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Your data matches 34 different statistics following compositions of up to 3 maps.
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Matching statistic: St000260
Mp00223: Permutations —runsort⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ([],1)
=> 0
[1,2] => [1,2] => [2,1] => ([(0,1)],2)
=> 1
[2,1] => [1,2] => [2,1] => ([(0,1)],2)
=> 1
[1,2,3] => [1,2,3] => [2,3,1] => ([(0,2),(1,2)],3)
=> 1
[2,3,1] => [1,2,3] => [2,3,1] => ([(0,2),(1,2)],3)
=> 1
[3,1,2] => [1,2,3] => [2,3,1] => ([(0,2),(1,2)],3)
=> 1
[3,2,1] => [1,2,3] => [2,3,1] => ([(0,2),(1,2)],3)
=> 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,3,2,4] => [1,3,2,4] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,4,2,3] => [1,4,2,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2
[1,4,3,2] => [1,4,2,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2
[2,1,4,3] => [1,4,2,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2
[2,3,1,4] => [1,4,2,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[2,4,1,3] => [1,3,2,4] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,1,4,2] => [1,4,2,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2
[3,2,1,4] => [1,4,2,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2
[3,4,1,2] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[3,4,2,1] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4,1,2,3] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4,1,3,2] => [1,3,2,4] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,2,1,3] => [1,3,2,4] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,2,3,1] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4,3,1,2] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4,3,2,1] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,2,5,3,4] => [1,2,5,3,4] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[1,2,5,4,3] => [1,2,5,3,4] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[1,3,2,4,5] => [1,3,2,4,5] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,4,2,5] => [1,3,4,2,5] => [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,5,2,4] => [1,3,5,2,4] => [2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,5,4,2] => [1,3,5,2,4] => [2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,2,3,5] => [1,4,2,3,5] => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,3,5,2] => [1,4,2,3,5] => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,5,2,3] => [1,4,5,2,3] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[1,4,5,3,2] => [1,4,5,2,3] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[1,5,2,3,4] => [1,5,2,3,4] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[1,5,2,4,3] => [1,5,2,4,3] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[1,5,3,2,4] => [1,5,2,4,3] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[1,5,3,4,2] => [1,5,2,3,4] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[1,5,4,2,3] => [1,5,2,3,4] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[1,5,4,3,2] => [1,5,2,3,4] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[2,1,3,5,4] => [1,3,5,2,4] => [2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[2,1,4,3,5] => [1,4,2,3,5] => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,1,4,5,3] => [1,4,5,2,3] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[2,1,5,3,4] => [1,5,2,3,4] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[2,1,5,4,3] => [1,5,2,3,4] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[2,3,1,4,5] => [1,4,5,2,3] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[2,3,1,5,4] => [1,5,2,3,4] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St000326
Mp00223: Permutations —runsort⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00136: Binary words —rotate back-to-front⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 67% ●values known / values provided: 100%●distinct values known / distinct values provided: 67%
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00136: Binary words —rotate back-to-front⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 67% ●values known / values provided: 100%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => => ? => ? = 0
[1,2] => [1,2] => 1 => 1 => 1
[2,1] => [1,2] => 1 => 1 => 1
[1,2,3] => [1,2,3] => 11 => 11 => 1
[2,3,1] => [1,2,3] => 11 => 11 => 1
[3,1,2] => [1,2,3] => 11 => 11 => 1
[3,2,1] => [1,2,3] => 11 => 11 => 1
[1,2,3,4] => [1,2,3,4] => 111 => 111 => 1
[1,3,2,4] => [1,3,2,4] => 101 => 110 => 1
[1,4,2,3] => [1,4,2,3] => 100 => 010 => 2
[1,4,3,2] => [1,4,2,3] => 100 => 010 => 2
[2,1,4,3] => [1,4,2,3] => 100 => 010 => 2
[2,3,1,4] => [1,4,2,3] => 100 => 010 => 2
[2,3,4,1] => [1,2,3,4] => 111 => 111 => 1
[2,4,1,3] => [1,3,2,4] => 101 => 110 => 1
[3,1,4,2] => [1,4,2,3] => 100 => 010 => 2
[3,2,1,4] => [1,4,2,3] => 100 => 010 => 2
[3,4,1,2] => [1,2,3,4] => 111 => 111 => 1
[3,4,2,1] => [1,2,3,4] => 111 => 111 => 1
[4,1,2,3] => [1,2,3,4] => 111 => 111 => 1
[4,1,3,2] => [1,3,2,4] => 101 => 110 => 1
[4,2,1,3] => [1,3,2,4] => 101 => 110 => 1
[4,2,3,1] => [1,2,3,4] => 111 => 111 => 1
[4,3,1,2] => [1,2,3,4] => 111 => 111 => 1
[4,3,2,1] => [1,2,3,4] => 111 => 111 => 1
[1,2,3,4,5] => [1,2,3,4,5] => 1111 => 1111 => 1
[1,2,4,3,5] => [1,2,4,3,5] => 1101 => 1110 => 1
[1,2,5,3,4] => [1,2,5,3,4] => 1100 => 0110 => 2
[1,2,5,4,3] => [1,2,5,3,4] => 1100 => 0110 => 2
[1,3,2,4,5] => [1,3,2,4,5] => 1011 => 1101 => 1
[1,3,4,2,5] => [1,3,4,2,5] => 1001 => 1100 => 1
[1,3,5,2,4] => [1,3,5,2,4] => 1000 => 0100 => 2
[1,3,5,4,2] => [1,3,5,2,4] => 1000 => 0100 => 2
[1,4,2,3,5] => [1,4,2,3,5] => 1001 => 1100 => 1
[1,4,3,5,2] => [1,4,2,3,5] => 1001 => 1100 => 1
[1,4,5,2,3] => [1,4,5,2,3] => 1000 => 0100 => 2
[1,4,5,3,2] => [1,4,5,2,3] => 1000 => 0100 => 2
[1,5,2,3,4] => [1,5,2,3,4] => 1000 => 0100 => 2
[1,5,2,4,3] => [1,5,2,4,3] => 1000 => 0100 => 2
[1,5,3,2,4] => [1,5,2,4,3] => 1000 => 0100 => 2
[1,5,3,4,2] => [1,5,2,3,4] => 1000 => 0100 => 2
[1,5,4,2,3] => [1,5,2,3,4] => 1000 => 0100 => 2
[1,5,4,3,2] => [1,5,2,3,4] => 1000 => 0100 => 2
[2,1,3,5,4] => [1,3,5,2,4] => 1000 => 0100 => 2
[2,1,4,3,5] => [1,4,2,3,5] => 1001 => 1100 => 1
[2,1,4,5,3] => [1,4,5,2,3] => 1000 => 0100 => 2
[2,1,5,3,4] => [1,5,2,3,4] => 1000 => 0100 => 2
[2,1,5,4,3] => [1,5,2,3,4] => 1000 => 0100 => 2
[2,3,1,4,5] => [1,4,5,2,3] => 1000 => 0100 => 2
[2,3,1,5,4] => [1,5,2,3,4] => 1000 => 0100 => 2
[2,3,4,1,5] => [1,5,2,3,4] => 1000 => 0100 => 2
Description
The position of the first one in a binary word after appending a 1 at the end.
Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Matching statistic: St000297
Mp00223: Permutations —runsort⟶ Permutations
Mp00130: Permutations —descent tops⟶ Binary words
Mp00136: Binary words —rotate back-to-front⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 67% ●values known / values provided: 100%●distinct values known / distinct values provided: 67%
Mp00130: Permutations —descent tops⟶ Binary words
Mp00136: Binary words —rotate back-to-front⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 67% ●values known / values provided: 100%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => => ? => ? = 0 - 1
[1,2] => [1,2] => 0 => 0 => 0 = 1 - 1
[2,1] => [1,2] => 0 => 0 => 0 = 1 - 1
[1,2,3] => [1,2,3] => 00 => 00 => 0 = 1 - 1
[2,3,1] => [1,2,3] => 00 => 00 => 0 = 1 - 1
[3,1,2] => [1,2,3] => 00 => 00 => 0 = 1 - 1
[3,2,1] => [1,2,3] => 00 => 00 => 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => 000 => 000 => 0 = 1 - 1
[1,3,2,4] => [1,3,2,4] => 010 => 001 => 0 = 1 - 1
[1,4,2,3] => [1,4,2,3] => 001 => 100 => 1 = 2 - 1
[1,4,3,2] => [1,4,2,3] => 001 => 100 => 1 = 2 - 1
[2,1,4,3] => [1,4,2,3] => 001 => 100 => 1 = 2 - 1
[2,3,1,4] => [1,4,2,3] => 001 => 100 => 1 = 2 - 1
[2,3,4,1] => [1,2,3,4] => 000 => 000 => 0 = 1 - 1
[2,4,1,3] => [1,3,2,4] => 010 => 001 => 0 = 1 - 1
[3,1,4,2] => [1,4,2,3] => 001 => 100 => 1 = 2 - 1
[3,2,1,4] => [1,4,2,3] => 001 => 100 => 1 = 2 - 1
[3,4,1,2] => [1,2,3,4] => 000 => 000 => 0 = 1 - 1
[3,4,2,1] => [1,2,3,4] => 000 => 000 => 0 = 1 - 1
[4,1,2,3] => [1,2,3,4] => 000 => 000 => 0 = 1 - 1
[4,1,3,2] => [1,3,2,4] => 010 => 001 => 0 = 1 - 1
[4,2,1,3] => [1,3,2,4] => 010 => 001 => 0 = 1 - 1
[4,2,3,1] => [1,2,3,4] => 000 => 000 => 0 = 1 - 1
[4,3,1,2] => [1,2,3,4] => 000 => 000 => 0 = 1 - 1
[4,3,2,1] => [1,2,3,4] => 000 => 000 => 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0000 => 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3,5] => 0010 => 0001 => 0 = 1 - 1
[1,2,5,3,4] => [1,2,5,3,4] => 0001 => 1000 => 1 = 2 - 1
[1,2,5,4,3] => [1,2,5,3,4] => 0001 => 1000 => 1 = 2 - 1
[1,3,2,4,5] => [1,3,2,4,5] => 0100 => 0010 => 0 = 1 - 1
[1,3,4,2,5] => [1,3,4,2,5] => 0010 => 0001 => 0 = 1 - 1
[1,3,5,2,4] => [1,3,5,2,4] => 0001 => 1000 => 1 = 2 - 1
[1,3,5,4,2] => [1,3,5,2,4] => 0001 => 1000 => 1 = 2 - 1
[1,4,2,3,5] => [1,4,2,3,5] => 0010 => 0001 => 0 = 1 - 1
[1,4,3,5,2] => [1,4,2,3,5] => 0010 => 0001 => 0 = 1 - 1
[1,4,5,2,3] => [1,4,5,2,3] => 0001 => 1000 => 1 = 2 - 1
[1,4,5,3,2] => [1,4,5,2,3] => 0001 => 1000 => 1 = 2 - 1
[1,5,2,3,4] => [1,5,2,3,4] => 0001 => 1000 => 1 = 2 - 1
[1,5,2,4,3] => [1,5,2,4,3] => 0011 => 1001 => 1 = 2 - 1
[1,5,3,2,4] => [1,5,2,4,3] => 0011 => 1001 => 1 = 2 - 1
[1,5,3,4,2] => [1,5,2,3,4] => 0001 => 1000 => 1 = 2 - 1
[1,5,4,2,3] => [1,5,2,3,4] => 0001 => 1000 => 1 = 2 - 1
[1,5,4,3,2] => [1,5,2,3,4] => 0001 => 1000 => 1 = 2 - 1
[2,1,3,5,4] => [1,3,5,2,4] => 0001 => 1000 => 1 = 2 - 1
[2,1,4,3,5] => [1,4,2,3,5] => 0010 => 0001 => 0 = 1 - 1
[2,1,4,5,3] => [1,4,5,2,3] => 0001 => 1000 => 1 = 2 - 1
[2,1,5,3,4] => [1,5,2,3,4] => 0001 => 1000 => 1 = 2 - 1
[2,1,5,4,3] => [1,5,2,3,4] => 0001 => 1000 => 1 = 2 - 1
[2,3,1,4,5] => [1,4,5,2,3] => 0001 => 1000 => 1 = 2 - 1
[2,3,1,5,4] => [1,5,2,3,4] => 0001 => 1000 => 1 = 2 - 1
[2,3,4,1,5] => [1,5,2,3,4] => 0001 => 1000 => 1 = 2 - 1
Description
The number of leading ones in a binary word.
Matching statistic: St001730
Mp00223: Permutations —runsort⟶ Permutations
Mp00130: Permutations —descent tops⟶ Binary words
Mp00136: Binary words —rotate back-to-front⟶ Binary words
St001730: Binary words ⟶ ℤResult quality: 67% ●values known / values provided: 100%●distinct values known / distinct values provided: 67%
Mp00130: Permutations —descent tops⟶ Binary words
Mp00136: Binary words —rotate back-to-front⟶ Binary words
St001730: Binary words ⟶ ℤResult quality: 67% ●values known / values provided: 100%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => => ? => ? = 0 - 1
[1,2] => [1,2] => 0 => 0 => 0 = 1 - 1
[2,1] => [1,2] => 0 => 0 => 0 = 1 - 1
[1,2,3] => [1,2,3] => 00 => 00 => 0 = 1 - 1
[2,3,1] => [1,2,3] => 00 => 00 => 0 = 1 - 1
[3,1,2] => [1,2,3] => 00 => 00 => 0 = 1 - 1
[3,2,1] => [1,2,3] => 00 => 00 => 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => 000 => 000 => 0 = 1 - 1
[1,3,2,4] => [1,3,2,4] => 010 => 001 => 0 = 1 - 1
[1,4,2,3] => [1,4,2,3] => 001 => 100 => 1 = 2 - 1
[1,4,3,2] => [1,4,2,3] => 001 => 100 => 1 = 2 - 1
[2,1,4,3] => [1,4,2,3] => 001 => 100 => 1 = 2 - 1
[2,3,1,4] => [1,4,2,3] => 001 => 100 => 1 = 2 - 1
[2,3,4,1] => [1,2,3,4] => 000 => 000 => 0 = 1 - 1
[2,4,1,3] => [1,3,2,4] => 010 => 001 => 0 = 1 - 1
[3,1,4,2] => [1,4,2,3] => 001 => 100 => 1 = 2 - 1
[3,2,1,4] => [1,4,2,3] => 001 => 100 => 1 = 2 - 1
[3,4,1,2] => [1,2,3,4] => 000 => 000 => 0 = 1 - 1
[3,4,2,1] => [1,2,3,4] => 000 => 000 => 0 = 1 - 1
[4,1,2,3] => [1,2,3,4] => 000 => 000 => 0 = 1 - 1
[4,1,3,2] => [1,3,2,4] => 010 => 001 => 0 = 1 - 1
[4,2,1,3] => [1,3,2,4] => 010 => 001 => 0 = 1 - 1
[4,2,3,1] => [1,2,3,4] => 000 => 000 => 0 = 1 - 1
[4,3,1,2] => [1,2,3,4] => 000 => 000 => 0 = 1 - 1
[4,3,2,1] => [1,2,3,4] => 000 => 000 => 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0000 => 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3,5] => 0010 => 0001 => 0 = 1 - 1
[1,2,5,3,4] => [1,2,5,3,4] => 0001 => 1000 => 1 = 2 - 1
[1,2,5,4,3] => [1,2,5,3,4] => 0001 => 1000 => 1 = 2 - 1
[1,3,2,4,5] => [1,3,2,4,5] => 0100 => 0010 => 0 = 1 - 1
[1,3,4,2,5] => [1,3,4,2,5] => 0010 => 0001 => 0 = 1 - 1
[1,3,5,2,4] => [1,3,5,2,4] => 0001 => 1000 => 1 = 2 - 1
[1,3,5,4,2] => [1,3,5,2,4] => 0001 => 1000 => 1 = 2 - 1
[1,4,2,3,5] => [1,4,2,3,5] => 0010 => 0001 => 0 = 1 - 1
[1,4,3,5,2] => [1,4,2,3,5] => 0010 => 0001 => 0 = 1 - 1
[1,4,5,2,3] => [1,4,5,2,3] => 0001 => 1000 => 1 = 2 - 1
[1,4,5,3,2] => [1,4,5,2,3] => 0001 => 1000 => 1 = 2 - 1
[1,5,2,3,4] => [1,5,2,3,4] => 0001 => 1000 => 1 = 2 - 1
[1,5,2,4,3] => [1,5,2,4,3] => 0011 => 1001 => 1 = 2 - 1
[1,5,3,2,4] => [1,5,2,4,3] => 0011 => 1001 => 1 = 2 - 1
[1,5,3,4,2] => [1,5,2,3,4] => 0001 => 1000 => 1 = 2 - 1
[1,5,4,2,3] => [1,5,2,3,4] => 0001 => 1000 => 1 = 2 - 1
[1,5,4,3,2] => [1,5,2,3,4] => 0001 => 1000 => 1 = 2 - 1
[2,1,3,5,4] => [1,3,5,2,4] => 0001 => 1000 => 1 = 2 - 1
[2,1,4,3,5] => [1,4,2,3,5] => 0010 => 0001 => 0 = 1 - 1
[2,1,4,5,3] => [1,4,5,2,3] => 0001 => 1000 => 1 = 2 - 1
[2,1,5,3,4] => [1,5,2,3,4] => 0001 => 1000 => 1 = 2 - 1
[2,1,5,4,3] => [1,5,2,3,4] => 0001 => 1000 => 1 = 2 - 1
[2,3,1,4,5] => [1,4,5,2,3] => 0001 => 1000 => 1 = 2 - 1
[2,3,1,5,4] => [1,5,2,3,4] => 0001 => 1000 => 1 = 2 - 1
[2,3,4,1,5] => [1,5,2,3,4] => 0001 => 1000 => 1 = 2 - 1
Description
The number of times the path corresponding to a binary word crosses the base line.
Interpret each $0$ as a step $(1,-1)$ and $1$ as a step $(1,1)$. Then this statistic counts the number of times the path crosses the $x$-axis.
