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Your data matches 39 different statistics following compositions of up to 3 maps.
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Matching statistic: St001645
(load all 237 compositions to match this statistic)
(load all 237 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ([],1)
=> 1
[3,1,2] => [1,3,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 4
[3,2,1] => [1,3,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 4
[3,1,4,2] => [1,3,4,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
[3,2,4,1] => [1,3,4,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
[3,1,5,2,4] => [1,3,5,4,2] => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 5
[3,2,5,1,4] => [1,3,5,4,2] => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 5
[4,1,2,5,3] => [1,4,5,3,2] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 5
[4,2,1,5,3] => [1,4,5,3,2] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 5
[5,1,2,3,4] => [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[5,2,1,3,4] => [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,4,2,5,6,3] => [1,2,4,5,6,3] => [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 6
[1,4,2,6,3,5] => [1,2,4,6,5,3] => [6,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 6
[1,4,3,5,6,2] => [1,2,4,5,6,3] => [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 6
[1,4,3,6,2,5] => [1,2,4,6,5,3] => [6,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 6
[1,5,2,3,6,4] => [1,2,5,6,4,3] => [5,6,4,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 6
[1,5,2,6,4,3] => [1,2,5,4,6,3] => [5,4,6,1,2,3] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,5,3,2,6,4] => [1,2,5,6,4,3] => [5,6,4,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 6
[1,5,3,6,4,2] => [1,2,5,4,6,3] => [5,4,6,1,2,3] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,6,2,3,4,5] => [1,2,6,5,4,3] => [6,5,4,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,6,2,5,3,4] => [1,2,6,4,5,3] => [4,6,5,1,2,3] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,6,3,2,4,5] => [1,2,6,5,4,3] => [6,5,4,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,6,3,5,2,4] => [1,2,6,4,5,3] => [4,6,5,1,2,3] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[2,4,1,5,6,3] => [1,2,4,5,6,3] => [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 6
[2,4,1,6,3,5] => [1,2,4,6,5,3] => [6,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 6
[2,4,3,5,6,1] => [1,2,4,5,6,3] => [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 6
[2,4,3,6,1,5] => [1,2,4,6,5,3] => [6,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 6
[2,5,1,3,6,4] => [1,2,5,6,4,3] => [5,6,4,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 6
[2,5,1,6,4,3] => [1,2,5,4,6,3] => [5,4,6,1,2,3] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[2,5,3,1,6,4] => [1,2,5,6,4,3] => [5,6,4,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 6
[2,5,3,6,4,1] => [1,2,5,4,6,3] => [5,4,6,1,2,3] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[2,6,1,3,4,5] => [1,2,6,5,4,3] => [6,5,4,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[2,6,1,5,3,4] => [1,2,6,4,5,3] => [4,6,5,1,2,3] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[2,6,3,1,4,5] => [1,2,6,5,4,3] => [6,5,4,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[2,6,3,5,1,4] => [1,2,6,4,5,3] => [4,6,5,1,2,3] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[3,1,4,6,2,5] => [1,3,4,6,5,2] => [6,3,4,5,1,2] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 6
[3,1,5,2,6,4] => [1,3,5,6,4,2] => [5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 6
[3,1,6,2,4,5] => [1,3,6,5,4,2] => [6,5,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[3,2,4,6,1,5] => [1,3,4,6,5,2] => [6,3,4,5,1,2] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 6
[3,2,5,1,6,4] => [1,3,5,6,4,2] => [5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 6
[3,2,6,1,4,5] => [1,3,6,5,4,2] => [6,5,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[4,1,2,5,6,3] => [1,4,5,6,3,2] => [4,5,6,3,1,2] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 6
[4,1,2,6,3,5] => [1,4,6,5,3,2] => [6,4,5,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[4,1,5,6,2,3] => [1,4,6,3,5,2] => [6,4,3,5,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 6
[4,1,6,3,2,5] => [1,4,3,6,5,2] => [4,6,3,5,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 6
[4,2,1,5,6,3] => [1,4,5,6,3,2] => [4,5,6,3,1,2] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 6
[4,2,1,6,3,5] => [1,4,6,5,3,2] => [6,4,5,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[4,2,5,6,1,3] => [1,4,6,3,5,2] => [6,4,3,5,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 6
[4,2,6,3,1,5] => [1,4,3,6,5,2] => [4,6,3,5,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 6
[5,1,2,3,6,4] => [1,5,6,4,3,2] => [5,6,4,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
Description
The pebbling number of a connected graph.
Matching statistic: St001721
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00200: Binary words —twist⟶ Binary words
Mp00269: Binary words —flag zeros to zeros⟶ Binary words
St001721: Binary words ⟶ ℤResult quality: 80% ●values known / values provided: 100%●distinct values known / distinct values provided: 80%
Mp00200: Binary words —twist⟶ Binary words
Mp00269: Binary words —flag zeros to zeros⟶ Binary words
St001721: Binary words ⟶ ℤResult quality: 80% ●values known / values provided: 100%●distinct values known / distinct values provided: 80%
Values
[1] => => ? => ? => ? = 1 - 1
[3,1,2] => 00 => 10 => 00 => 3 = 4 - 1
[3,2,1] => 00 => 10 => 00 => 3 = 4 - 1
[3,1,4,2] => 000 => 100 => 010 => 3 = 4 - 1
[3,2,4,1] => 000 => 100 => 010 => 3 = 4 - 1
[3,1,5,2,4] => 0000 => 1000 => 0110 => 4 = 5 - 1
[3,2,5,1,4] => 0000 => 1000 => 0110 => 4 = 5 - 1
[4,1,2,5,3] => 0000 => 1000 => 0110 => 4 = 5 - 1
[4,2,1,5,3] => 0000 => 1000 => 0110 => 4 = 5 - 1
[5,1,2,3,4] => 0000 => 1000 => 0110 => 4 = 5 - 1
[5,2,1,3,4] => 0000 => 1000 => 0110 => 4 = 5 - 1
[1,4,2,5,6,3] => 10000 => 00000 => 01111 => 5 = 6 - 1
[1,4,2,6,3,5] => 10000 => 00000 => 01111 => 5 = 6 - 1
[1,4,3,5,6,2] => 10000 => 00000 => 01111 => 5 = 6 - 1
[1,4,3,6,2,5] => 10000 => 00000 => 01111 => 5 = 6 - 1
[1,5,2,3,6,4] => 10000 => 00000 => 01111 => 5 = 6 - 1
[1,5,2,6,4,3] => 10000 => 00000 => 01111 => 5 = 6 - 1
[1,5,3,2,6,4] => 10000 => 00000 => 01111 => 5 = 6 - 1
[1,5,3,6,4,2] => 10000 => 00000 => 01111 => 5 = 6 - 1
[1,6,2,3,4,5] => 10000 => 00000 => 01111 => 5 = 6 - 1
[1,6,2,5,3,4] => 10000 => 00000 => 01111 => 5 = 6 - 1
[1,6,3,2,4,5] => 10000 => 00000 => 01111 => 5 = 6 - 1
[1,6,3,5,2,4] => 10000 => 00000 => 01111 => 5 = 6 - 1
[2,4,1,5,6,3] => 00000 => 10000 => 01110 => 5 = 6 - 1
[2,4,1,6,3,5] => 00000 => 10000 => 01110 => 5 = 6 - 1
[2,4,3,5,6,1] => 00000 => 10000 => 01110 => 5 = 6 - 1
[2,4,3,6,1,5] => 00000 => 10000 => 01110 => 5 = 6 - 1
[2,5,1,3,6,4] => 00000 => 10000 => 01110 => 5 = 6 - 1
[2,5,1,6,4,3] => 00000 => 10000 => 01110 => 5 = 6 - 1
[2,5,3,1,6,4] => 00000 => 10000 => 01110 => 5 = 6 - 1
[2,5,3,6,4,1] => 00000 => 10000 => 01110 => 5 = 6 - 1
[2,6,1,3,4,5] => 00000 => 10000 => 01110 => 5 = 6 - 1
[2,6,1,5,3,4] => 00000 => 10000 => 01110 => 5 = 6 - 1
[2,6,3,1,4,5] => 00000 => 10000 => 01110 => 5 = 6 - 1
[2,6,3,5,1,4] => 00000 => 10000 => 01110 => 5 = 6 - 1
[3,1,4,6,2,5] => 00000 => 10000 => 01110 => 5 = 6 - 1
[3,1,5,2,6,4] => 00000 => 10000 => 01110 => 5 = 6 - 1
[3,1,6,2,4,5] => 00000 => 10000 => 01110 => 5 = 6 - 1
[3,2,4,6,1,5] => 00000 => 10000 => 01110 => 5 = 6 - 1
[3,2,5,1,6,4] => 00000 => 10000 => 01110 => 5 = 6 - 1
[3,2,6,1,4,5] => 00000 => 10000 => 01110 => 5 = 6 - 1
[4,1,2,5,6,3] => 00000 => 10000 => 01110 => 5 = 6 - 1
[4,1,2,6,3,5] => 00000 => 10000 => 01110 => 5 = 6 - 1
[4,1,5,6,2,3] => 00000 => 10000 => 01110 => 5 = 6 - 1
[4,1,6,3,2,5] => 00000 => 10000 => 01110 => 5 = 6 - 1
[4,2,1,5,6,3] => 00000 => 10000 => 01110 => 5 = 6 - 1
[4,2,1,6,3,5] => 00000 => 10000 => 01110 => 5 = 6 - 1
[4,2,5,6,1,3] => 00000 => 10000 => 01110 => 5 = 6 - 1
[4,2,6,3,1,5] => 00000 => 10000 => 01110 => 5 = 6 - 1
[5,1,2,3,6,4] => 00000 => 10000 => 01110 => 5 = 6 - 1
[5,1,2,6,4,3] => 00000 => 10000 => 01110 => 5 = 6 - 1
Description
The degree of a binary word.
