1,2,3,4,7,5,6] => [5,2] => ? = 1 + 1
[1,2,3,6,7,4,5] => [5,4,7,6,3,2,1] => [1,2,3,4,7,5,6] => [5,2] => ? = 1 + 1
[1,2,3,6,7,5,4] => [4,5,7,6,3,2,1] => [1,2,3,4,5,7,6] => [6,1] => ? = 1 + 1
[1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => [1,2,3,4,7,5,6] => [5,2] => ? = 1 + 1
[1,2,3,7,4,6,5] => [5,6,4,7,3,2,1] => [1,2,3,4,7,5,6] => [5,2] => ? = 1 + 1
[1,2,3,7,5,4,6] => [6,4,5,7,3,2,1] => [1,2,3,4,5,7,6] => [6,1] => ? = 1 + 1
[1,2,3,7,5,6,4] => [4,6,5,7,3,2,1] => [1,2,3,4,6,5,7] => [5,2] => ? = 1 + 1
[1,2,3,7,6,4,5] => [5,4,6,7,3,2,1] => [1,2,3,4,6,7,5] => [6,1] => ? = 1 + 1
[1,2,3,7,6,5,4] => [4,5,6,7,3,2,1] => [1,2,3,4,5,6,7] => [7] => ? = 0 + 1
[1,2,4,3,5,6,7] => [7,6,5,3,4,2,1] => [1,2,3,4,5,6,7] => [7] => ? = 0 + 1
[1,2,4,3,5,7,6] => [6,7,5,3,4,2,1] => [1,2,3,4,5,6,7] => [7] => ? = 0 + 1
[1,2,4,3,6,5,7] => [7,5,6,3,4,2,1] => [1,2,3,4,5,6,7] => [7] => ? = 0 + 1
[1,2,4,3,6,7,5] => [5,7,6,3,4,2,1] => [1,2,3,4,5,7,6] => [6,1] => ? = 1 + 1
[1,2,4,3,7,5,6] => [6,5,7,3,4,2,1] => [1,2,3,4,5,7,6] => [6,1] => ? = 1 + 1
[1,2,4,3,7,6,5] => [5,6,7,3,4,2,1] => [1,2,3,4,5,6,7] => [7] => ? = 0 + 1
[1,2,4,5,3,6,7] => [7,6,3,5,4,2,1] => [1,2,3,5,4,6,7] => [4,3] => ? = 1 + 1
[1,2,4,5,3,7,6] => [6,7,3,5,4,2,1] => [1,2,3,5,4,6,7] => [4,3] => ? = 1 + 1
[1,2,4,5,6,3,7] => [7,3,6,5,4,2,1] => [1,2,3,6,4,5,7] => [4,3] => ? = 1 + 1
[1,2,4,5,6,7,3] => [3,7,6,5,4,2,1] => [1,2,3,7,4,5,6] => [4,3] => ? = 1 + 1
[1,2,4,5,7,3,6] => [6,3,7,5,4,2,1] => [1,2,3,7,4,5,6] => [4,3] => ? = 1 + 1
[1,2,4,5,7,6,3] => [3,6,7,5,4,2,1] => [1,2,3,6,7,4,5] => [5,2] => ? = 1 + 1
[1,2,4,6,3,5,7] => [7,5,3,6,4,2,1] => [1,2,3,6,4,5,7] => [4,3] => ? = 1 + 1
[1,2,4,6,3,7,5] => [5,7,3,6,4,2,1] => [1,2,3,6,4,5,7] => [4,3] => ? = 1 + 1
[1,2,4,6,5,3,7] => [7,3,5,6,4,2,1] => [1,2,3,5,6,4,7] => [5,2] => ? = 1 + 1
[1,2,4,6,5,7,3] => [3,7,5,6,4,2,1] => [1,2,3,7,4,5,6] => [4,3] => ? = 1 + 1
[1,2,4,6,7,3,5] => [5,3,7,6,4,2,1] => [1,2,3,7,4,5,6] => [4,3] => ? = 1 + 1
[1,2,4,6,7,5,3] => [3,5,7,6,4,2,1] => [1,2,3,5,7,4,6] => [5,2] => ? = 1 + 1
[1,2,4,7,3,5,6] => [6,5,3,7,4,2,1] => [1,2,3,7,4,5,6] => [4,3] => ? = 1 + 1
[1,2,4,7,3,6,5] => [5,6,3,7,4,2,1] => [1,2,3,7,4,5,6] => [4,3] => ? = 1 + 1
[1,2,4,7,5,3,6] => [6,3,5,7,4,2,1] => [1,2,3,5,7,4,6] => [5,2] => ? = 1 + 1
[1,2,4,7,5,6,3] => [3,6,5,7,4,2,1] => [1,2,3,6,4,5,7] => [4,3] => ? = 1 + 1
[1,2,4,7,6,3,5] => [5,3,6,7,4,2,1] => [1,2,3,6,7,4,5] => [5,2] => ? = 1 + 1
[1,2,4,7,6,5,3] => [3,5,6,7,4,2,1] => [1,2,3,5,6,7,4] => [6,1] => ? = 1 + 1
[1,2,5,3,4,6,7] => [7,6,4,3,5,2,1] => [1,2,3,5,4,6,7] => [4,3] => ? = 1 + 1
[1,2,5,3,4,7,6] => [6,7,4,3,5,2,1] => [1,2,3,5,4,6,7] => [4,3] => ? = 1 + 1
Description
The global dimension of the corresponding Comp-Nakayama algebra.