Matching statistic: St001498
Mp00223: Permutations —runsort⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 65%●distinct values known / distinct values provided: 33%
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 65%●distinct values known / distinct values provided: 33%
Values
[1] => [1] => [1] => [1,0]
=> ? = 0 - 2
[1,2] => [1,2] => [2,1] => [1,1,0,0]
=> ? = 1 - 2
[2,1] => [1,2] => [2,1] => [1,1,0,0]
=> ? = 1 - 2
[1,2,3] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? = 1 - 2
[2,3,1] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? = 1 - 2
[3,1,2] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? = 1 - 2
[3,2,1] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? = 1 - 2
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 1 - 2
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 1 - 2
[1,4,2,3] => [1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 0 = 2 - 2
[1,4,3,2] => [1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 0 = 2 - 2
[2,1,4,3] => [1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 0 = 2 - 2
[2,3,1,4] => [1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 0 = 2 - 2
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 1 - 2
[2,4,1,3] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 1 - 2
[3,1,4,2] => [1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 0 = 2 - 2
[3,2,1,4] => [1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 0 = 2 - 2
[3,4,1,2] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 1 - 2
[3,4,2,1] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 1 - 2
[4,1,2,3] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 1 - 2
[4,1,3,2] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 1 - 2
[4,2,1,3] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 1 - 2
[4,2,3,1] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 1 - 2
[4,3,1,2] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 1 - 2
[4,3,2,1] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 1 - 2
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 2
[1,2,4,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 2
[1,2,5,3,4] => [1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 0 = 2 - 2
[1,2,5,4,3] => [1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 0 = 2 - 2
[1,3,2,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 2
[1,3,4,2,5] => [1,3,4,2,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 2
[1,3,5,2,4] => [1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> 0 = 2 - 2
[1,3,5,4,2] => [1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> 0 = 2 - 2
[1,4,2,3,5] => [1,4,2,3,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 2
[1,4,3,5,2] => [1,4,2,3,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 2
[1,4,5,2,3] => [1,4,5,2,3] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> 0 = 2 - 2
[1,4,5,3,2] => [1,4,5,2,3] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> 0 = 2 - 2
[1,5,2,3,4] => [1,5,2,3,4] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 0 = 2 - 2
[1,5,2,4,3] => [1,5,2,4,3] => [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> 0 = 2 - 2
[1,5,3,2,4] => [1,5,2,4,3] => [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> 0 = 2 - 2
[1,5,3,4,2] => [1,5,2,3,4] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 0 = 2 - 2
[1,5,4,2,3] => [1,5,2,3,4] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 0 = 2 - 2
[1,5,4,3,2] => [1,5,2,3,4] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 0 = 2 - 2
[2,1,3,5,4] => [1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> 0 = 2 - 2
[2,1,4,3,5] => [1,4,2,3,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 2
[2,1,4,5,3] => [1,4,5,2,3] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> 0 = 2 - 2
[2,1,5,3,4] => [1,5,2,3,4] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 0 = 2 - 2
[2,1,5,4,3] => [1,5,2,3,4] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 0 = 2 - 2
[2,3,1,4,5] => [1,4,5,2,3] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> 0 = 2 - 2
[2,3,1,5,4] => [1,5,2,3,4] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 0 = 2 - 2
[2,3,4,1,5] => [1,5,2,3,4] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 0 = 2 - 2
[2,3,4,5,1] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 2
[2,3,5,1,4] => [1,4,2,3,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 2
[2,4,1,3,5] => [1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> 0 = 2 - 2
[2,4,1,5,3] => [1,5,2,4,3] => [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> 0 = 2 - 2
[2,4,3,1,5] => [1,5,2,4,3] => [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> 0 = 2 - 2
[2,4,3,5,1] => [1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 2
[2,4,5,1,3] => [1,3,2,4,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 2
[2,5,1,3,4] => [1,3,4,2,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 2
[2,5,3,4,1] => [1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 0 = 2 - 2
[2,5,4,3,1] => [1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 0 = 2 - 2
[3,1,2,5,4] => [1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 0 = 2 - 2
[3,1,4,5,2] => [1,4,5,2,3] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> 0 = 2 - 2
[3,1,5,2,4] => [1,5,2,4,3] => [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> 0 = 2 - 2
[3,1,5,4,2] => [1,5,2,3,4] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 0 = 2 - 2
[3,2,1,4,5] => [1,4,5,2,3] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> 0 = 2 - 2
[3,2,1,5,4] => [1,5,2,3,4] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 0 = 2 - 2
[3,2,4,1,5] => [1,5,2,4,3] => [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> 0 = 2 - 2
[3,2,5,4,1] => [1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 0 = 2 - 2
[3,4,1,2,5] => [1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 0 = 2 - 2
[3,4,1,5,2] => [1,5,2,3,4] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 0 = 2 - 2
[3,4,2,1,5] => [1,5,2,3,4] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 0 = 2 - 2
[3,4,2,5,1] => [1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 0 = 2 - 2
[3,4,5,1,2] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 2
[3,4,5,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 2
[3,5,1,2,4] => [1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 2
[3,5,1,4,2] => [1,4,2,3,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 2
[3,5,2,1,4] => [1,4,2,3,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 2
[3,5,2,4,1] => [1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 2
[4,1,2,5,3] => [1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 0 = 2 - 2
[4,1,3,5,2] => [1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> 0 = 2 - 2
[4,1,5,2,3] => [1,5,2,3,4] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 0 = 2 - 2
[4,1,5,3,2] => [1,5,2,3,4] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 0 = 2 - 2
[4,2,1,3,5] => [1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> 0 = 2 - 2
[4,2,1,5,3] => [1,5,2,3,4] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 0 = 2 - 2
[4,2,3,1,5] => [1,5,2,3,4] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 0 = 2 - 2
[4,2,5,3,1] => [1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 0 = 2 - 2
[4,5,1,2,3] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 2
[4,5,1,3,2] => [1,3,2,4,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 2
[4,5,2,1,3] => [1,3,2,4,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 2