A valley in a binary word is a letter $0$ which is not immediately followed by a $1$. A peak is a letter $1$ which is not immediately followed by a $0$.
Let $f$ be the map that replaces every valley with a peak. The degree of a binary word $w$ is the number of times $f$ has to be applied to obtain a binary word without zeros.
Matching statistic: St000393
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00280: Binary words —path rowmotion⟶ Binary words
Mp00135: Binary words —rotate front-to-back⟶ Binary words
St000393: Binary words ⟶ ℤResult quality: 80% ●values known / values provided: 100%●distinct values known / distinct values provided: 80%
Mp00280: Binary words —path rowmotion⟶ Binary words
Mp00135: Binary words —rotate front-to-back⟶ Binary words
St000393: Binary words ⟶ ℤResult quality: 80% ●values known / values provided: 100%●distinct values known / distinct values provided: 80%
Values
[1] => => => ? => ? = 1 - 2
[3,1,2] => 00 => 01 => 10 => 2 = 4 - 2
[3,2,1] => 00 => 01 => 10 => 2 = 4 - 2
[3,1,4,2] => 000 => 001 => 010 => 2 = 4 - 2
[3,2,4,1] => 000 => 001 => 010 => 2 = 4 - 2
[3,1,5,2,4] => 0000 => 0001 => 0010 => 3 = 5 - 2
[3,2,5,1,4] => 0000 => 0001 => 0010 => 3 = 5 - 2
[4,1,2,5,3] => 0000 => 0001 => 0010 => 3 = 5 - 2
[4,2,1,5,3] => 0000 => 0001 => 0010 => 3 = 5 - 2
[5,1,2,3,4] => 0000 => 0001 => 0010 => 3 = 5 - 2
[5,2,1,3,4] => 0000 => 0001 => 0010 => 3 = 5 - 2
[1,4,2,5,6,3] => 10000 => 00011 => 00110 => 4 = 6 - 2
[1,4,2,6,3,5] => 10000 => 00011 => 00110 => 4 = 6 - 2
[1,4,3,5,6,2] => 10000 => 00011 => 00110 => 4 = 6 - 2
[1,4,3,6,2,5] => 10000 => 00011 => 00110 => 4 = 6 - 2
[1,5,2,3,6,4] => 10000 => 00011 => 00110 => 4 = 6 - 2
[1,5,2,6,4,3] => 10000 => 00011 => 00110 => 4 = 6 - 2
[1,5,3,2,6,4] => 10000 => 00011 => 00110 => 4 = 6 - 2
[1,5,3,6,4,2] => 10000 => 00011 => 00110 => 4 = 6 - 2
[1,6,2,3,4,5] => 10000 => 00011 => 00110 => 4 = 6 - 2
[1,6,2,5,3,4] => 10000 => 00011 => 00110 => 4 = 6 - 2
[1,6,3,2,4,5] => 10000 => 00011 => 00110 => 4 = 6 - 2
[1,6,3,5,2,4] => 10000 => 00011 => 00110 => 4 = 6 - 2
[2,4,1,5,6,3] => 00000 => 00001 => 00010 => 4 = 6 - 2
[2,4,1,6,3,5] => 00000 => 00001 => 00010 => 4 = 6 - 2
[2,4,3,5,6,1] => 00000 => 00001 => 00010 => 4 = 6 - 2
[2,4,3,6,1,5] => 00000 => 00001 => 00010 => 4 = 6 - 2
[2,5,1,3,6,4] => 00000 => 00001 => 00010 => 4 = 6 - 2
[2,5,1,6,4,3] => 00000 => 00001 => 00010 => 4 = 6 - 2
[2,5,3,1,6,4] => 00000 => 00001 => 00010 => 4 = 6 - 2
[2,5,3,6,4,1] => 00000 => 00001 => 00010 => 4 = 6 - 2
[2,6,1,3,4,5] => 00000 => 00001 => 00010 => 4 = 6 - 2
[2,6,1,5,3,4] => 00000 => 00001 => 00010 => 4 = 6 - 2
[2,6,3,1,4,5] => 00000 => 00001 => 00010 => 4 = 6 - 2
[2,6,3,5,1,4] => 00000 => 00001 => 00010 => 4 = 6 - 2
[3,1,4,6,2,5] => 00000 => 00001 => 00010 => 4 = 6 - 2
[3,1,5,2,6,4] => 00000 => 00001 => 00010 => 4 = 6 - 2
[3,1,6,2,4,5] => 00000 => 00001 => 00010 => 4 = 6 - 2
[3,2,4,6,1,5] => 00000 => 00001 => 00010 => 4 = 6 - 2
[3,2,5,1,6,4] => 00000 => 00001 => 00010 => 4 = 6 - 2
[3,2,6,1,4,5] => 00000 => 00001 => 00010 => 4 = 6 - 2
[4,1,2,5,6,3] => 00000 => 00001 => 00010 => 4 = 6 - 2
[4,1,2,6,3,5] => 00000 => 00001 => 00010 => 4 = 6 - 2
[4,1,5,6,2,3] => 00000 => 00001 => 00010 => 4 = 6 - 2
[4,1,6,3,2,5] => 00000 => 00001 => 00010 => 4 = 6 - 2
[4,2,1,5,6,3] => 00000 => 00001 => 00010 => 4 = 6 - 2
[4,2,1,6,3,5] => 00000 => 00001 => 00010 => 4 = 6 - 2
[4,2,5,6,1,3] => 00000 => 00001 => 00010 => 4 = 6 - 2
[4,2,6,3,1,5] => 00000 => 00001 => 00010 => 4 = 6 - 2
[5,1,2,3,6,4] => 00000 => 00001 => 00010 => 4 = 6 - 2
[5,1,2,6,4,3] => 00000 => 00001 => 00010 => 4 = 6 - 2
Description
The number of strictly increasing runs in a binary word.
Matching statistic: St000626
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00136: Binary words —rotate back-to-front⟶ Binary words
St000626: Binary words ⟶ ℤResult quality: 80% ●values known / values provided: 100%●distinct values known / distinct values provided: 80%
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00136: Binary words —rotate back-to-front⟶ Binary words
St000626: Binary words ⟶ ℤResult quality: 80% ●values known / values provided: 100%●distinct values known / distinct values provided: 80%
Values
[1] => [1] => => ? => ? = 1 - 2
[3,1,2] => [1,3,2] => 10 => 01 => 2 = 4 - 2
[3,2,1] => [1,3,2] => 10 => 01 => 2 = 4 - 2
[3,1,4,2] => [1,3,4,2] => 100 => 010 => 2 = 4 - 2
[3,2,4,1] => [1,3,4,2] => 100 => 010 => 2 = 4 - 2
[3,1,5,2,4] => [1,3,5,4,2] => 1000 => 0100 => 3 = 5 - 2
[3,2,5,1,4] => [1,3,5,4,2] => 1000 => 0100 => 3 = 5 - 2
[4,1,2,5,3] => [1,4,5,3,2] => 1000 => 0100 => 3 = 5 - 2
[4,2,1,5,3] => [1,4,5,3,2] => 1000 => 0100 => 3 = 5 - 2
[5,1,2,3,4] => [1,5,4,3,2] => 1000 => 0100 => 3 = 5 - 2
[5,2,1,3,4] => [1,5,4,3,2] => 1000 => 0100 => 3 = 5 - 2
[1,4,2,5,6,3] => [1,2,4,5,6,3] => 11000 => 01100 => 4 = 6 - 2
[1,4,2,6,3,5] => [1,2,4,6,5,3] => 11000 => 01100 => 4 = 6 - 2
[1,4,3,5,6,2] => [1,2,4,5,6,3] => 11000 => 01100 => 4 = 6 - 2
[1,4,3,6,2,5] => [1,2,4,6,5,3] => 11000 => 01100 => 4 = 6 - 2
[1,5,2,3,6,4] => [1,2,5,6,4,3] => 11000 => 01100 => 4 = 6 - 2
[1,5,2,6,4,3] => [1,2,5,4,6,3] => 11000 => 01100 => 4 = 6 - 2
[1,5,3,2,6,4] => [1,2,5,6,4,3] => 11000 => 01100 => 4 = 6 - 2
[1,5,3,6,4,2] => [1,2,5,4,6,3] => 11000 => 01100 => 4 = 6 - 2
[1,6,2,3,4,5] => [1,2,6,5,4,3] => 11000 => 01100 => 4 = 6 - 2
[1,6,2,5,3,4] => [1,2,6,4,5,3] => 11000 => 01100 => 4 = 6 - 2
[1,6,3,2,4,5] => [1,2,6,5,4,3] => 11000 => 01100 => 4 = 6 - 2
[1,6,3,5,2,4] => [1,2,6,4,5,3] => 11000 => 01100 => 4 = 6 - 2
[2,4,1,5,6,3] => [1,2,4,5,6,3] => 11000 => 01100 => 4 = 6 - 2
[2,4,1,6,3,5] => [1,2,4,6,5,3] => 11000 => 01100 => 4 = 6 - 2
[2,4,3,5,6,1] => [1,2,4,5,6,3] => 11000 => 01100 => 4 = 6 - 2
[2,4,3,6,1,5] => [1,2,4,6,5,3] => 11000 => 01100 => 4 = 6 - 2
[2,5,1,3,6,4] => [1,2,5,6,4,3] => 11000 => 01100 => 4 = 6 - 2
[2,5,1,6,4,3] => [1,2,5,4,6,3] => 11000 => 01100 => 4 = 6 - 2
[2,5,3,1,6,4] => [1,2,5,6,4,3] => 11000 => 01100 => 4 = 6 - 2
[2,5,3,6,4,1] => [1,2,5,4,6,3] => 11000 => 01100 => 4 = 6 - 2
[2,6,1,3,4,5] => [1,2,6,5,4,3] => 11000 => 01100 => 4 = 6 - 2
[2,6,1,5,3,4] => [1,2,6,4,5,3] => 11000 => 01100 => 4 = 6 - 2
[2,6,3,1,4,5] => [1,2,6,5,4,3] => 11000 => 01100 => 4 = 6 - 2
[2,6,3,5,1,4] => [1,2,6,4,5,3] => 11000 => 01100 => 4 = 6 - 2
[3,1,4,6,2,5] => [1,3,4,6,5,2] => 10000 => 01000 => 4 = 6 - 2
[3,1,5,2,6,4] => [1,3,5,6,4,2] => 10000 => 01000 => 4 = 6 - 2
[3,1,6,2,4,5] => [1,3,6,5,4,2] => 10000 => 01000 => 4 = 6 - 2
[3,2,4,6,1,5] => [1,3,4,6,5,2] => 10000 => 01000 => 4 = 6 - 2
[3,2,5,1,6,4] => [1,3,5,6,4,2] => 10000 => 01000 => 4 = 6 - 2
[3,2,6,1,4,5] => [1,3,6,5,4,2] => 10000 => 01000 => 4 = 6 - 2
[4,1,2,5,6,3] => [1,4,5,6,3,2] => 10000 => 01000 => 4 = 6 - 2
[4,1,2,6,3,5] => [1,4,6,5,3,2] => 10000 => 01000 => 4 = 6 - 2
[4,1,5,6,2,3] => [1,4,6,3,5,2] => 10000 => 01000 => 4 = 6 - 2
[4,1,6,3,2,5] => [1,4,3,6,5,2] => 10000 => 01000 => 4 = 6 - 2
[4,2,1,5,6,3] => [1,4,5,6,3,2] => 10000 => 01000 => 4 = 6 - 2
[4,2,1,6,3,5] => [1,4,6,5,3,2] => 10000 => 01000 => 4 = 6 - 2
[4,2,5,6,1,3] => [1,4,6,3,5,2] => 10000 => 01000 => 4 = 6 - 2
[4,2,6,3,1,5] => [1,4,3,6,5,2] => 10000 => 01000 => 4 = 6 - 2
[5,1,2,3,6,4] => [1,5,6,4,3,2] => 10000 => 01000 => 4 = 6 - 2
[5,1,2,6,4,3] => [1,5,4,6,3,2] => 10000 => 01000 => 4 = 6 - 2
Description
The minimal period of a binary word.