We identify the composition [n1-1,n2-1,...,nr-1] with the Nakayama algebra with Kupisch series [n1,n1-1,...,2,n2,n2-1,...,2,...,nr,nr-1,...,3,2,1]. We call such Nakayama algebras with Kupisch series corresponding to a integer composition "Comp-Nakayama algebra".
Matching statistic: St001570
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00237: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St001570: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 33%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St001570: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 33%
Values
[1,2] => [1,2] => ([],2)
=> ([],1)
=> ? = 0 - 1
[2,1] => [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 0 - 1
[1,2,3] => [1,2,3] => ([],3)
=> ([],1)
=> ? = 0 - 1
[1,3,2] => [1,3,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 0 - 1
[2,1,3] => [2,1,3] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 0 - 1
[2,3,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? = 1 - 1
[3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? = 0 - 1
[1,2,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? = 0 - 1
[1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 0 - 1
[1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 0 - 1
[1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 - 1
[1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 0 - 1
[2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 0 - 1
[2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> ? = 0 - 1
[2,3,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[2,3,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[2,4,3,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[3,1,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 - 1
[3,1,4,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ? = 1 - 1
[3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 0 - 1
[3,2,4,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[3,4,2,1] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[4,1,2,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 - 1
[4,1,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 - 1
[4,2,3,1] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[4,3,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 - 1
[4,3,2,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 0 - 1
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? = 0 - 1
[1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 0 - 1
[1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 0 - 1
[1,2,4,5,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[1,2,5,3,4] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
[1,2,5,4,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 0 - 1
[1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 0 - 1
[1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ? = 0 - 1
[1,3,4,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[1,3,4,5,2] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[1,3,5,2,4] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[1,3,5,4,2] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[1,4,2,3,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
[1,4,2,5,3] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
[1,4,3,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 0 - 1
[1,4,3,5,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,4,5,2,3] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[1,4,5,3,2] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[1,5,2,3,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
[1,5,2,4,3] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[1,5,3,2,4] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
[1,5,3,4,2] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[1,5,4,2,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
[1,5,4,3,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 0 - 1
[2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 0 - 1
[2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ? = 0 - 1
[2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ? = 0 - 1
[2,1,4,5,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[2,1,5,3,4] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
[2,1,5,4,3] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 0 - 1
[2,3,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[2,3,1,5,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[2,3,4,1,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[2,3,4,5,1] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[2,3,5,1,4] => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[2,3,5,4,1] => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[2,4,1,3,5] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[2,4,1,5,3] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[2,4,3,1,5] => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[2,4,3,5,1] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[2,4,5,1,3] => [5,2,1,4,3] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[2,4,5,3,1] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[2,5,1,3,4] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[2,5,1,4,3] => [5,2,4,1,3] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[2,5,3,1,4] => [3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[2,5,3,4,1] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[2,5,4,1,3] => [4,2,1,5,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[2,5,4,3,1] => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[3,1,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
[3,1,2,5,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
[3,1,4,2,5] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
[3,1,4,5,2] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[3,1,5,2,4] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
[3,1,5,4,2] => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
[3,2,1,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 0 - 1
[3,2,1,5,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 0 - 1
[3,2,4,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[3,2,4,5,1] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[3,2,5,1,4] => [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[3,2,5,4,1] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[3,4,1,2,5] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[3,4,1,5,2] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[3,4,2,1,5] => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[3,4,2,5,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[3,4,5,1,2] => [5,1,3,4,2] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[4,1,2,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
[4,1,2,5,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
[4,2,1,3,5] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
Description
The minimal number of edges to add to make a graph Hamiltonian.