[4,5,2,3,1] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 2
[4,5,3,1,2] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 2
[4,5,3,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 2
[5,1,2,3,4] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 2
[5,1,2,4,3] => [1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 2
[5,1,3,2,4] => [1,3,2,4,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 2
[5,1,3,4,2] => [1,3,4,2,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 2
[5,1,4,2,3] => [1,4,2,3,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 2
[5,1,4,3,2] => [1,4,2,3,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 2
[5,2,1,3,4] => [1,3,4,2,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 2
Description
The normalised height of a Nakayama algebra with magnitude 1.
We use the bijection (see code) suggested by Christian Stump, to have a bijection between such Nakayama algebras with magnitude 1 and Dyck paths. The normalised height is the height of the (periodic) Dyck path given by the top of the Auslander-Reiten quiver. Thus when having a CNakayama algebra it is the Loewy length minus the number of simple modules and for the LNakayama algebras it is the usual height.
Matching statistic: St000541
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00325: Permutations —ones to leading⟶ Permutations
St000541: Permutations ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 67%
Mp00325: Permutations —ones to leading⟶ Permutations
St000541: Permutations ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1] => ? = 0 - 1
[1,2] => [1,2] => [1,2] => 0 = 1 - 1
[2,1] => [1,2] => [1,2] => 0 = 1 - 1
[1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[2,3,1] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[3,1,2] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[3,2,1] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,3,2,4] => [1,3,2,4] => [1,2,4,3] => 0 = 1 - 1
[1,4,2,3] => [1,4,2,3] => [3,4,1,2] => 1 = 2 - 1
[1,4,3,2] => [1,4,2,3] => [3,4,1,2] => 1 = 2 - 1
[2,1,4,3] => [1,4,2,3] => [3,4,1,2] => 1 = 2 - 1
[2,3,1,4] => [1,4,2,3] => [3,4,1,2] => 1 = 2 - 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[2,4,1,3] => [1,3,2,4] => [1,2,4,3] => 0 = 1 - 1
[3,1,4,2] => [1,4,2,3] => [3,4,1,2] => 1 = 2 - 1
[3,2,1,4] => [1,4,2,3] => [3,4,1,2] => 1 = 2 - 1
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[4,1,3,2] => [1,3,2,4] => [1,2,4,3] => 0 = 1 - 1
[4,2,1,3] => [1,3,2,4] => [1,2,4,3] => 0 = 1 - 1
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,3,5,4] => 0 = 1 - 1
[1,2,5,3,4] => [1,2,5,3,4] => [3,4,5,1,2] => 1 = 2 - 1
[1,2,5,4,3] => [1,2,5,3,4] => [3,4,5,1,2] => 1 = 2 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,2,4,3,5] => 0 = 1 - 1
[1,3,4,2,5] => [1,3,4,2,5] => [1,2,5,3,4] => 0 = 1 - 1
[1,3,5,2,4] => [1,3,5,2,4] => [3,4,1,2,5] => 1 = 2 - 1
[1,3,5,4,2] => [1,3,5,2,4] => [3,4,1,2,5] => 1 = 2 - 1
[1,4,2,3,5] => [1,4,2,3,5] => [1,2,4,5,3] => 0 = 1 - 1
[1,4,3,5,2] => [1,4,2,3,5] => [1,2,4,5,3] => 0 = 1 - 1
[1,4,5,2,3] => [1,4,5,2,3] => [3,4,1,5,2] => 1 = 2 - 1
[1,4,5,3,2] => [1,4,5,2,3] => [3,4,1,5,2] => 1 = 2 - 1
[1,5,2,3,4] => [1,5,2,3,4] => [4,5,1,3,2] => 1 = 2 - 1
[1,5,2,4,3] => [1,5,2,4,3] => [4,5,1,2,3] => 1 = 2 - 1
[1,5,3,2,4] => [1,5,2,4,3] => [4,5,1,2,3] => 1 = 2 - 1
[1,5,3,4,2] => [1,5,2,3,4] => [4,5,1,3,2] => 1 = 2 - 1
[1,5,4,2,3] => [1,5,2,3,4] => [4,5,1,3,2] => 1 = 2 - 1
[1,5,4,3,2] => [1,5,2,3,4] => [4,5,1,3,2] => 1 = 2 - 1
[2,1,3,5,4] => [1,3,5,2,4] => [3,4,1,2,5] => 1 = 2 - 1
[2,1,4,3,5] => [1,4,2,3,5] => [1,2,4,5,3] => 0 = 1 - 1
[2,1,4,5,3] => [1,4,5,2,3] => [3,4,1,5,2] => 1 = 2 - 1
[2,1,5,3,4] => [1,5,2,3,4] => [4,5,1,3,2] => 1 = 2 - 1
[2,1,5,4,3] => [1,5,2,3,4] => [4,5,1,3,2] => 1 = 2 - 1
[2,3,1,4,5] => [1,4,5,2,3] => [3,4,1,5,2] => 1 = 2 - 1
[2,3,1,5,4] => [1,5,2,3,4] => [4,5,1,3,2] => 1 = 2 - 1
[2,3,4,1,5] => [1,5,2,3,4] => [4,5,1,3,2] => 1 = 2 - 1
[1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => [3,4,5,6,7,1,2] => ? = 2 - 1
[1,2,3,4,7,6,5] => [1,2,3,4,7,5,6] => [3,4,5,6,7,1,2] => ? = 2 - 1
[1,2,3,5,7,4,6] => [1,2,3,5,7,4,6] => [3,4,5,6,1,2,7] => ? = 2 - 1
[1,2,3,5,7,6,4] => [1,2,3,5,7,4,6] => [3,4,5,6,1,2,7] => ? = 2 - 1
[1,2,3,6,7,4,5] => [1,2,3,6,7,4,5] => [3,4,5,6,1,7,2] => ? = 2 - 1
[1,2,3,6,7,5,4] => [1,2,3,6,7,4,5] => [3,4,5,6,1,7,2] => ? = 2 - 1
[1,2,3,7,4,5,6] => [1,2,3,7,4,5,6] => [4,5,6,7,1,3,2] => ? = 2 - 1
[1,2,3,7,4,6,5] => [1,2,3,7,4,6,5] => [4,5,6,7,1,2,3] => ? = 2 - 1
[1,2,3,7,5,4,6] => [1,2,3,7,4,6,5] => [4,5,6,7,1,2,3] => ? = 2 - 1
[1,2,3,7,5,6,4] => [1,2,3,7,4,5,6] => [4,5,6,7,1,3,2] => ? = 2 - 1
[1,2,3,7,6,4,5] => [1,2,3,7,4,5,6] => [4,5,6,7,1,3,2] => ? = 2 - 1
[1,2,3,7,6,5,4] => [1,2,3,7,4,5,6] => [4,5,6,7,1,3,2] => ? = 2 - 1
[1,2,4,3,7,5,6] => [1,2,4,3,7,5,6] => [3,4,5,7,6,1,2] => ? = 2 - 1
[1,2,4,3,7,6,5] => [1,2,4,3,7,5,6] => [3,4,5,7,6,1,2] => ? = 2 - 1
[1,2,4,5,7,3,6] => [1,2,4,5,7,3,6] => [3,4,5,1,2,6,7] => ? = 2 - 1
[1,2,4,5,7,6,3] => [1,2,4,5,7,3,6] => [3,4,5,1,2,6,7] => ? = 2 - 1
[1,2,4,6,7,3,5] => [1,2,4,6,7,3,5] => [3,4,5,1,7,2,6] => ? = 2 - 1
[1,2,4,6,7,5,3] => [1,2,4,6,7,3,5] => [3,4,5,1,7,2,6] => ? = 2 - 1
[1,2,4,7,3,5,6] => [1,2,4,7,3,5,6] => [4,5,6,1,3,7,2] => ? = 2 - 1
[1,2,4,7,3,6,5] => [1,2,4,7,3,6,5] => [4,5,6,1,2,7,3] => ? = 2 - 1
[1,2,4,7,5,3,6] => [1,2,4,7,3,6,5] => [4,5,6,1,2,7,3] => ? = 2 - 1
[1,2,4,7,5,6,3] => [1,2,4,7,3,5,6] => [4,5,6,1,3,7,2] => ? = 2 - 1
[1,2,4,7,6,3,5] => [1,2,4,7,3,5,6] => [4,5,6,1,3,7,2] => ? = 2 - 1
[1,2,4,7,6,5,3] => [1,2,4,7,3,5,6] => [4,5,6,1,3,7,2] => ? = 2 - 1
[1,2,5,3,7,4,6] => [1,2,5,3,7,4,6] => [3,4,5,7,1,2,6] => ? = 2 - 1
[1,2,5,3,7,6,4] => [1,2,5,3,7,4,6] => [3,4,5,7,1,2,6] => ? = 2 - 1
[1,2,5,4,3,7,6] => [1,2,5,3,7,4,6] => [3,4,5,7,1,2,6] => ? = 2 - 1
[1,2,5,4,6,3,7] => [1,2,5,3,7,4,6] => [3,4,5,7,1,2,6] => ? = 2 - 1
[1,2,5,6,7,3,4] => [1,2,5,6,7,3,4] => [3,4,5,1,7,6,2] => ? = 2 - 1
[1,2,5,6,7,4,3] => [1,2,5,6,7,3,4] => [3,4,5,1,7,6,2] => ? = 2 - 1
[1,2,5,7,3,4,6] => [1,2,5,7,3,4,6] => [4,5,6,1,3,2,7] => ? = 2 - 1
[1,2,5,7,3,6,4] => [1,2,5,7,3,6,4] => [4,5,6,1,2,3,7] => ? = 2 - 1
[1,2,5,7,4,3,6] => [1,2,5,7,3,6,4] => [4,5,6,1,2,3,7] => ? = 2 - 1
[1,2,5,7,4,6,3] => [1,2,5,7,3,4,6] => [4,5,6,1,3,2,7] => ? = 2 - 1
[1,2,5,7,6,3,4] => [1,2,5,7,3,4,6] => [4,5,6,1,3,2,7] => ? = 2 - 1
[1,2,5,7,6,4,3] => [1,2,5,7,3,4,6] => [4,5,6,1,3,2,7] => ? = 2 - 1
[1,2,6,3,7,4,5] => [1,2,6,3,7,4,5] => [3,4,5,7,1,6,2] => ? = 2 - 1
[1,2,6,3,7,5,4] => [1,2,6,3,7,4,5] => [3,4,5,7,1,6,2] => ? = 2 - 1
[1,2,6,4,3,7,5] => [1,2,6,3,7,4,5] => [3,4,5,7,1,6,2] => ? = 2 - 1
[1,2,6,4,5,3,7] => [1,2,6,3,7,4,5] => [3,4,5,7,1,6,2] => ? = 2 - 1
[1,2,6,5,3,7,4] => [1,2,6,3,7,4,5] => [3,4,5,7,1,6,2] => ? = 2 - 1
[1,2,6,5,4,3,7] => [1,2,6,3,7,4,5] => [3,4,5,7,1,6,2] => ? = 2 - 1
[1,2,6,7,3,4,5] => [1,2,6,7,3,4,5] => [4,5,6,1,7,3,2] => ? = 2 - 1
[1,2,6,7,3,5,4] => [1,2,6,7,3,5,4] => [4,5,6,1,7,2,3] => ? = 2 - 1
[1,2,6,7,4,3,5] => [1,2,6,7,3,5,4] => [4,5,6,1,7,2,3] => ? = 2 - 1
[1,2,6,7,4,5,3] => [1,2,6,7,3,4,5] => [4,5,6,1,7,3,2] => ? = 2 - 1
[1,2,6,7,5,3,4] => [1,2,6,7,3,4,5] => [4,5,6,1,7,3,2] => ? = 2 - 1
[1,2,6,7,5,4,3] => [1,2,6,7,3,4,5] => [4,5,6,1,7,3,2] => ? = 2 - 1
[1,2,7,3,4,5,6] => [1,2,7,3,4,5,6] => [5,6,7,1,4,2,3] => ? = 2 - 1
Description
The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right.
For a permutation $\pi$ of length $n$, this is the number of indices $2 \leq j \leq n$ such that for all $1 \leq i < j$, the pair $(i,j)$ is an inversion of $\pi$.