This is the smallest natural number $p$ such that $w_i=w_{i+p}$ for all $i\in\{1,\dots,|w|-p\}$.
Matching statistic: St001437
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
St001437: Binary words ⟶ ℤResult quality: 80% ●values known / values provided: 100%●distinct values known / distinct values provided: 80%
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
St001437: Binary words ⟶ ℤResult quality: 80% ●values known / values provided: 100%●distinct values known / distinct values provided: 80%
Values
[1] => [1] => => ? => ? = 1 - 2
[3,1,2] => [1,3,2] => 10 => 11 => 2 = 4 - 2
[3,2,1] => [1,3,2] => 10 => 11 => 2 = 4 - 2
[3,1,4,2] => [1,3,4,2] => 100 => 101 => 2 = 4 - 2
[3,2,4,1] => [1,3,4,2] => 100 => 101 => 2 = 4 - 2
[3,1,5,2,4] => [1,3,5,4,2] => 1000 => 1001 => 3 = 5 - 2
[3,2,5,1,4] => [1,3,5,4,2] => 1000 => 1001 => 3 = 5 - 2
[4,1,2,5,3] => [1,4,5,3,2] => 1000 => 1001 => 3 = 5 - 2
[4,2,1,5,3] => [1,4,5,3,2] => 1000 => 1001 => 3 = 5 - 2
[5,1,2,3,4] => [1,5,4,3,2] => 1000 => 1001 => 3 = 5 - 2
[5,2,1,3,4] => [1,5,4,3,2] => 1000 => 1001 => 3 = 5 - 2
[1,4,2,5,6,3] => [1,2,4,5,6,3] => 11000 => 11001 => 4 = 6 - 2
[1,4,2,6,3,5] => [1,2,4,6,5,3] => 11000 => 11001 => 4 = 6 - 2
[1,4,3,5,6,2] => [1,2,4,5,6,3] => 11000 => 11001 => 4 = 6 - 2
[1,4,3,6,2,5] => [1,2,4,6,5,3] => 11000 => 11001 => 4 = 6 - 2
[1,5,2,3,6,4] => [1,2,5,6,4,3] => 11000 => 11001 => 4 = 6 - 2
[1,5,2,6,4,3] => [1,2,5,4,6,3] => 11000 => 11001 => 4 = 6 - 2
[1,5,3,2,6,4] => [1,2,5,6,4,3] => 11000 => 11001 => 4 = 6 - 2
[1,5,3,6,4,2] => [1,2,5,4,6,3] => 11000 => 11001 => 4 = 6 - 2
[1,6,2,3,4,5] => [1,2,6,5,4,3] => 11000 => 11001 => 4 = 6 - 2
[1,6,2,5,3,4] => [1,2,6,4,5,3] => 11000 => 11001 => 4 = 6 - 2
[1,6,3,2,4,5] => [1,2,6,5,4,3] => 11000 => 11001 => 4 = 6 - 2
[1,6,3,5,2,4] => [1,2,6,4,5,3] => 11000 => 11001 => 4 = 6 - 2
[2,4,1,5,6,3] => [1,2,4,5,6,3] => 11000 => 11001 => 4 = 6 - 2
[2,4,1,6,3,5] => [1,2,4,6,5,3] => 11000 => 11001 => 4 = 6 - 2
[2,4,3,5,6,1] => [1,2,4,5,6,3] => 11000 => 11001 => 4 = 6 - 2
[2,4,3,6,1,5] => [1,2,4,6,5,3] => 11000 => 11001 => 4 = 6 - 2
[2,5,1,3,6,4] => [1,2,5,6,4,3] => 11000 => 11001 => 4 = 6 - 2
[2,5,1,6,4,3] => [1,2,5,4,6,3] => 11000 => 11001 => 4 = 6 - 2
[2,5,3,1,6,4] => [1,2,5,6,4,3] => 11000 => 11001 => 4 = 6 - 2
[2,5,3,6,4,1] => [1,2,5,4,6,3] => 11000 => 11001 => 4 = 6 - 2
[2,6,1,3,4,5] => [1,2,6,5,4,3] => 11000 => 11001 => 4 = 6 - 2
[2,6,1,5,3,4] => [1,2,6,4,5,3] => 11000 => 11001 => 4 = 6 - 2
[2,6,3,1,4,5] => [1,2,6,5,4,3] => 11000 => 11001 => 4 = 6 - 2
[2,6,3,5,1,4] => [1,2,6,4,5,3] => 11000 => 11001 => 4 = 6 - 2
[3,1,4,6,2,5] => [1,3,4,6,5,2] => 10000 => 10001 => 4 = 6 - 2
[3,1,5,2,6,4] => [1,3,5,6,4,2] => 10000 => 10001 => 4 = 6 - 2
[3,1,6,2,4,5] => [1,3,6,5,4,2] => 10000 => 10001 => 4 = 6 - 2
[3,2,4,6,1,5] => [1,3,4,6,5,2] => 10000 => 10001 => 4 = 6 - 2
[3,2,5,1,6,4] => [1,3,5,6,4,2] => 10000 => 10001 => 4 = 6 - 2
[3,2,6,1,4,5] => [1,3,6,5,4,2] => 10000 => 10001 => 4 = 6 - 2
[4,1,2,5,6,3] => [1,4,5,6,3,2] => 10000 => 10001 => 4 = 6 - 2
[4,1,2,6,3,5] => [1,4,6,5,3,2] => 10000 => 10001 => 4 = 6 - 2
[4,1,5,6,2,3] => [1,4,6,3,5,2] => 10000 => 10001 => 4 = 6 - 2
[4,1,6,3,2,5] => [1,4,3,6,5,2] => 10000 => 10001 => 4 = 6 - 2
[4,2,1,5,6,3] => [1,4,5,6,3,2] => 10000 => 10001 => 4 = 6 - 2
[4,2,1,6,3,5] => [1,4,6,5,3,2] => 10000 => 10001 => 4 = 6 - 2
[4,2,5,6,1,3] => [1,4,6,3,5,2] => 10000 => 10001 => 4 = 6 - 2
[4,2,6,3,1,5] => [1,4,3,6,5,2] => 10000 => 10001 => 4 = 6 - 2
[5,1,2,3,6,4] => [1,5,6,4,3,2] => 10000 => 10001 => 4 = 6 - 2
[5,1,2,6,4,3] => [1,5,4,6,3,2] => 10000 => 10001 => 4 = 6 - 2
Description
The flex of a binary word.
This is the product of the lex statistic ([[St001436]], augmented by 1) and its frequency ([[St000627]]), see [1, §8].