A graph is Hamiltonian if it contains a cycle as a subgraph, which contains all vertices.
Matching statistic: St000264
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00237: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 33%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 33%
Values
[1,2] => [1,2] => ([],2)
=> ([],1)
=> ? = 0 + 2
[2,1] => [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 0 + 2
[1,2,3] => [1,2,3] => ([],3)
=> ([],1)
=> ? = 0 + 2
[1,3,2] => [1,3,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 0 + 2
[2,1,3] => [2,1,3] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 0 + 2
[2,3,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? = 1 + 2
[3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? = 0 + 2
[1,2,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? = 0 + 2
[1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 0 + 2
[1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 0 + 2
[1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 2
[1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 0 + 2
[2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 0 + 2
[2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> ? = 0 + 2
[2,3,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[2,3,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[2,4,3,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[3,1,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 2
[3,1,4,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 2
[3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 0 + 2
[3,2,4,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[3,4,2,1] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[4,1,2,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 2
[4,1,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 2
[4,2,3,1] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[4,3,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 2
[4,3,2,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 0 + 2
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? = 0 + 2
[1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 0 + 2
[1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 0 + 2
[1,2,4,5,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[1,2,5,3,4] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 2
[1,2,5,4,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 0 + 2
[1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 0 + 2
[1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ? = 0 + 2
[1,3,4,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[1,3,4,5,2] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[1,3,5,2,4] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[1,3,5,4,2] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[1,4,2,3,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 2
[1,4,2,5,3] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 2
[1,4,3,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 0 + 2
[1,4,3,5,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[1,4,5,2,3] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[1,4,5,3,2] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[1,5,2,3,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 2
[1,5,2,4,3] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[1,5,3,2,4] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 2
[1,5,3,4,2] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[1,5,4,2,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 2
[1,5,4,3,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 0 + 2
[2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 0 + 2
[2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ? = 0 + 2
[2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ? = 0 + 2
[2,1,4,5,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[2,1,5,3,4] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 2
[2,1,5,4,3] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 0 + 2
[2,3,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[2,3,1,5,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[2,3,4,1,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[2,3,4,5,1] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[2,3,5,1,4] => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[2,3,5,4,1] => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[2,4,1,3,5] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[2,4,1,5,3] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[2,4,3,1,5] => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[2,4,3,5,1] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[2,4,5,1,3] => [5,2,1,4,3] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[2,4,5,3,1] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[2,5,1,3,4] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[2,5,1,4,3] => [5,2,4,1,3] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[2,5,3,1,4] => [3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[2,5,3,4,1] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[2,5,4,1,3] => [4,2,1,5,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[2,5,4,3,1] => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[3,1,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 2
[3,1,2,5,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 2
[3,1,4,2,5] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 2
[3,1,4,5,2] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[3,1,5,2,4] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 2
[3,1,5,4,2] => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 2
[3,2,1,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 0 + 2
[3,2,1,5,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 0 + 2
[3,2,4,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[3,2,4,5,1] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[3,2,5,1,4] => [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[3,2,5,4,1] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[3,4,1,2,5] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[3,4,1,5,2] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[3,4,2,1,5] => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[3,4,2,5,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[3,4,5,1,2] => [5,1,3,4,2] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[4,1,2,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 2
[4,1,2,5,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 2
[4,2,1,3,5] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 2
Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
Matching statistic: St000298
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000298: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 67%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000298: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 67%
Values
[1,2] => [1,2] => [1,0,1,0]
=> ([(0,1)],2)
=> 1 = 0 + 1
[2,1] => [2,1] => [1,1,0,0]
=> ([(0,1)],2)
=> 1 = 0 + 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[2,3,1] => [3,2,1] => [1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,2,1] => [2,3,1] => [1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 1 + 1
[1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 1 + 1
[1,4,3,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[2,3,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[2,4,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[2,4,3,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[3,1,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[3,1,4,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[3,2,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[3,2,4,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[3,4,1,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[3,4,2,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 1 + 1
[4,1,2,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[4,1,3,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[4,2,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 1 + 1
[4,2,3,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[4,3,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[4,3,2,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,4,5,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 2 = 1 + 1
[1,2,5,3,4] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 2 = 1 + 1
[1,2,5,4,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 1 + 1
[1,3,4,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[1,3,5,2,4] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[1,3,5,4,2] => [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 1 + 1
[1,4,2,3,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 1 + 1
[1,4,2,5,3] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[1,4,3,2,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,4,3,5,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[1,4,5,2,3] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[1,4,5,3,2] => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 2 = 1 + 1