Matching statistic: St000035
Mp00223: Permutations —runsort⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000035: Permutations ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 100%
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000035: Permutations ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => [2,1] => 1
[2,1] => [1,2] => [2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => [2,3,1] => [2,3,1] => 1
[2,3,1] => [1,2,3] => [2,3,1] => [2,3,1] => 1
[3,1,2] => [1,2,3] => [2,3,1] => [2,3,1] => 1
[3,2,1] => [1,2,3] => [2,3,1] => [2,3,1] => 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => [2,4,3,1] => 1
[1,3,2,4] => [1,3,2,4] => [2,4,3,1] => [2,4,3,1] => 1
[1,4,2,3] => [1,4,2,3] => [2,1,4,3] => [2,1,4,3] => 2
[1,4,3,2] => [1,4,2,3] => [2,1,4,3] => [2,1,4,3] => 2
[2,1,4,3] => [1,4,2,3] => [2,1,4,3] => [2,1,4,3] => 2
[2,3,1,4] => [1,4,2,3] => [2,1,4,3] => [2,1,4,3] => 2
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => [2,4,3,1] => 1
[2,4,1,3] => [1,3,2,4] => [2,4,3,1] => [2,4,3,1] => 1
[3,1,4,2] => [1,4,2,3] => [2,1,4,3] => [2,1,4,3] => 2
[3,2,1,4] => [1,4,2,3] => [2,1,4,3] => [2,1,4,3] => 2
[3,4,1,2] => [1,2,3,4] => [2,3,4,1] => [2,4,3,1] => 1
[3,4,2,1] => [1,2,3,4] => [2,3,4,1] => [2,4,3,1] => 1
[4,1,2,3] => [1,2,3,4] => [2,3,4,1] => [2,4,3,1] => 1
[4,1,3,2] => [1,3,2,4] => [2,4,3,1] => [2,4,3,1] => 1
[4,2,1,3] => [1,3,2,4] => [2,4,3,1] => [2,4,3,1] => 1
[4,2,3,1] => [1,2,3,4] => [2,3,4,1] => [2,4,3,1] => 1
[4,3,1,2] => [1,2,3,4] => [2,3,4,1] => [2,4,3,1] => 1
[4,3,2,1] => [1,2,3,4] => [2,3,4,1] => [2,4,3,1] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [2,5,4,3,1] => 1
[1,2,4,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => [2,5,4,3,1] => 1
[1,2,5,3,4] => [1,2,5,3,4] => [2,3,1,5,4] => [2,5,1,4,3] => 2
[1,2,5,4,3] => [1,2,5,3,4] => [2,3,1,5,4] => [2,5,1,4,3] => 2
[1,3,2,4,5] => [1,3,2,4,5] => [2,4,3,5,1] => [2,5,4,3,1] => 1
[1,3,4,2,5] => [1,3,4,2,5] => [2,4,5,3,1] => [2,5,4,3,1] => 1
[1,3,5,2,4] => [1,3,5,2,4] => [2,4,1,5,3] => [2,5,1,4,3] => 2
[1,3,5,4,2] => [1,3,5,2,4] => [2,4,1,5,3] => [2,5,1,4,3] => 2
[1,4,2,3,5] => [1,4,2,3,5] => [2,5,3,4,1] => [2,5,4,3,1] => 1
[1,4,3,5,2] => [1,4,2,3,5] => [2,5,3,4,1] => [2,5,4,3,1] => 1
[1,4,5,2,3] => [1,4,5,2,3] => [2,5,1,4,3] => [2,5,1,4,3] => 2
[1,4,5,3,2] => [1,4,5,2,3] => [2,5,1,4,3] => [2,5,1,4,3] => 2
[1,5,2,3,4] => [1,5,2,3,4] => [2,1,4,5,3] => [2,1,5,4,3] => 2
[1,5,2,4,3] => [1,5,2,4,3] => [2,1,4,3,5] => [2,1,5,4,3] => 2
[1,5,3,2,4] => [1,5,2,4,3] => [2,1,4,3,5] => [2,1,5,4,3] => 2
[1,5,3,4,2] => [1,5,2,3,4] => [2,1,4,5,3] => [2,1,5,4,3] => 2
[1,5,4,2,3] => [1,5,2,3,4] => [2,1,4,5,3] => [2,1,5,4,3] => 2
[1,5,4,3,2] => [1,5,2,3,4] => [2,1,4,5,3] => [2,1,5,4,3] => 2
[2,1,3,5,4] => [1,3,5,2,4] => [2,4,1,5,3] => [2,5,1,4,3] => 2
[2,1,4,3,5] => [1,4,2,3,5] => [2,5,3,4,1] => [2,5,4,3,1] => 1
[2,1,4,5,3] => [1,4,5,2,3] => [2,5,1,4,3] => [2,5,1,4,3] => 2
[2,1,5,3,4] => [1,5,2,3,4] => [2,1,4,5,3] => [2,1,5,4,3] => 2
[2,1,5,4,3] => [1,5,2,3,4] => [2,1,4,5,3] => [2,1,5,4,3] => 2
[2,3,1,4,5] => [1,4,5,2,3] => [2,5,1,4,3] => [2,5,1,4,3] => 2
[2,3,1,5,4] => [1,5,2,3,4] => [2,1,4,5,3] => [2,1,5,4,3] => 2
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [2,7,6,5,4,3,1] => ? = 1
[1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [2,3,4,5,7,6,1] => [2,7,6,5,4,3,1] => ? = 1
[1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => [2,3,4,5,1,7,6] => [2,7,6,5,1,4,3] => ? = 2
[1,2,3,4,7,6,5] => [1,2,3,4,7,5,6] => [2,3,4,5,1,7,6] => [2,7,6,5,1,4,3] => ? = 2
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [2,3,4,6,5,7,1] => [2,7,6,5,4,3,1] => ? = 1
[1,2,3,5,6,4,7] => [1,2,3,5,6,4,7] => [2,3,4,6,7,5,1] => [2,7,6,5,4,3,1] => ? = 1
[1,2,3,5,7,4,6] => [1,2,3,5,7,4,6] => [2,3,4,6,1,7,5] => [2,7,6,5,1,4,3] => ? = 2
[1,2,3,5,7,6,4] => [1,2,3,5,7,4,6] => [2,3,4,6,1,7,5] => [2,7,6,5,1,4,3] => ? = 2
[1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [2,3,4,7,5,6,1] => [2,7,6,5,4,3,1] => ? = 1
[1,2,3,6,5,7,4] => [1,2,3,6,4,5,7] => [2,3,4,7,5,6,1] => [2,7,6,5,4,3,1] => ? = 1
[1,2,3,6,7,4,5] => [1,2,3,6,7,4,5] => [2,3,4,7,1,6,5] => [2,7,6,5,1,4,3] => ? = 2
[1,2,3,6,7,5,4] => [1,2,3,6,7,4,5] => [2,3,4,7,1,6,5] => [2,7,6,5,1,4,3] => ? = 2
[1,2,3,7,4,5,6] => [1,2,3,7,4,5,6] => [2,3,4,1,6,7,5] => [2,7,6,1,5,4,3] => ? = 2
[1,2,3,7,4,6,5] => [1,2,3,7,4,6,5] => [2,3,4,1,6,5,7] => [2,7,6,1,5,4,3] => ? = 2
[1,2,3,7,5,4,6] => [1,2,3,7,4,6,5] => [2,3,4,1,6,5,7] => [2,7,6,1,5,4,3] => ? = 2
[1,2,3,7,5,6,4] => [1,2,3,7,4,5,6] => [2,3,4,1,6,7,5] => [2,7,6,1,5,4,3] => ? = 2
[1,2,3,7,6,4,5] => [1,2,3,7,4,5,6] => [2,3,4,1,6,7,5] => [2,7,6,1,5,4,3] => ? = 2
[1,2,3,7,6,5,4] => [1,2,3,7,4,5,6] => [2,3,4,1,6,7,5] => [2,7,6,1,5,4,3] => ? = 2
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [2,3,5,4,6,7,1] => [2,7,6,5,4,3,1] => ? = 1
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [2,3,5,4,7,6,1] => [2,7,6,5,4,3,1] => ? = 1
[1,2,4,3,7,5,6] => [1,2,4,3,7,5,6] => [2,3,5,4,1,7,6] => [2,7,6,5,1,4,3] => ? = 2
[1,2,4,3,7,6,5] => [1,2,4,3,7,5,6] => [2,3,5,4,1,7,6] => [2,7,6,5,1,4,3] => ? = 2
[1,2,4,5,3,6,7] => [1,2,4,5,3,6,7] => [2,3,5,6,4,7,1] => [2,7,6,5,4,3,1] => ? = 1