Matching statistic: St000829
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000829: Permutations ⟶ ℤResult quality: 80% ●values known / values provided: 100%●distinct values known / distinct values provided: 80%
Mp00252: Permutations —restriction⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000829: Permutations ⟶ ℤResult quality: 80% ●values known / values provided: 100%●distinct values known / distinct values provided: 80%
Values
[1] => [1] => [] => [] => ? = 1 - 3
[3,1,2] => [2,3,1] => [2,1] => [2,1] => 1 = 4 - 3
[3,2,1] => [3,2,1] => [2,1] => [2,1] => 1 = 4 - 3
[3,1,4,2] => [4,2,1,3] => [2,1,3] => [2,1,3] => 1 = 4 - 3
[3,2,4,1] => [2,1,4,3] => [2,1,3] => [2,1,3] => 1 = 4 - 3
[3,1,5,2,4] => [4,1,5,2,3] => [4,1,2,3] => [4,1,3,2] => 2 = 5 - 3
[3,2,5,1,4] => [2,1,4,5,3] => [2,1,4,3] => [2,1,4,3] => 2 = 5 - 3
[4,1,2,5,3] => [5,2,3,1,4] => [2,3,1,4] => [2,4,1,3] => 2 = 5 - 3
[4,2,1,5,3] => [2,5,3,1,4] => [2,3,1,4] => [2,4,1,3] => 2 = 5 - 3
[5,1,2,3,4] => [2,3,4,5,1] => [2,3,4,1] => [2,4,3,1] => 2 = 5 - 3
[5,2,1,3,4] => [3,2,4,5,1] => [3,2,4,1] => [3,2,4,1] => 2 = 5 - 3
[1,4,2,5,6,3] => [4,2,6,1,3,5] => [4,2,1,3,5] => [4,2,1,5,3] => 3 = 6 - 3
[1,4,2,6,3,5] => [5,2,6,1,3,4] => [5,2,1,3,4] => [5,2,1,4,3] => 3 = 6 - 3
[1,4,3,5,6,2] => [3,2,4,6,1,5] => [3,2,4,1,5] => [3,2,5,1,4] => 3 = 6 - 3
[1,4,3,6,2,5] => [3,2,5,6,1,4] => [3,2,5,1,4] => [3,2,5,1,4] => 3 = 6 - 3
[1,5,2,3,6,4] => [6,3,4,1,2,5] => [3,4,1,2,5] => [3,5,1,4,2] => 3 = 6 - 3
[1,5,2,6,4,3] => [6,5,3,1,2,4] => [5,3,1,2,4] => [5,3,1,4,2] => 3 = 6 - 3
[1,5,3,2,6,4] => [3,6,4,1,2,5] => [3,4,1,2,5] => [3,5,1,4,2] => 3 = 6 - 3
[1,5,3,6,4,2] => [6,3,2,5,1,4] => [3,2,5,1,4] => [3,2,5,1,4] => 3 = 6 - 3
[1,6,2,3,4,5] => [3,4,5,6,1,2] => [3,4,5,1,2] => [3,5,4,1,2] => 3 = 6 - 3
[1,6,2,5,3,4] => [5,6,3,4,1,2] => [5,3,4,1,2] => [5,3,4,1,2] => 3 = 6 - 3
[1,6,3,2,4,5] => [4,3,5,6,1,2] => [4,3,5,1,2] => [4,3,5,1,2] => 3 = 6 - 3
[1,6,3,5,2,4] => [3,5,6,4,1,2] => [3,5,4,1,2] => [3,5,4,1,2] => 3 = 6 - 3
[2,4,1,5,6,3] => [4,1,2,6,3,5] => [4,1,2,3,5] => [4,1,5,3,2] => 3 = 6 - 3
[2,4,1,6,3,5] => [5,1,2,6,3,4] => [5,1,2,3,4] => [5,1,4,3,2] => 3 = 6 - 3
[2,4,3,5,6,1] => [3,1,2,4,6,5] => [3,1,2,4,5] => [3,1,5,4,2] => 3 = 6 - 3
[2,4,3,6,1,5] => [3,1,2,5,6,4] => [3,1,2,5,4] => [3,1,5,4,2] => 3 = 6 - 3
[2,5,1,3,6,4] => [6,1,3,4,2,5] => [1,3,4,2,5] => [1,5,4,3,2] => 3 = 6 - 3
[2,5,1,6,4,3] => [6,5,1,3,2,4] => [5,1,3,2,4] => [5,1,4,3,2] => 3 = 6 - 3
[2,5,3,1,6,4] => [3,6,1,4,2,5] => [3,1,4,2,5] => [3,1,5,4,2] => 3 = 6 - 3
[2,5,3,6,4,1] => [6,3,1,2,5,4] => [3,1,2,5,4] => [3,1,5,4,2] => 3 = 6 - 3
[2,6,1,3,4,5] => [1,3,4,5,6,2] => [1,3,4,5,2] => [1,5,4,3,2] => 3 = 6 - 3
[2,6,1,5,3,4] => [5,6,1,3,4,2] => [5,1,3,4,2] => [5,1,4,3,2] => 3 = 6 - 3
[2,6,3,1,4,5] => [4,1,3,5,6,2] => [4,1,3,5,2] => [4,1,5,3,2] => 3 = 6 - 3
[2,6,3,5,1,4] => [3,5,6,1,4,2] => [3,5,1,4,2] => [3,5,1,4,2] => 3 = 6 - 3
[3,1,4,6,2,5] => [3,1,5,6,2,4] => [3,1,5,2,4] => [3,1,5,4,2] => 3 = 6 - 3
[3,1,5,2,6,4] => [1,6,4,2,3,5] => [1,4,2,3,5] => [1,5,4,3,2] => 3 = 6 - 3
[3,1,6,2,4,5] => [4,1,5,6,2,3] => [4,1,5,2,3] => [4,1,5,3,2] => 3 = 6 - 3
[3,2,4,6,1,5] => [2,1,3,5,6,4] => [2,1,3,5,4] => [2,1,5,4,3] => 3 = 6 - 3
[3,2,5,1,6,4] => [1,6,2,4,3,5] => [1,2,4,3,5] => [1,5,4,3,2] => 3 = 6 - 3
[3,2,6,1,4,5] => [2,1,4,5,6,3] => [2,1,4,5,3] => [2,1,5,4,3] => 3 = 6 - 3
[4,1,2,5,6,3] => [4,6,2,1,3,5] => [4,2,1,3,5] => [4,2,1,5,3] => 3 = 6 - 3
[4,1,2,6,3,5] => [5,6,2,1,3,4] => [5,2,1,3,4] => [5,2,1,4,3] => 3 = 6 - 3
[4,1,5,6,2,3] => [5,3,1,6,2,4] => [5,3,1,2,4] => [5,3,1,4,2] => 3 = 6 - 3
[4,1,6,3,2,5] => [5,4,1,6,2,3] => [5,4,1,2,3] => [5,4,1,3,2] => 3 = 6 - 3
[4,2,1,5,6,3] => [2,4,1,6,3,5] => [2,4,1,3,5] => [2,5,1,4,3] => 3 = 6 - 3
[4,2,1,6,3,5] => [2,5,1,6,3,4] => [2,5,1,3,4] => [2,5,1,4,3] => 3 = 6 - 3
[4,2,5,6,1,3] => [5,2,1,3,6,4] => [5,2,1,3,4] => [5,2,1,4,3] => 3 = 6 - 3
[4,2,6,3,1,5] => [5,2,1,4,6,3] => [5,2,1,4,3] => [5,2,1,4,3] => 3 = 6 - 3
[5,1,2,3,6,4] => [6,2,3,4,1,5] => [2,3,4,1,5] => [2,5,4,1,3] => 3 = 6 - 3
[5,1,2,6,4,3] => [6,5,2,3,1,4] => [5,2,3,1,4] => [5,2,4,1,3] => 3 = 6 - 3
Description
The Ulam distance of a permutation to the identity permutation.
This is, for a permutation $\pi$ of $n$, given by $n$ minus the length of the longest increasing subsequence of $\pi^{-1}$.
In other words, this statistic plus [[St000062]] equals $n$.