[1,5,2,3,4] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[1,5,2,4,3] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[1,5,3,2,4] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 2 = 1 + 1
[1,5,3,4,2] => [1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[1,5,4,2,3] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 1 + 1
[1,5,4,3,2] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,4,1,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[2,3,4,5,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[2,3,5,1,4] => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[2,3,5,4,1] => [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[2,4,1,3,5] => [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[2,4,1,5,3] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 1
[2,4,3,5,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[2,4,5,1,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[2,5,1,3,4] => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[2,5,1,4,3] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[2,5,3,4,1] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 1
[2,5,4,1,3] => [4,2,1,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[3,1,4,2,5] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[3,1,4,5,2] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 1
[3,1,5,2,4] => [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 1
[3,1,5,4,2] => [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[3,2,4,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[3,2,4,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[3,2,5,1,4] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[3,2,5,4,1] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[3,4,1,2,5] => [4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[3,4,1,5,2] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[3,4,2,5,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[3,4,5,1,2] => [5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[3,4,5,2,1] => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[3,5,1,2,4] => [5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[3,5,1,4,2] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[3,5,2,1,4] => [2,5,3,1,4] => [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[3,5,2,4,1] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[3,5,4,1,2] => [4,1,3,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[4,1,2,3,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[4,1,2,5,3] => [4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 1
[4,1,3,2,5] => [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[4,1,3,5,2] => [4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[4,1,5,2,3] => [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[4,1,5,3,2] => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 1
[4,2,1,5,3] => [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[4,2,3,1,5] => [3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[4,2,3,5,1] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[4,2,5,1,3] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[4,2,5,3,1] => [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 1
[4,3,1,5,2] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
Description
The order dimension or Dushnik-Miller dimension of a poset.
This is the minimal number of linear orderings whose intersection is the given poset.
Matching statistic: St000845
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000845: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 67%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000845: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 67%
Values
[1,2] => [1,2] => [1,0,1,0]
=> ([(0,1)],2)
=> 1 = 0 + 1
[2,1] => [2,1] => [1,1,0,0]
=> ([(0,1)],2)
=> 1 = 0 + 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[2,3,1] => [3,2,1] => [1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,2,1] => [2,3,1] => [1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 1 + 1
[1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 1 + 1
[1,4,3,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[2,3,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[2,4,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[2,4,3,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[3,1,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[3,1,4,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[3,2,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[3,2,4,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[3,4,1,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[3,4,2,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 1 + 1
[4,1,2,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[4,1,3,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[4,2,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 1 + 1
[4,2,3,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[4,3,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[4,3,2,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,4,5,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 2 = 1 + 1
[1,2,5,3,4] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 2 = 1 + 1
[1,2,5,4,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 1 + 1
[1,3,4,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[1,3,5,2,4] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[1,3,5,4,2] => [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 1 + 1
[1,4,2,3,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 1 + 1
[1,4,2,5,3] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[1,4,3,2,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,4,3,5,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[1,4,5,2,3] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[1,4,5,3,2] => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 2 = 1 + 1
[1,5,2,3,4] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[1,5,2,4,3] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[1,5,3,2,4] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 2 = 1 + 1
[1,5,3,4,2] => [1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[1,5,4,2,3] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 1 + 1
[1,5,4,3,2] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,4,1,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[2,3,4,5,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[2,3,5,1,4] => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[2,3,5,4,1] => [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[2,4,1,3,5] => [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[2,4,1,5,3] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 1
[2,4,3,5,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[2,4,5,1,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[2,5,1,3,4] => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[2,5,1,4,3] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[2,5,3,4,1] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 1
[2,5,4,1,3] => [4,2,1,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[3,1,4,2,5] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[3,1,4,5,2] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 1
[3,1,5,2,4] => [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 1
[3,1,5,4,2] => [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[3,2,4,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[3,2,4,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[3,2,5,1,4] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[3,2,5,4,1] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[3,4,1,2,5] => [4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[3,4,1,5,2] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[3,4,2,5,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[3,4,5,1,2] => [5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[3,4,5,2,1] => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[3,5,1,2,4] => [5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[3,5,1,4,2] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[3,5,2,1,4] => [2,5,3,1,4] => [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[3,5,2,4,1] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[3,5,4,1,2] => [4,1,3,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[4,1,2,3,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[4,1,2,5,3] => [4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 1
[4,1,3,2,5] => [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[4,1,3,5,2] => [4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[4,1,5,2,3] => [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[4,1,5,3,2] => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 1
[4,2,1,5,3] => [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[4,2,3,1,5] => [3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[4,2,3,5,1] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[4,2,5,1,3] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[4,2,5,3,1] => [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 1
[4,3,1,5,2] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
Description
The maximal number of elements covered by an element in a poset.