[1,2,4,5,6,3,7] => [1,2,4,5,6,3,7] => [2,3,5,6,7,4,1] => [2,7,6,5,4,3,1] => ? = 1
[1,2,4,5,7,3,6] => [1,2,4,5,7,3,6] => [2,3,5,6,1,7,4] => [2,7,6,5,1,4,3] => ? = 2
[1,2,4,5,7,6,3] => [1,2,4,5,7,3,6] => [2,3,5,6,1,7,4] => [2,7,6,5,1,4,3] => ? = 2
[1,2,4,6,3,5,7] => [1,2,4,6,3,5,7] => [2,3,5,7,4,6,1] => [2,7,6,5,4,3,1] => ? = 1
[1,2,4,6,5,7,3] => [1,2,4,6,3,5,7] => [2,3,5,7,4,6,1] => [2,7,6,5,4,3,1] => ? = 1
[1,2,4,6,7,3,5] => [1,2,4,6,7,3,5] => [2,3,5,7,1,6,4] => [2,7,6,5,1,4,3] => ? = 2
[1,2,4,6,7,5,3] => [1,2,4,6,7,3,5] => [2,3,5,7,1,6,4] => [2,7,6,5,1,4,3] => ? = 2
[1,2,4,7,3,5,6] => [1,2,4,7,3,5,6] => [2,3,5,1,6,7,4] => [2,7,6,1,5,4,3] => ? = 2
[1,2,4,7,3,6,5] => [1,2,4,7,3,6,5] => [2,3,5,1,6,4,7] => [2,7,6,1,5,4,3] => ? = 2
[1,2,4,7,5,3,6] => [1,2,4,7,3,6,5] => [2,3,5,1,6,4,7] => [2,7,6,1,5,4,3] => ? = 2
[1,2,4,7,5,6,3] => [1,2,4,7,3,5,6] => [2,3,5,1,6,7,4] => [2,7,6,1,5,4,3] => ? = 2
[1,2,4,7,6,3,5] => [1,2,4,7,3,5,6] => [2,3,5,1,6,7,4] => [2,7,6,1,5,4,3] => ? = 2
[1,2,4,7,6,5,3] => [1,2,4,7,3,5,6] => [2,3,5,1,6,7,4] => [2,7,6,1,5,4,3] => ? = 2
[1,2,5,3,4,6,7] => [1,2,5,3,4,6,7] => [2,3,6,4,5,7,1] => [2,7,6,5,4,3,1] => ? = 1
[1,2,5,3,6,4,7] => [1,2,5,3,6,4,7] => [2,3,6,4,7,5,1] => [2,7,6,5,4,3,1] => ? = 1
[1,2,5,3,7,4,6] => [1,2,5,3,7,4,6] => [2,3,6,4,1,7,5] => [2,7,6,5,1,4,3] => ? = 2
[1,2,5,3,7,6,4] => [1,2,5,3,7,4,6] => [2,3,6,4,1,7,5] => [2,7,6,5,1,4,3] => ? = 2
[1,2,5,4,3,7,6] => [1,2,5,3,7,4,6] => [2,3,6,4,1,7,5] => [2,7,6,5,1,4,3] => ? = 2
[1,2,5,4,6,3,7] => [1,2,5,3,7,4,6] => [2,3,6,4,1,7,5] => [2,7,6,5,1,4,3] => ? = 2
[1,2,5,4,6,7,3] => [1,2,5,3,4,6,7] => [2,3,6,4,5,7,1] => [2,7,6,5,4,3,1] => ? = 1
[1,2,5,4,7,3,6] => [1,2,5,3,6,4,7] => [2,3,6,4,7,5,1] => [2,7,6,5,4,3,1] => ? = 1
[1,2,5,6,3,4,7] => [1,2,5,6,3,4,7] => [2,3,6,7,4,5,1] => [2,7,6,5,4,3,1] => ? = 1
[1,2,5,6,4,7,3] => [1,2,5,6,3,4,7] => [2,3,6,7,4,5,1] => [2,7,6,5,4,3,1] => ? = 1
[1,2,5,6,7,3,4] => [1,2,5,6,7,3,4] => [2,3,6,7,1,5,4] => [2,7,6,5,1,4,3] => ? = 2
[1,2,5,6,7,4,3] => [1,2,5,6,7,3,4] => [2,3,6,7,1,5,4] => [2,7,6,5,1,4,3] => ? = 2
[1,2,5,7,3,4,6] => [1,2,5,7,3,4,6] => [2,3,6,1,5,7,4] => [2,7,6,1,5,4,3] => ? = 2
[1,2,5,7,3,6,4] => [1,2,5,7,3,6,4] => [2,3,6,1,5,4,7] => [2,7,6,1,5,4,3] => ? = 2
Description
The number of left outer peaks of a permutation.
A left outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $1$ if $w_1 > w_2$.
In other words, it is a peak in the word $[0,w_1,..., w_n]$.
This appears in [1, def.3.1]. The joint distribution with [[St000366]] is studied in [3], where left outer peaks are called ''exterior peaks''.
Matching statistic: St000374
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 100%
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => [2,1] => 1
[2,1] => [1,2] => [2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => [2,3,1] => [2,3,1] => 1
[2,3,1] => [1,2,3] => [2,3,1] => [2,3,1] => 1
[3,1,2] => [1,2,3] => [2,3,1] => [2,3,1] => 1
[3,2,1] => [1,2,3] => [2,3,1] => [2,3,1] => 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => [2,4,3,1] => 1
[1,3,2,4] => [1,3,2,4] => [2,4,3,1] => [2,4,3,1] => 1
[1,4,2,3] => [1,4,2,3] => [2,1,4,3] => [2,1,4,3] => 2
[1,4,3,2] => [1,4,2,3] => [2,1,4,3] => [2,1,4,3] => 2
[2,1,4,3] => [1,4,2,3] => [2,1,4,3] => [2,1,4,3] => 2
[2,3,1,4] => [1,4,2,3] => [2,1,4,3] => [2,1,4,3] => 2
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => [2,4,3,1] => 1
[2,4,1,3] => [1,3,2,4] => [2,4,3,1] => [2,4,3,1] => 1
[3,1,4,2] => [1,4,2,3] => [2,1,4,3] => [2,1,4,3] => 2
[3,2,1,4] => [1,4,2,3] => [2,1,4,3] => [2,1,4,3] => 2
[3,4,1,2] => [1,2,3,4] => [2,3,4,1] => [2,4,3,1] => 1
[3,4,2,1] => [1,2,3,4] => [2,3,4,1] => [2,4,3,1] => 1
[4,1,2,3] => [1,2,3,4] => [2,3,4,1] => [2,4,3,1] => 1
[4,1,3,2] => [1,3,2,4] => [2,4,3,1] => [2,4,3,1] => 1
[4,2,1,3] => [1,3,2,4] => [2,4,3,1] => [2,4,3,1] => 1
[4,2,3,1] => [1,2,3,4] => [2,3,4,1] => [2,4,3,1] => 1
[4,3,1,2] => [1,2,3,4] => [2,3,4,1] => [2,4,3,1] => 1
[4,3,2,1] => [1,2,3,4] => [2,3,4,1] => [2,4,3,1] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [2,5,4,3,1] => 1
[1,2,4,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => [2,5,4,3,1] => 1
[1,2,5,3,4] => [1,2,5,3,4] => [2,3,1,5,4] => [2,5,1,4,3] => 2
[1,2,5,4,3] => [1,2,5,3,4] => [2,3,1,5,4] => [2,5,1,4,3] => 2
[1,3,2,4,5] => [1,3,2,4,5] => [2,4,3,5,1] => [2,5,4,3,1] => 1
[1,3,4,2,5] => [1,3,4,2,5] => [2,4,5,3,1] => [2,5,4,3,1] => 1
[1,3,5,2,4] => [1,3,5,2,4] => [2,4,1,5,3] => [2,5,1,4,3] => 2
[1,3,5,4,2] => [1,3,5,2,4] => [2,4,1,5,3] => [2,5,1,4,3] => 2
[1,4,2,3,5] => [1,4,2,3,5] => [2,5,3,4,1] => [2,5,4,3,1] => 1
[1,4,3,5,2] => [1,4,2,3,5] => [2,5,3,4,1] => [2,5,4,3,1] => 1
[1,4,5,2,3] => [1,4,5,2,3] => [2,5,1,4,3] => [2,5,1,4,3] => 2
[1,4,5,3,2] => [1,4,5,2,3] => [2,5,1,4,3] => [2,5,1,4,3] => 2
[1,5,2,3,4] => [1,5,2,3,4] => [2,1,4,5,3] => [2,1,5,4,3] => 2
[1,5,2,4,3] => [1,5,2,4,3] => [2,1,4,3,5] => [2,1,5,4,3] => 2
[1,5,3,2,4] => [1,5,2,4,3] => [2,1,4,3,5] => [2,1,5,4,3] => 2
[1,5,3,4,2] => [1,5,2,3,4] => [2,1,4,5,3] => [2,1,5,4,3] => 2
[1,5,4,2,3] => [1,5,2,3,4] => [2,1,4,5,3] => [2,1,5,4,3] => 2
[1,5,4,3,2] => [1,5,2,3,4] => [2,1,4,5,3] => [2,1,5,4,3] => 2