Matching statistic: St000876
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
Mp00200: Binary words —twist⟶ Binary words
St000876: Binary words ⟶ ℤResult quality: 80% ●values known / values provided: 100%●distinct values known / distinct values provided: 80%
Mp00224: Binary words —runsort⟶ Binary words
Mp00200: Binary words —twist⟶ Binary words
St000876: Binary words ⟶ ℤResult quality: 80% ●values known / values provided: 100%●distinct values known / distinct values provided: 80%
Values
[1] => => => ? => ? = 1 - 3
[3,1,2] => 00 => 00 => 10 => 1 = 4 - 3
[3,2,1] => 00 => 00 => 10 => 1 = 4 - 3
[3,1,4,2] => 000 => 000 => 100 => 1 = 4 - 3
[3,2,4,1] => 000 => 000 => 100 => 1 = 4 - 3
[3,1,5,2,4] => 0000 => 0000 => 1000 => 2 = 5 - 3
[3,2,5,1,4] => 0000 => 0000 => 1000 => 2 = 5 - 3
[4,1,2,5,3] => 0000 => 0000 => 1000 => 2 = 5 - 3
[4,2,1,5,3] => 0000 => 0000 => 1000 => 2 = 5 - 3
[5,1,2,3,4] => 0000 => 0000 => 1000 => 2 = 5 - 3
[5,2,1,3,4] => 0000 => 0000 => 1000 => 2 = 5 - 3
[1,4,2,5,6,3] => 10000 => 00001 => 10001 => 3 = 6 - 3
[1,4,2,6,3,5] => 10000 => 00001 => 10001 => 3 = 6 - 3
[1,4,3,5,6,2] => 10000 => 00001 => 10001 => 3 = 6 - 3
[1,4,3,6,2,5] => 10000 => 00001 => 10001 => 3 = 6 - 3
[1,5,2,3,6,4] => 10000 => 00001 => 10001 => 3 = 6 - 3
[1,5,2,6,4,3] => 10000 => 00001 => 10001 => 3 = 6 - 3
[1,5,3,2,6,4] => 10000 => 00001 => 10001 => 3 = 6 - 3
[1,5,3,6,4,2] => 10000 => 00001 => 10001 => 3 = 6 - 3
[1,6,2,3,4,5] => 10000 => 00001 => 10001 => 3 = 6 - 3
[1,6,2,5,3,4] => 10000 => 00001 => 10001 => 3 = 6 - 3
[1,6,3,2,4,5] => 10000 => 00001 => 10001 => 3 = 6 - 3
[1,6,3,5,2,4] => 10000 => 00001 => 10001 => 3 = 6 - 3
[2,4,1,5,6,3] => 00000 => 00000 => 10000 => 3 = 6 - 3
[2,4,1,6,3,5] => 00000 => 00000 => 10000 => 3 = 6 - 3
[2,4,3,5,6,1] => 00000 => 00000 => 10000 => 3 = 6 - 3
[2,4,3,6,1,5] => 00000 => 00000 => 10000 => 3 = 6 - 3
[2,5,1,3,6,4] => 00000 => 00000 => 10000 => 3 = 6 - 3
[2,5,1,6,4,3] => 00000 => 00000 => 10000 => 3 = 6 - 3
[2,5,3,1,6,4] => 00000 => 00000 => 10000 => 3 = 6 - 3
[2,5,3,6,4,1] => 00000 => 00000 => 10000 => 3 = 6 - 3
[2,6,1,3,4,5] => 00000 => 00000 => 10000 => 3 = 6 - 3
[2,6,1,5,3,4] => 00000 => 00000 => 10000 => 3 = 6 - 3
[2,6,3,1,4,5] => 00000 => 00000 => 10000 => 3 = 6 - 3
[2,6,3,5,1,4] => 00000 => 00000 => 10000 => 3 = 6 - 3
[3,1,4,6,2,5] => 00000 => 00000 => 10000 => 3 = 6 - 3
[3,1,5,2,6,4] => 00000 => 00000 => 10000 => 3 = 6 - 3
[3,1,6,2,4,5] => 00000 => 00000 => 10000 => 3 = 6 - 3
[3,2,4,6,1,5] => 00000 => 00000 => 10000 => 3 = 6 - 3
[3,2,5,1,6,4] => 00000 => 00000 => 10000 => 3 = 6 - 3
[3,2,6,1,4,5] => 00000 => 00000 => 10000 => 3 = 6 - 3
[4,1,2,5,6,3] => 00000 => 00000 => 10000 => 3 = 6 - 3
[4,1,2,6,3,5] => 00000 => 00000 => 10000 => 3 = 6 - 3
[4,1,5,6,2,3] => 00000 => 00000 => 10000 => 3 = 6 - 3
[4,1,6,3,2,5] => 00000 => 00000 => 10000 => 3 = 6 - 3
[4,2,1,5,6,3] => 00000 => 00000 => 10000 => 3 = 6 - 3
[4,2,1,6,3,5] => 00000 => 00000 => 10000 => 3 = 6 - 3
[4,2,5,6,1,3] => 00000 => 00000 => 10000 => 3 = 6 - 3
[4,2,6,3,1,5] => 00000 => 00000 => 10000 => 3 = 6 - 3
[5,1,2,3,6,4] => 00000 => 00000 => 10000 => 3 = 6 - 3
[5,1,2,6,4,3] => 00000 => 00000 => 10000 => 3 = 6 - 3
Description
The number of factors in the Catalan decomposition of a binary word.
Every binary word can be written in a unique way as $(\mathcal D 0)^\ell \mathcal D (1 \mathcal D)^m$, where $\mathcal D$ is the set of Dyck words. This is the Catalan factorisation, see [1, sec.9.1.2].
This statistic records the number of factors in the Catalan factorisation, that is, $\ell + m$ if the middle Dyck word is empty and $\ell + 1 + m$ otherwise.
Matching statistic: St001232
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> [1,0]
=> 0 = 1 - 1
[3,1,2] => [1,2,3] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> ? = 4 - 1
[3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> ? = 4 - 1
[3,1,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[3,2,4,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> ? = 4 - 1
[3,1,5,2,4] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[3,2,5,1,4] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4 = 5 - 1
[4,1,2,5,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> ? = 5 - 1
[4,2,1,5,3] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[5,1,2,3,4] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 5 - 1
[5,2,1,3,4] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ? = 5 - 1
[1,4,2,5,6,3] => [1,4,2,5,6,3] => [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 5 = 6 - 1
[1,4,2,6,3,5] => [1,4,2,6,3,5] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 5 = 6 - 1
[1,4,3,5,6,2] => [1,4,2,3,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 6 - 1
[1,4,3,6,2,5] => [1,4,2,5,3,6] => [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 6 - 1
[1,5,2,3,6,4] => [1,5,2,3,6,4] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 5 = 6 - 1
[1,5,2,6,4,3] => [1,5,2,6,3,4] => [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 5 = 6 - 1
[1,5,3,2,6,4] => [1,5,2,6,3,4] => [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 5 = 6 - 1
[1,5,3,6,4,2] => [1,5,2,3,6,4] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 5 = 6 - 1
[1,6,2,3,4,5] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,6,2,5,3,4] => [1,6,2,5,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,6,3,2,4,5] => [1,6,2,4,5,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,6,3,5,2,4] => [1,6,2,4,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[2,4,1,5,6,3] => [1,5,6,2,4,3] => [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 6 - 1
[2,4,1,6,3,5] => [1,6,2,4,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[2,4,3,5,6,1] => [1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 6 - 1
[2,4,3,6,1,5] => [1,5,2,4,3,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5 = 6 - 1
[2,5,1,3,6,4] => [1,3,6,2,5,4] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 6 - 1
[2,5,1,6,4,3] => [1,6,2,5,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[2,5,3,1,6,4] => [1,6,2,5,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[2,5,3,6,4,1] => [1,2,5,3,6,4] => [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 6 - 1
[2,6,1,3,4,5] => [1,3,4,5,2,6] => [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 6 - 1
[2,6,1,5,3,4] => [1,5,2,6,3,4] => [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 5 = 6 - 1
[2,6,3,1,4,5] => [1,4,5,2,6,3] => [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 5 = 6 - 1
[2,6,3,5,1,4] => [1,4,2,6,3,5] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 5 = 6 - 1
[3,1,4,6,2,5] => [1,4,6,2,5,3] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 6 - 1
[3,1,5,2,6,4] => [1,5,2,6,3,4] => [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 5 = 6 - 1
[3,1,6,2,4,5] => [1,6,2,4,5,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[3,2,4,6,1,5] => [1,5,2,4,6,3] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 5 = 6 - 1
[3,2,5,1,6,4] => [1,6,2,5,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[3,2,6,1,4,5] => [1,4,5,2,6,3] => [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 5 = 6 - 1
[4,1,2,5,6,3] => [1,2,5,6,3,4] => [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 6 - 1