Matching statistic: St000846
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000846: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 67%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000846: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 67%
Values
[1,2] => [1,2] => [1,0,1,0]
=> ([(0,1)],2)
=> 1 = 0 + 1
[2,1] => [2,1] => [1,1,0,0]
=> ([(0,1)],2)
=> 1 = 0 + 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[2,3,1] => [3,2,1] => [1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,2,1] => [2,3,1] => [1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 1 + 1
[1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 1 + 1
[1,4,3,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[2,3,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[2,4,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[2,4,3,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[3,1,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[3,1,4,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[3,2,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[3,2,4,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[3,4,1,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[3,4,2,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 1 + 1
[4,1,2,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[4,1,3,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[4,2,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 1 + 1
[4,2,3,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[4,3,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[4,3,2,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,4,5,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 2 = 1 + 1
[1,2,5,3,4] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 2 = 1 + 1
[1,2,5,4,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 1 + 1
[1,3,4,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[1,3,5,2,4] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[1,3,5,4,2] => [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 1 + 1
[1,4,2,3,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 1 + 1
[1,4,2,5,3] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[1,4,3,2,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,4,3,5,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[1,4,5,2,3] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[1,4,5,3,2] => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 2 = 1 + 1
[1,5,2,3,4] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[1,5,2,4,3] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[1,5,3,2,4] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 2 = 1 + 1
[1,5,3,4,2] => [1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[1,5,4,2,3] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 1 + 1
[1,5,4,3,2] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,4,1,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[2,3,4,5,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[2,3,5,1,4] => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[2,3,5,4,1] => [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[2,4,1,3,5] => [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[2,4,1,5,3] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 1
[2,4,3,5,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[2,4,5,1,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[2,5,1,3,4] => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[2,5,1,4,3] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[2,5,3,4,1] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 1
[2,5,4,1,3] => [4,2,1,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[3,1,4,2,5] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[3,1,4,5,2] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 1
[3,1,5,2,4] => [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 1
[3,1,5,4,2] => [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[3,2,4,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[3,2,4,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[3,2,5,1,4] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[3,2,5,4,1] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[3,4,1,2,5] => [4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[3,4,1,5,2] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[3,4,2,5,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[3,4,5,1,2] => [5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[3,4,5,2,1] => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[3,5,1,2,4] => [5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[3,5,1,4,2] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[3,5,2,1,4] => [2,5,3,1,4] => [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[3,5,2,4,1] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[3,5,4,1,2] => [4,1,3,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[4,1,2,3,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[4,1,2,5,3] => [4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 1
[4,1,3,2,5] => [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[4,1,3,5,2] => [4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[4,1,5,2,3] => [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[4,1,5,3,2] => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 1
[4,2,1,5,3] => [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[4,2,3,1,5] => [3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[4,2,3,5,1] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[4,2,5,1,3] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[4,2,5,3,1] => [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 1
[4,3,1,5,2] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
Description
The maximal number of elements covering an element of a poset.
The following 8 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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. St000640The rank of the largest boolean interval in a poset. St001060The distinguishing index of a graph. St000699The toughness times the least common multiple of 1,. 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.
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