[2,1,3,5,4] => [1,3,5,2,4] => [2,4,1,5,3] => [2,5,1,4,3] => 2
[2,1,4,3,5] => [1,4,2,3,5] => [2,5,3,4,1] => [2,5,4,3,1] => 1
[2,1,4,5,3] => [1,4,5,2,3] => [2,5,1,4,3] => [2,5,1,4,3] => 2
[2,1,5,3,4] => [1,5,2,3,4] => [2,1,4,5,3] => [2,1,5,4,3] => 2
[2,1,5,4,3] => [1,5,2,3,4] => [2,1,4,5,3] => [2,1,5,4,3] => 2
[2,3,1,4,5] => [1,4,5,2,3] => [2,5,1,4,3] => [2,5,1,4,3] => 2
[2,3,1,5,4] => [1,5,2,3,4] => [2,1,4,5,3] => [2,1,5,4,3] => 2
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [2,7,6,5,4,3,1] => ? = 1
[1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [2,3,4,5,7,6,1] => [2,7,6,5,4,3,1] => ? = 1
[1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => [2,3,4,5,1,7,6] => [2,7,6,5,1,4,3] => ? = 2
[1,2,3,4,7,6,5] => [1,2,3,4,7,5,6] => [2,3,4,5,1,7,6] => [2,7,6,5,1,4,3] => ? = 2
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [2,3,4,6,5,7,1] => [2,7,6,5,4,3,1] => ? = 1
[1,2,3,5,6,4,7] => [1,2,3,5,6,4,7] => [2,3,4,6,7,5,1] => [2,7,6,5,4,3,1] => ? = 1
[1,2,3,5,7,4,6] => [1,2,3,5,7,4,6] => [2,3,4,6,1,7,5] => [2,7,6,5,1,4,3] => ? = 2
[1,2,3,5,7,6,4] => [1,2,3,5,7,4,6] => [2,3,4,6,1,7,5] => [2,7,6,5,1,4,3] => ? = 2
[1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [2,3,4,7,5,6,1] => [2,7,6,5,4,3,1] => ? = 1
[1,2,3,6,5,7,4] => [1,2,3,6,4,5,7] => [2,3,4,7,5,6,1] => [2,7,6,5,4,3,1] => ? = 1
[1,2,3,6,7,4,5] => [1,2,3,6,7,4,5] => [2,3,4,7,1,6,5] => [2,7,6,5,1,4,3] => ? = 2
[1,2,3,6,7,5,4] => [1,2,3,6,7,4,5] => [2,3,4,7,1,6,5] => [2,7,6,5,1,4,3] => ? = 2
[1,2,3,7,4,5,6] => [1,2,3,7,4,5,6] => [2,3,4,1,6,7,5] => [2,7,6,1,5,4,3] => ? = 2
[1,2,3,7,4,6,5] => [1,2,3,7,4,6,5] => [2,3,4,1,6,5,7] => [2,7,6,1,5,4,3] => ? = 2
[1,2,3,7,5,4,6] => [1,2,3,7,4,6,5] => [2,3,4,1,6,5,7] => [2,7,6,1,5,4,3] => ? = 2
[1,2,3,7,5,6,4] => [1,2,3,7,4,5,6] => [2,3,4,1,6,7,5] => [2,7,6,1,5,4,3] => ? = 2
[1,2,3,7,6,4,5] => [1,2,3,7,4,5,6] => [2,3,4,1,6,7,5] => [2,7,6,1,5,4,3] => ? = 2
[1,2,3,7,6,5,4] => [1,2,3,7,4,5,6] => [2,3,4,1,6,7,5] => [2,7,6,1,5,4,3] => ? = 2
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [2,3,5,4,6,7,1] => [2,7,6,5,4,3,1] => ? = 1
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [2,3,5,4,7,6,1] => [2,7,6,5,4,3,1] => ? = 1
[1,2,4,3,7,5,6] => [1,2,4,3,7,5,6] => [2,3,5,4,1,7,6] => [2,7,6,5,1,4,3] => ? = 2
[1,2,4,3,7,6,5] => [1,2,4,3,7,5,6] => [2,3,5,4,1,7,6] => [2,7,6,5,1,4,3] => ? = 2
[1,2,4,5,3,6,7] => [1,2,4,5,3,6,7] => [2,3,5,6,4,7,1] => [2,7,6,5,4,3,1] => ? = 1
[1,2,4,5,6,3,7] => [1,2,4,5,6,3,7] => [2,3,5,6,7,4,1] => [2,7,6,5,4,3,1] => ? = 1
[1,2,4,5,7,3,6] => [1,2,4,5,7,3,6] => [2,3,5,6,1,7,4] => [2,7,6,5,1,4,3] => ? = 2
[1,2,4,5,7,6,3] => [1,2,4,5,7,3,6] => [2,3,5,6,1,7,4] => [2,7,6,5,1,4,3] => ? = 2
[1,2,4,6,3,5,7] => [1,2,4,6,3,5,7] => [2,3,5,7,4,6,1] => [2,7,6,5,4,3,1] => ? = 1
[1,2,4,6,5,7,3] => [1,2,4,6,3,5,7] => [2,3,5,7,4,6,1] => [2,7,6,5,4,3,1] => ? = 1
[1,2,4,6,7,3,5] => [1,2,4,6,7,3,5] => [2,3,5,7,1,6,4] => [2,7,6,5,1,4,3] => ? = 2
[1,2,4,6,7,5,3] => [1,2,4,6,7,3,5] => [2,3,5,7,1,6,4] => [2,7,6,5,1,4,3] => ? = 2
[1,2,4,7,3,5,6] => [1,2,4,7,3,5,6] => [2,3,5,1,6,7,4] => [2,7,6,1,5,4,3] => ? = 2
[1,2,4,7,3,6,5] => [1,2,4,7,3,6,5] => [2,3,5,1,6,4,7] => [2,7,6,1,5,4,3] => ? = 2
[1,2,4,7,5,3,6] => [1,2,4,7,3,6,5] => [2,3,5,1,6,4,7] => [2,7,6,1,5,4,3] => ? = 2
[1,2,4,7,5,6,3] => [1,2,4,7,3,5,6] => [2,3,5,1,6,7,4] => [2,7,6,1,5,4,3] => ? = 2
[1,2,4,7,6,3,5] => [1,2,4,7,3,5,6] => [2,3,5,1,6,7,4] => [2,7,6,1,5,4,3] => ? = 2
[1,2,4,7,6,5,3] => [1,2,4,7,3,5,6] => [2,3,5,1,6,7,4] => [2,7,6,1,5,4,3] => ? = 2
[1,2,5,3,4,6,7] => [1,2,5,3,4,6,7] => [2,3,6,4,5,7,1] => [2,7,6,5,4,3,1] => ? = 1
[1,2,5,3,6,4,7] => [1,2,5,3,6,4,7] => [2,3,6,4,7,5,1] => [2,7,6,5,4,3,1] => ? = 1
[1,2,5,3,7,4,6] => [1,2,5,3,7,4,6] => [2,3,6,4,1,7,5] => [2,7,6,5,1,4,3] => ? = 2
[1,2,5,3,7,6,4] => [1,2,5,3,7,4,6] => [2,3,6,4,1,7,5] => [2,7,6,5,1,4,3] => ? = 2
[1,2,5,4,3,7,6] => [1,2,5,3,7,4,6] => [2,3,6,4,1,7,5] => [2,7,6,5,1,4,3] => ? = 2
[1,2,5,4,6,3,7] => [1,2,5,3,7,4,6] => [2,3,6,4,1,7,5] => [2,7,6,5,1,4,3] => ? = 2
[1,2,5,4,6,7,3] => [1,2,5,3,4,6,7] => [2,3,6,4,5,7,1] => [2,7,6,5,4,3,1] => ? = 1
[1,2,5,4,7,3,6] => [1,2,5,3,6,4,7] => [2,3,6,4,7,5,1] => [2,7,6,5,4,3,1] => ? = 1
[1,2,5,6,3,4,7] => [1,2,5,6,3,4,7] => [2,3,6,7,4,5,1] => [2,7,6,5,4,3,1] => ? = 1
[1,2,5,6,4,7,3] => [1,2,5,6,3,4,7] => [2,3,6,7,4,5,1] => [2,7,6,5,4,3,1] => ? = 1
[1,2,5,6,7,3,4] => [1,2,5,6,7,3,4] => [2,3,6,7,1,5,4] => [2,7,6,5,1,4,3] => ? = 2
[1,2,5,6,7,4,3] => [1,2,5,6,7,3,4] => [2,3,6,7,1,5,4] => [2,7,6,5,1,4,3] => ? = 2
[1,2,5,7,3,4,6] => [1,2,5,7,3,4,6] => [2,3,6,1,5,7,4] => [2,7,6,1,5,4,3] => ? = 2
[1,2,5,7,3,6,4] => [1,2,5,7,3,6,4] => [2,3,6,1,5,4,7] => [2,7,6,1,5,4,3] => ? = 2
Description
The number of exclusive right-to-left minima of a permutation.
This is the number of right-to-left minima that are not left-to-right maxima.
This is also the number of non weak exceedences of a permutation that are also not mid-points of a decreasing subsequence of length 3.
Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j < j$ and there do not exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$.
See also [[St000213]] and [[St000119]].