[4,1,2,6,3,5] => [1,2,6,3,5,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 6 - 1
[4,1,5,6,2,3] => [1,5,6,2,3,4] => [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 6 - 1
[4,1,6,3,2,5] => [1,6,2,5,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[4,2,1,5,6,3] => [1,5,6,2,3,4] => [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 6 - 1
[4,2,1,6,3,5] => [1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[4,2,5,6,1,3] => [1,3,2,5,6,4] => [1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 6 - 1
[4,2,6,3,1,5] => [1,5,2,6,3,4] => [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 5 = 6 - 1
[5,1,2,3,6,4] => [1,2,3,6,4,5] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 6 - 1
[5,1,2,6,4,3] => [1,2,6,3,4,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 6 - 1
[5,1,4,2,6,3] => [1,4,2,6,3,5] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 5 = 6 - 1
[5,2,1,3,6,4] => [1,3,6,2,4,5] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 6 - 1
[5,2,1,6,4,3] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[5,2,4,1,6,3] => [1,6,2,4,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[6,1,2,3,4,5] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6 - 1
[6,1,2,5,3,4] => [1,2,5,3,4,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 6 - 1
[6,1,5,2,4,3] => [1,5,2,4,3,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5 = 6 - 1
[6,2,1,3,4,5] => [1,3,4,5,2,6] => [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 6 - 1
[6,2,1,5,3,4] => [1,5,2,3,4,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5 = 6 - 1
[6,2,5,1,4,3] => [1,4,2,5,3,6] => [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 6 - 1
[1,2,5,3,6,7,4] => [1,2,5,3,6,7,4] => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 7 - 1
[1,2,5,3,7,4,6] => [1,2,5,3,7,4,6] => [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 7 - 1
[1,2,5,4,6,7,3] => [1,2,5,3,4,6,7] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 7 - 1
[1,2,5,4,7,3,6] => [1,2,5,3,6,4,7] => [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 7 - 1
[1,2,6,3,4,7,5] => [1,2,6,3,4,7,5] => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 7 - 1
[1,2,6,3,7,5,4] => [1,2,6,3,7,4,5] => [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 7 - 1
[1,2,6,4,3,7,5] => [1,2,6,3,7,4,5] => [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 7 - 1
[1,2,6,4,7,5,3] => [1,2,6,3,4,7,5] => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 7 - 1
[1,2,7,3,4,5,6] => [1,2,7,3,4,5,6] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 7 - 1
[1,2,7,3,6,4,5] => [1,2,7,3,6,4,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 7 - 1
[1,2,7,4,3,5,6] => [1,2,7,3,5,6,4] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 7 - 1
[1,2,7,4,6,3,5] => [1,2,7,3,5,4,6] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 7 - 1
[1,3,5,2,6,7,4] => [1,3,5,2,6,7,4] => [1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 7 - 1
[1,3,5,2,7,4,6] => [1,3,5,2,7,4,6] => [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 7 - 1
[1,3,5,4,6,7,2] => [1,3,5,2,4,6,7] => [1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 7 - 1
[1,3,5,4,7,2,6] => [1,3,5,2,6,4,7] => [1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 7 - 1
[1,3,6,2,4,7,5] => [1,3,6,2,4,7,5] => [1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 7 - 1
[1,3,6,2,7,5,4] => [1,3,6,2,7,4,5] => [1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 7 - 1
[1,3,6,4,2,7,5] => [1,3,6,2,7,4,5] => [1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 7 - 1
[1,3,6,4,7,5,2] => [1,3,6,2,4,7,5] => [1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 7 - 1
[1,3,7,2,4,5,6] => [1,3,7,2,4,5,6] => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 7 - 1
[1,3,7,2,6,4,5] => [1,3,7,2,6,4,5] => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 7 - 1
[1,3,7,4,2,5,6] => [1,3,7,2,5,6,4] => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 7 - 1
[1,3,7,4,6,2,5] => [1,3,7,2,5,4,6] => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 7 - 1
[1,4,2,5,6,7,3] => [1,4,2,5,6,7,3] => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> 6 = 7 - 1
[1,4,2,5,7,3,6] => [1,4,2,5,7,3,6] => [1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> 6 = 7 - 1
[1,4,2,6,3,7,5] => [1,4,2,6,3,7,5] => [1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> 6 = 7 - 1
[1,4,2,6,7,5,3] => [1,4,2,6,7,3,5] => [1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> 6 = 7 - 1
[1,4,2,7,3,5,6] => [1,4,2,7,3,5,6] => [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> 6 = 7 - 1
[1,4,2,7,6,3,5] => [1,4,2,7,3,5,6] => [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> 6 = 7 - 1
[1,4,3,5,6,7,2] => [1,4,2,3,5,6,7] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 7 - 1
[1,4,3,5,7,2,6] => [1,4,2,6,3,5,7] => [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> 6 = 7 - 1
[1,4,3,6,2,7,5] => [1,4,2,7,3,6,5] => [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> 6 = 7 - 1
[1,4,3,6,7,5,2] => [1,4,2,3,6,7,5] => [1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 7 - 1
[1,4,3,7,2,5,6] => [1,4,2,5,6,3,7] => [1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> 6 = 7 - 1
[1,4,3,7,6,2,5] => [1,4,2,5,3,7,6] => [1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> 6 = 7 - 1
[1,5,2,3,6,7,4] => [1,5,2,3,6,7,4] => [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> 6 = 7 - 1
[1,5,3,6,7,2,4] => [1,5,2,4,3,6,7] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 7 - 1
[2,1,5,4,6,7,3] => [1,5,2,3,4,6,7] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 7 - 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000837
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000837: Permutations ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 80%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000837: Permutations ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 80%
Values
[1] => [1] => [1] => [1] => ? = 1 - 3
[3,1,2] => [1,3,2] => [1,3,2] => [1,3,2] => 1 = 4 - 3
[3,2,1] => [1,3,2] => [1,3,2] => [1,3,2] => 1 = 4 - 3
[3,1,4,2] => [1,3,4,2] => [1,4,3,2] => [1,4,2,3] => 1 = 4 - 3
[3,2,4,1] => [1,3,4,2] => [1,4,3,2] => [1,4,2,3] => 1 = 4 - 3
[3,1,5,2,4] => [1,3,5,4,2] => [1,5,4,3,2] => [1,5,2,3,4] => 2 = 5 - 3
[3,2,5,1,4] => [1,3,5,4,2] => [1,5,4,3,2] => [1,5,2,3,4] => 2 = 5 - 3
[4,1,2,5,3] => [1,4,5,3,2] => [1,5,4,3,2] => [1,5,2,3,4] => 2 = 5 - 3
[4,2,1,5,3] => [1,4,5,3,2] => [1,5,4,3,2] => [1,5,2,3,4] => 2 = 5 - 3
[5,1,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,2,3,4] => 2 = 5 - 3
[5,2,1,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,2,3,4] => 2 = 5 - 3
[1,4,2,5,6,3] => [1,2,4,5,6,3] => [1,2,6,4,5,3] => [1,2,6,3,4,5] => 3 = 6 - 3
[1,4,2,6,3,5] => [1,2,4,6,5,3] => [1,2,6,5,4,3] => [1,2,6,3,4,5] => 3 = 6 - 3
[1,4,3,5,6,2] => [1,2,4,5,6,3] => [1,2,6,4,5,3] => [1,2,6,3,4,5] => 3 = 6 - 3
[1,4,3,6,2,5] => [1,2,4,6,5,3] => [1,2,6,5,4,3] => [1,2,6,3,4,5] => 3 = 6 - 3
[1,5,2,3,6,4] => [1,2,5,6,4,3] => [1,2,6,5,4,3] => [1,2,6,3,4,5] => 3 = 6 - 3
[1,5,2,6,4,3] => [1,2,5,4,6,3] => [1,2,6,4,5,3] => [1,2,6,3,4,5] => 3 = 6 - 3
[1,5,3,2,6,4] => [1,2,5,6,4,3] => [1,2,6,5,4,3] => [1,2,6,3,4,5] => 3 = 6 - 3
[1,5,3,6,4,2] => [1,2,5,4,6,3] => [1,2,6,4,5,3] => [1,2,6,3,4,5] => 3 = 6 - 3
[1,6,2,3,4,5] => [1,2,6,5,4,3] => [1,2,6,5,4,3] => [1,2,6,3,4,5] => 3 = 6 - 3
[1,6,2,5,3,4] => [1,2,6,4,5,3] => [1,2,6,5,4,3] => [1,2,6,3,4,5] => 3 = 6 - 3
[1,6,3,2,4,5] => [1,2,6,5,4,3] => [1,2,6,5,4,3] => [1,2,6,3,4,5] => 3 = 6 - 3
[1,6,3,5,2,4] => [1,2,6,4,5,3] => [1,2,6,5,4,3] => [1,2,6,3,4,5] => 3 = 6 - 3
[2,4,1,5,6,3] => [1,2,4,5,6,3] => [1,2,6,4,5,3] => [1,2,6,3,4,5] => 3 = 6 - 3
[2,4,1,6,3,5] => [1,2,4,6,5,3] => [1,2,6,5,4,3] => [1,2,6,3,4,5] => 3 = 6 - 3
[2,4,3,5,6,1] => [1,2,4,5,6,3] => [1,2,6,4,5,3] => [1,2,6,3,4,5] => 3 = 6 - 3
[2,4,3,6,1,5] => [1,2,4,6,5,3] => [1,2,6,5,4,3] => [1,2,6,3,4,5] => 3 = 6 - 3
[2,5,1,3,6,4] => [1,2,5,6,4,3] => [1,2,6,5,4,3] => [1,2,6,3,4,5] => 3 = 6 - 3
[2,5,1,6,4,3] => [1,2,5,4,6,3] => [1,2,6,4,5,3] => [1,2,6,3,4,5] => 3 = 6 - 3
[2,5,3,1,6,4] => [1,2,5,6,4,3] => [1,2,6,5,4,3] => [1,2,6,3,4,5] => 3 = 6 - 3
[2,5,3,6,4,1] => [1,2,5,4,6,3] => [1,2,6,4,5,3] => [1,2,6,3,4,5] => 3 = 6 - 3
[2,6,1,3,4,5] => [1,2,6,5,4,3] => [1,2,6,5,4,3] => [1,2,6,3,4,5] => 3 = 6 - 3
[2,6,1,5,3,4] => [1,2,6,4,5,3] => [1,2,6,5,4,3] => [1,2,6,3,4,5] => 3 = 6 - 3
[2,6,3,1,4,5] => [1,2,6,5,4,3] => [1,2,6,5,4,3] => [1,2,6,3,4,5] => 3 = 6 - 3
[2,6,3,5,1,4] => [1,2,6,4,5,3] => [1,2,6,5,4,3] => [1,2,6,3,4,5] => 3 = 6 - 3
[3,1,4,6,2,5] => [1,3,4,6,5,2] => [1,6,3,5,4,2] => [1,6,2,3,5,4] => 3 = 6 - 3
[3,1,5,2,6,4] => [1,3,5,6,4,2] => [1,6,5,4,3,2] => [1,6,2,3,4,5] => 3 = 6 - 3
[3,1,6,2,4,5] => [1,3,6,5,4,2] => [1,6,5,4,3,2] => [1,6,2,3,4,5] => 3 = 6 - 3
[3,2,4,6,1,5] => [1,3,4,6,5,2] => [1,6,3,5,4,2] => [1,6,2,3,5,4] => 3 = 6 - 3
[3,2,5,1,6,4] => [1,3,5,6,4,2] => [1,6,5,4,3,2] => [1,6,2,3,4,5] => 3 = 6 - 3
[3,2,6,1,4,5] => [1,3,6,5,4,2] => [1,6,5,4,3,2] => [1,6,2,3,4,5] => 3 = 6 - 3
[4,1,2,5,6,3] => [1,4,5,6,3,2] => [1,6,5,4,3,2] => [1,6,2,3,4,5] => 3 = 6 - 3
[4,1,2,6,3,5] => [1,4,6,5,3,2] => [1,6,5,4,3,2] => [1,6,2,3,4,5] => 3 = 6 - 3
[4,1,5,6,2,3] => [1,4,6,3,5,2] => [1,6,5,4,3,2] => [1,6,2,3,4,5] => 3 = 6 - 3
[4,1,6,3,2,5] => [1,4,3,6,5,2] => [1,6,3,5,4,2] => [1,6,2,3,5,4] => 3 = 6 - 3
[4,2,1,5,6,3] => [1,4,5,6,3,2] => [1,6,5,4,3,2] => [1,6,2,3,4,5] => 3 = 6 - 3
[4,2,1,6,3,5] => [1,4,6,5,3,2] => [1,6,5,4,3,2] => [1,6,2,3,4,5] => 3 = 6 - 3
[4,2,5,6,1,3] => [1,4,6,3,5,2] => [1,6,5,4,3,2] => [1,6,2,3,4,5] => 3 = 6 - 3
[4,2,6,3,1,5] => [1,4,3,6,5,2] => [1,6,3,5,4,2] => [1,6,2,3,5,4] => 3 = 6 - 3
[5,1,2,3,6,4] => [1,5,6,4,3,2] => [1,6,5,4,3,2] => [1,6,2,3,4,5] => 3 = 6 - 3
[5,1,2,6,4,3] => [1,5,4,6,3,2] => [1,6,5,4,3,2] => [1,6,2,3,4,5] => 3 = 6 - 3
[3,1,4,5,7,2,6] => [1,3,4,5,7,6,2] => [1,7,3,4,6,5,2] => [1,7,2,3,4,6,5] => ? = 7 - 3
[3,1,4,6,2,7,5] => [1,3,4,6,7,5,2] => [1,7,3,6,5,4,2] => [1,7,2,3,6,4,5] => ? = 7 - 3
[3,1,4,7,2,5,6] => [1,3,4,7,6,5,2] => [1,7,3,6,5,4,2] => [1,7,2,3,6,4,5] => ? = 7 - 3
[3,1,5,2,6,7,4] => [1,3,5,6,7,4,2] => [1,7,6,4,5,3,2] => [1,7,2,3,4,5,6] => ? = 7 - 3
[3,1,5,2,7,4,6] => [1,3,5,7,6,4,2] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => ? = 7 - 3
[3,1,5,4,6,7,2] => [1,3,5,6,7,2,4] => [1,6,7,4,5,2,3] => [1,6,7,2,3,4,5] => ? = 7 - 3
[3,1,5,4,7,2,6] => [1,3,5,7,6,2,4] => [1,6,7,5,4,2,3] => [1,6,7,2,3,4,5] => ? = 7 - 3
[3,1,5,6,7,2,4] => [1,3,5,7,4,6,2] => [1,7,5,6,3,4,2] => [1,7,2,3,4,5,6] => ? = 7 - 3
[3,1,5,7,4,2,6] => [1,3,5,4,7,6,2] => [1,7,4,3,6,5,2] => [1,7,2,3,6,4,5] => ? = 7 - 3
[3,1,6,2,4,7,5] => [1,3,6,7,5,4,2] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => ? = 7 - 3
[3,1,6,2,7,5,4] => [1,3,6,5,7,4,2] => [1,7,6,4,5,3,2] => [1,7,2,3,4,5,6] => ? = 7 - 3
[3,1,6,4,2,7,5] => [1,3,6,7,5,2,4] => [1,6,7,5,4,2,3] => [1,6,7,2,3,4,5] => ? = 7 - 3
[3,1,6,4,7,5,2] => [1,3,6,5,7,2,4] => [1,6,7,4,5,2,3] => [1,6,7,2,3,4,5] => ? = 7 - 3
[3,1,6,5,2,7,4] => [1,3,6,7,4,5,2] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => ? = 7 - 3
[3,1,7,2,4,5,6] => [1,3,7,6,5,4,2] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => ? = 7 - 3
[3,1,7,2,6,4,5] => [1,3,7,5,6,4,2] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => ? = 7 - 3
[3,1,7,4,2,5,6] => [1,3,7,6,5,2,4] => [1,6,7,5,4,2,3] => [1,6,7,2,3,4,5] => ? = 7 - 3
[3,1,7,4,6,2,5] => [1,3,7,5,6,2,4] => [1,6,7,5,4,2,3] => [1,6,7,2,3,4,5] => ? = 7 - 3
[3,1,7,6,2,5,4] => [1,3,7,4,6,5,2] => [1,7,6,4,5,3,2] => [1,7,2,3,4,5,6] => ? = 7 - 3
[3,1,7,6,4,2,5] => [1,3,7,5,4,6,2] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => ? = 7 - 3
[3,2,4,5,7,1,6] => [1,3,4,5,7,6,2] => [1,7,3,4,6,5,2] => [1,7,2,3,4,6,5] => ? = 7 - 3
[3,2,4,6,1,7,5] => [1,3,4,6,7,5,2] => [1,7,3,6,5,4,2] => [1,7,2,3,6,4,5] => ? = 7 - 3
[3,2,4,7,1,5,6] => [1,3,4,7,6,5,2] => [1,7,3,6,5,4,2] => [1,7,2,3,6,4,5] => ? = 7 - 3
[3,2,5,1,6,7,4] => [1,3,5,6,7,4,2] => [1,7,6,4,5,3,2] => [1,7,2,3,4,5,6] => ? = 7 - 3
[3,2,5,1,7,4,6] => [1,3,5,7,6,4,2] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => ? = 7 - 3
[3,2,5,4,6,7,1] => [1,3,5,6,7,2,4] => [1,6,7,4,5,2,3] => [1,6,7,2,3,4,5] => ? = 7 - 3
[3,2,5,4,7,1,6] => [1,3,5,7,6,2,4] => [1,6,7,5,4,2,3] => [1,6,7,2,3,4,5] => ? = 7 - 3
[3,2,5,6,7,1,4] => [1,3,5,7,4,6,2] => [1,7,5,6,3,4,2] => [1,7,2,3,4,5,6] => ? = 7 - 3
[3,2,5,7,4,1,6] => [1,3,5,4,7,6,2] => [1,7,4,3,6,5,2] => [1,7,2,3,6,4,5] => ? = 7 - 3
[3,2,6,1,4,7,5] => [1,3,6,7,5,4,2] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => ? = 7 - 3
[3,2,6,1,7,5,4] => [1,3,6,5,7,4,2] => [1,7,6,4,5,3,2] => [1,7,2,3,4,5,6] => ? = 7 - 3
[3,2,6,4,1,7,5] => [1,3,6,7,5,2,4] => [1,6,7,5,4,2,3] => [1,6,7,2,3,4,5] => ? = 7 - 3
[3,2,6,4,7,5,1] => [1,3,6,5,7,2,4] => [1,6,7,4,5,2,3] => [1,6,7,2,3,4,5] => ? = 7 - 3
[3,2,6,5,1,7,4] => [1,3,6,7,4,5,2] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => ? = 7 - 3
[3,2,7,1,4,5,6] => [1,3,7,6,5,4,2] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => ? = 7 - 3
[3,2,7,1,6,4,5] => [1,3,7,5,6,4,2] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => ? = 7 - 3
[3,2,7,4,1,5,6] => [1,3,7,6,5,2,4] => [1,6,7,5,4,2,3] => [1,6,7,2,3,4,5] => ? = 7 - 3
[3,2,7,4,6,1,5] => [1,3,7,5,6,2,4] => [1,6,7,5,4,2,3] => [1,6,7,2,3,4,5] => ? = 7 - 3
[3,2,7,6,1,5,4] => [1,3,7,4,6,5,2] => [1,7,6,4,5,3,2] => [1,7,2,3,4,5,6] => ? = 7 - 3
[3,2,7,6,4,1,5] => [1,3,7,5,4,6,2] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => ? = 7 - 3
[3,4,5,1,6,7,2] => [1,3,5,6,7,2,4] => [1,6,7,4,5,2,3] => [1,6,7,2,3,4,5] => ? = 7 - 3
[3,4,5,1,7,2,6] => [1,3,5,7,6,2,4] => [1,6,7,5,4,2,3] => [1,6,7,2,3,4,5] => ? = 7 - 3
[3,4,5,2,6,7,1] => [1,3,5,6,7,2,4] => [1,6,7,4,5,2,3] => [1,6,7,2,3,4,5] => ? = 7 - 3
[3,4,5,2,7,1,6] => [1,3,5,7,6,2,4] => [1,6,7,5,4,2,3] => [1,6,7,2,3,4,5] => ? = 7 - 3
[3,4,6,1,2,7,5] => [1,3,6,7,5,2,4] => [1,6,7,5,4,2,3] => [1,6,7,2,3,4,5] => ? = 7 - 3
[3,4,6,1,7,5,2] => [1,3,6,5,7,2,4] => [1,6,7,4,5,2,3] => [1,6,7,2,3,4,5] => ? = 7 - 3
[3,4,6,2,1,7,5] => [1,3,6,7,5,2,4] => [1,6,7,5,4,2,3] => [1,6,7,2,3,4,5] => ? = 7 - 3
[3,4,6,2,7,5,1] => [1,3,6,5,7,2,4] => [1,6,7,4,5,2,3] => [1,6,7,2,3,4,5] => ? = 7 - 3
[3,4,7,1,2,5,6] => [1,3,7,6,5,2,4] => [1,6,7,5,4,2,3] => [1,6,7,2,3,4,5] => ? = 7 - 3
Description
The number of ascents of distance 2 of a permutation.
This is, $\operatorname{asc}_2(\pi) = | \{ i : \pi(i) < \pi(i+2) \} |$.
Matching statistic: St001198
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 20%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 20%
Values
[1] => [1]
=> []
=> []
=> ? = 1 - 5
[3,1,2] => [3]
=> []
=> []
=> ? = 4 - 5
[3,2,1] => [2,1]
=> [1]
=> [1,0]
=> ? = 4 - 5
[3,1,4,2] => [4]
=> []
=> []
=> ? = 4 - 5
[3,2,4,1] => [3,1]
=> [1]
=> [1,0]
=> ? = 4 - 5
[3,1,5,2,4] => [5]
=> []
=> []
=> ? = 5 - 5
[3,2,5,1,4] => [4,1]
=> [1]
=> [1,0]
=> ? = 5 - 5
[4,1,2,5,3] => [5]
=> []
=> []
=> ? = 5 - 5
[4,2,1,5,3] => [4,1]
=> [1]
=> [1,0]
=> ? = 5 - 5
[5,1,2,3,4] => [5]
=> []
=> []
=> ? = 5 - 5
[5,2,1,3,4] => [4,1]
=> [1]
=> [1,0]
=> ? = 5 - 5
[1,4,2,5,6,3] => [5,1]
=> [1]
=> [1,0]
=> ? = 6 - 5
[1,4,2,6,3,5] => [5,1]
=> [1]
=> [1,0]
=> ? = 6 - 5
[1,4,3,5,6,2] => [4,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? = 6 - 5
[1,4,3,6,2,5] => [4,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? = 6 - 5
[1,5,2,3,6,4] => [5,1]
=> [1]
=> [1,0]
=> ? = 6 - 5
[1,5,2,6,4,3] => [5,1]
=> [1]
=> [1,0]
=> ? = 6 - 5
[1,5,3,2,6,4] => [4,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? = 6 - 5
[1,5,3,6,4,2] => [4,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? = 6 - 5
[1,6,2,3,4,5] => [5,1]
=> [1]
=> [1,0]
=> ? = 6 - 5
[1,6,2,5,3,4] => [5,1]
=> [1]
=> [1,0]
=> ? = 6 - 5
[1,6,3,2,4,5] => [4,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? = 6 - 5
[1,6,3,5,2,4] => [4,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? = 6 - 5
[2,4,1,5,6,3] => [6]
=> []
=> []
=> ? = 6 - 5
[2,4,1,6,3,5] => [6]
=> []
=> []
=> ? = 6 - 5
[2,4,3,5,6,1] => [5,1]
=> [1]
=> [1,0]
=> ? = 6 - 5
[2,4,3,6,1,5] => [5,1]
=> [1]
=> [1,0]
=> ? = 6 - 5
[2,5,1,3,6,4] => [6]
=> []
=> []
=> ? = 6 - 5
[2,5,1,6,4,3] => [6]
=> []
=> []
=> ? = 6 - 5
[2,5,3,1,6,4] => [5,1]
=> [1]
=> [1,0]
=> ? = 6 - 5
[2,5,3,6,4,1] => [5,1]
=> [1]
=> [1,0]
=> ? = 6 - 5
[2,6,1,3,4,5] => [6]
=> []
=> []
=> ? = 6 - 5
[2,6,1,5,3,4] => [6]
=> []
=> []
=> ? = 6 - 5
[2,6,3,1,4,5] => [5,1]
=> [1]
=> [1,0]
=> ? = 6 - 5
[2,6,3,5,1,4] => [5,1]
=> [1]
=> [1,0]
=> ? = 6 - 5
[3,1,4,6,2,5] => [6]
=> []
=> []
=> ? = 6 - 5
[3,1,5,2,6,4] => [6]
=> []
=> []
=> ? = 6 - 5
[3,1,6,2,4,5] => [6]
=> []
=> []
=> ? = 6 - 5
[3,2,4,6,1,5] => [5,1]
=> [1]
=> [1,0]
=> ? = 6 - 5
[3,2,5,1,6,4] => [5,1]
=> [1]
=> [1,0]
=> ? = 6 - 5
[3,2,6,1,4,5] => [5,1]
=> [1]
=> [1,0]
=> ? = 6 - 5
[4,1,2,5,6,3] => [6]
=> []
=> []
=> ? = 6 - 5
[4,1,2,6,3,5] => [6]
=> []
=> []
=> ? = 6 - 5
[4,1,5,6,2,3] => [6]
=> []
=> []
=> ? = 6 - 5
[4,1,6,3,2,5] => [6]
=> []
=> []
=> ? = 6 - 5
[4,2,1,5,6,3] => [5,1]
=> [1]
=> [1,0]
=> ? = 6 - 5
[4,2,1,6,3,5] => [5,1]
=> [1]
=> [1,0]
=> ? = 6 - 5
[4,2,5,6,1,3] => [5,1]
=> [1]
=> [1,0]
=> ? = 6 - 5
[4,2,6,3,1,5] => [5,1]
=> [1]
=> [1,0]
=> ? = 6 - 5
[5,1,2,3,6,4] => [6]
=> []
=> []
=> ? = 6 - 5
[1,2,5,4,6,7,3] => [4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 7 - 5
[1,2,5,4,7,3,6] => [4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 7 - 5
[1,2,6,4,3,7,5] => [4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 7 - 5
[1,2,6,4,7,5,3] => [4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 7 - 5
[1,2,7,4,3,5,6] => [4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 7 - 5
[1,2,7,4,6,3,5] => [4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 7 - 5
[2,1,5,3,6,7,4] => [5,2]
=> [2]
=> [1,0,1,0]
=> 2 = 7 - 5
[2,1,5,3,7,4,6] => [5,2]
=> [2]
=> [1,0,1,0]
=> 2 = 7 - 5
[2,1,5,4,6,7,3] => [4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2 = 7 - 5
[2,1,5,4,7,3,6] => [4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2 = 7 - 5
[2,1,6,3,4,7,5] => [5,2]
=> [2]
=> [1,0,1,0]
=> 2 = 7 - 5
[2,1,6,3,7,5,4] => [5,2]
=> [2]
=> [1,0,1,0]
=> 2 = 7 - 5
[2,1,6,4,3,7,5] => [4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2 = 7 - 5
[2,1,6,4,7,5,3] => [4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2 = 7 - 5
[2,1,7,3,4,5,6] => [5,2]
=> [2]
=> [1,0,1,0]
=> 2 = 7 - 5
[2,1,7,3,6,4,5] => [5,2]
=> [2]
=> [1,0,1,0]
=> 2 = 7 - 5
[2,1,7,4,3,5,6] => [4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2 = 7 - 5
[2,1,7,4,6,3,5] => [4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2 = 7 - 5
[3,4,5,2,6,7,1] => [5,2]
=> [2]
=> [1,0,1,0]
=> 2 = 7 - 5
[3,4,5,2,7,1,6] => [5,2]
=> [2]
=> [1,0,1,0]
=> 2 = 7 - 5
[3,4,6,2,1,7,5] => [5,2]
=> [2]
=> [1,0,1,0]
=> 2 = 7 - 5
[3,4,6,2,7,5,1] => [5,2]
=> [2]
=> [1,0,1,0]
=> 2 = 7 - 5
[3,4,7,2,1,5,6] => [5,2]
=> [2]
=> [1,0,1,0]
=> 2 = 7 - 5
[3,4,7,2,6,1,5] => [5,2]
=> [2]
=> [1,0,1,0]
=> 2 = 7 - 5
[3,5,1,2,7,4,6] => [5,2]
=> [2]
=> [1,0,1,0]
=> 2 = 7 - 5
[3,5,1,4,7,2,6] => [4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2 = 7 - 5
[3,5,6,2,4,7,1] => [4,3]
=> [3]
=> [1,0,1,0,1,0]
=> 2 = 7 - 5
[3,5,6,4,2,7,1] => [4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2 = 7 - 5
[3,6,1,2,4,7,5] => [5,2]
=> [2]
=> [1,0,1,0]
=> 2 = 7 - 5
[3,6,1,2,7,5,4] => [5,2]
=> [2]
=> [1,0,1,0]
=> 2 = 7 - 5
[3,6,1,4,2,7,5] => [4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2 = 7 - 5
[3,6,1,4,7,5,2] => [4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2 = 7 - 5
[3,6,5,2,1,7,4] => [4,3]
=> [3]
=> [1,0,1,0,1,0]
=> 2 = 7 - 5
[3,6,5,2,7,4,1] => [4,3]
=> [3]
=> [1,0,1,0,1,0]
=> 2 = 7 - 5
[3,6,5,4,1,7,2] => [3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 7 - 5
[3,6,5,4,7,2,1] => [4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2 = 7 - 5
[3,6,7,2,1,4,5] => [4,3]
=> [3]
=> [1,0,1,0,1,0]
=> 2 = 7 - 5
[3,6,7,2,4,5,1] => [4,3]
=> [3]
=> [1,0,1,0,1,0]
=> 2 = 7 - 5
[3,6,7,4,1,2,5] => [4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2 = 7 - 5
[3,6,7,4,2,5,1] => [3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 7 - 5
[3,7,1,2,4,5,6] => [5,2]
=> [2]
=> [1,0,1,0]
=> 2 = 7 - 5
[3,7,1,4,2,5,6] => [4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2 = 7 - 5
[3,7,5,2,1,4,6] => [4,3]
=> [3]
=> [1,0,1,0,1,0]
=> 2 = 7 - 5
[3,7,5,2,6,1,4] => [4,3]
=> [3]
=> [1,0,1,0,1,0]
=> 2 = 7 - 5
[3,7,5,4,1,2,6] => [3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 7 - 5
[3,7,5,4,6,1,2] => [4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2 = 7 - 5
[3,7,6,2,1,5,4] => [4,3]
=> [3]
=> [1,0,1,0,1,0]
=> 2 = 7 - 5
[3,7,6,4,1,5,2] => [4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2 = 7 - 5
[4,3,2,5,7,1,6] => [5,2]
=> [2]
=> [1,0,1,0]
=> 2 = 7 - 5
[4,3,2,6,1,7,5] => [5,2]
=> [2]
=> [1,0,1,0]
=> 2 = 7 - 5
Description
The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
The following 29 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000422The energy of a graph, if it is integral. St001118The acyclic chromatic index of a graph. St000702The number of weak deficiencies of a permutation. St000956The maximal displacement of a permutation. St001246The maximal difference between two consecutive entries of a permutation. St000836The number of descents of distance 2 of a permutation. St001520The number of strict 3-descents. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001060The distinguishing index of a graph. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition.
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