Matching statistic: St000996
Mp00223: Permutations —runsort⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => [2,1] => 1
[2,1] => [1,2] => [2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => [3,2,1] => [3,2,1] => 1
[2,3,1] => [1,2,3] => [3,2,1] => [3,2,1] => 1
[3,1,2] => [1,2,3] => [3,2,1] => [3,2,1] => 1
[3,2,1] => [1,2,3] => [3,2,1] => [3,2,1] => 1
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 1
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => 1
[1,4,2,3] => [1,4,2,3] => [3,2,4,1] => [3,2,4,1] => 2
[1,4,3,2] => [1,4,2,3] => [3,2,4,1] => [3,2,4,1] => 2
[2,1,4,3] => [1,4,2,3] => [3,2,4,1] => [3,2,4,1] => 2
[2,3,1,4] => [1,4,2,3] => [3,2,4,1] => [3,2,4,1] => 2
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 1
[2,4,1,3] => [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => 1
[3,1,4,2] => [1,4,2,3] => [3,2,4,1] => [3,2,4,1] => 2
[3,2,1,4] => [1,4,2,3] => [3,2,4,1] => [3,2,4,1] => 2
[3,4,1,2] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 1
[3,4,2,1] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 1
[4,1,2,3] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 1
[4,1,3,2] => [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => 1
[4,2,1,3] => [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => 1
[4,2,3,1] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 1
[4,3,1,2] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 1
[4,3,2,1] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => 1
[1,2,4,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => [5,3,4,2,1] => 1
[1,2,5,3,4] => [1,2,5,3,4] => [4,3,5,2,1] => [4,3,5,2,1] => 2
[1,2,5,4,3] => [1,2,5,3,4] => [4,3,5,2,1] => [4,3,5,2,1] => 2
[1,3,2,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => [5,4,2,3,1] => 1
[1,3,4,2,5] => [1,3,4,2,5] => [5,2,4,3,1] => [5,2,4,3,1] => 1
[1,3,5,2,4] => [1,3,5,2,4] => [4,2,5,3,1] => [4,2,5,3,1] => 2
[1,3,5,4,2] => [1,3,5,2,4] => [4,2,5,3,1] => [4,2,5,3,1] => 2
[1,4,2,3,5] => [1,4,2,3,5] => [5,3,2,4,1] => [5,3,2,4,1] => 1
[1,4,3,5,2] => [1,4,2,3,5] => [5,3,2,4,1] => [5,3,2,4,1] => 1
[1,4,5,2,3] => [1,4,5,2,3] => [3,2,5,4,1] => [3,2,5,4,1] => 2
[1,4,5,3,2] => [1,4,5,2,3] => [3,2,5,4,1] => [3,2,5,4,1] => 2
[1,5,2,3,4] => [1,5,2,3,4] => [4,3,2,5,1] => [4,3,2,5,1] => 2
[1,5,2,4,3] => [1,5,2,4,3] => [3,4,2,5,1] => [3,5,2,4,1] => 2
[1,5,3,2,4] => [1,5,2,4,3] => [3,4,2,5,1] => [3,5,2,4,1] => 2
[1,5,3,4,2] => [1,5,2,3,4] => [4,3,2,5,1] => [4,3,2,5,1] => 2
[1,5,4,2,3] => [1,5,2,3,4] => [4,3,2,5,1] => [4,3,2,5,1] => 2
[1,5,4,3,2] => [1,5,2,3,4] => [4,3,2,5,1] => [4,3,2,5,1] => 2
[2,1,3,5,4] => [1,3,5,2,4] => [4,2,5,3,1] => [4,2,5,3,1] => 2
[2,1,4,3,5] => [1,4,2,3,5] => [5,3,2,4,1] => [5,3,2,4,1] => 1
[2,1,4,5,3] => [1,4,5,2,3] => [3,2,5,4,1] => [3,2,5,4,1] => 2
[2,1,5,3,4] => [1,5,2,3,4] => [4,3,2,5,1] => [4,3,2,5,1] => 2
[2,1,5,4,3] => [1,5,2,3,4] => [4,3,2,5,1] => [4,3,2,5,1] => 2
[2,3,1,4,5] => [1,4,5,2,3] => [3,2,5,4,1] => [3,2,5,4,1] => 2
[2,3,1,5,4] => [1,5,2,3,4] => [4,3,2,5,1] => [4,3,2,5,1] => 2
[1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [7,5,6,4,3,2,1] => ? = 1
[1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => [6,5,7,4,3,2,1] => ? = 2
[1,2,3,4,7,6,5] => [1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => [6,5,7,4,3,2,1] => ? = 2
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => [7,6,4,5,3,2,1] => ? = 1
[1,2,3,5,6,4,7] => [1,2,3,5,6,4,7] => [7,4,6,5,3,2,1] => [7,4,6,5,3,2,1] => ? = 1
[1,2,3,5,7,4,6] => [1,2,3,5,7,4,6] => [6,4,7,5,3,2,1] => [6,4,7,5,3,2,1] => ? = 2
[1,2,3,5,7,6,4] => [1,2,3,5,7,4,6] => [6,4,7,5,3,2,1] => [6,4,7,5,3,2,1] => ? = 2
[1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [7,5,4,6,3,2,1] => [7,5,4,6,3,2,1] => ? = 1
[1,2,3,6,5,7,4] => [1,2,3,6,4,5,7] => [7,5,4,6,3,2,1] => [7,5,4,6,3,2,1] => ? = 1
[1,2,3,6,7,4,5] => [1,2,3,6,7,4,5] => [5,4,7,6,3,2,1] => [5,4,7,6,3,2,1] => ? = 2
[1,2,3,6,7,5,4] => [1,2,3,6,7,4,5] => [5,4,7,6,3,2,1] => [5,4,7,6,3,2,1] => ? = 2
[1,2,3,7,4,5,6] => [1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => [6,5,4,7,3,2,1] => ? = 2
[1,2,3,7,4,6,5] => [1,2,3,7,4,6,5] => [5,6,4,7,3,2,1] => [5,7,4,6,3,2,1] => ? = 2
[1,2,3,7,5,4,6] => [1,2,3,7,4,6,5] => [5,6,4,7,3,2,1] => [5,7,4,6,3,2,1] => ? = 2
[1,2,3,7,5,6,4] => [1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => [6,5,4,7,3,2,1] => ? = 2
[1,2,3,7,6,4,5] => [1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => [6,5,4,7,3,2,1] => ? = 2
[1,2,3,7,6,5,4] => [1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => [6,5,4,7,3,2,1] => ? = 2
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [7,6,5,3,4,2,1] => [7,6,5,3,4,2,1] => ? = 1
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [7,5,6,3,4,2,1] => [7,5,6,3,4,2,1] => ? = 1
[1,2,4,3,7,5,6] => [1,2,4,3,7,5,6] => [6,5,7,3,4,2,1] => [6,5,7,3,4,2,1] => ? = 2
[1,2,4,3,7,6,5] => [1,2,4,3,7,5,6] => [6,5,7,3,4,2,1] => [6,5,7,3,4,2,1] => ? = 2
[1,2,4,5,3,6,7] => [1,2,4,5,3,6,7] => [7,6,3,5,4,2,1] => [7,6,3,5,4,2,1] => ? = 1
[1,2,4,5,6,3,7] => [1,2,4,5,6,3,7] => [7,3,6,5,4,2,1] => [7,3,6,5,4,2,1] => ? = 1
[1,2,4,5,7,3,6] => [1,2,4,5,7,3,6] => [6,3,7,5,4,2,1] => [6,3,7,5,4,2,1] => ? = 2
[1,2,4,5,7,6,3] => [1,2,4,5,7,3,6] => [6,3,7,5,4,2,1] => [6,3,7,5,4,2,1] => ? = 2
[1,2,4,6,3,5,7] => [1,2,4,6,3,5,7] => [7,5,3,6,4,2,1] => [7,5,3,6,4,2,1] => ? = 1
[1,2,4,6,5,7,3] => [1,2,4,6,3,5,7] => [7,5,3,6,4,2,1] => [7,5,3,6,4,2,1] => ? = 1
[1,2,4,6,7,3,5] => [1,2,4,6,7,3,5] => [5,3,7,6,4,2,1] => [5,3,7,6,4,2,1] => ? = 2
[1,2,4,6,7,5,3] => [1,2,4,6,7,3,5] => [5,3,7,6,4,2,1] => [5,3,7,6,4,2,1] => ? = 2
[1,2,4,7,3,5,6] => [1,2,4,7,3,5,6] => [6,5,3,7,4,2,1] => [6,5,3,7,4,2,1] => ? = 2
[1,2,4,7,3,6,5] => [1,2,4,7,3,6,5] => [5,6,3,7,4,2,1] => [5,7,3,6,4,2,1] => ? = 2
[1,2,4,7,5,3,6] => [1,2,4,7,3,6,5] => [5,6,3,7,4,2,1] => [5,7,3,6,4,2,1] => ? = 2
[1,2,4,7,5,6,3] => [1,2,4,7,3,5,6] => [6,5,3,7,4,2,1] => [6,5,3,7,4,2,1] => ? = 2
[1,2,4,7,6,3,5] => [1,2,4,7,3,5,6] => [6,5,3,7,4,2,1] => [6,5,3,7,4,2,1] => ? = 2
[1,2,4,7,6,5,3] => [1,2,4,7,3,5,6] => [6,5,3,7,4,2,1] => [6,5,3,7,4,2,1] => ? = 2
[1,2,5,3,4,6,7] => [1,2,5,3,4,6,7] => [7,6,4,3,5,2,1] => [7,6,4,3,5,2,1] => ? = 1
[1,2,5,3,6,4,7] => [1,2,5,3,6,4,7] => [7,4,6,3,5,2,1] => [7,4,6,3,5,2,1] => ? = 1
[1,2,5,3,7,4,6] => [1,2,5,3,7,4,6] => [6,4,7,3,5,2,1] => [6,4,7,3,5,2,1] => ? = 2
[1,2,5,3,7,6,4] => [1,2,5,3,7,4,6] => [6,4,7,3,5,2,1] => [6,4,7,3,5,2,1] => ? = 2
[1,2,5,4,3,7,6] => [1,2,5,3,7,4,6] => [6,4,7,3,5,2,1] => [6,4,7,3,5,2,1] => ? = 2
[1,2,5,4,6,3,7] => [1,2,5,3,7,4,6] => [6,4,7,3,5,2,1] => [6,4,7,3,5,2,1] => ? = 2
[1,2,5,4,6,7,3] => [1,2,5,3,4,6,7] => [7,6,4,3,5,2,1] => [7,6,4,3,5,2,1] => ? = 1
[1,2,5,4,7,3,6] => [1,2,5,3,6,4,7] => [7,4,6,3,5,2,1] => [7,4,6,3,5,2,1] => ? = 1
[1,2,5,6,3,4,7] => [1,2,5,6,3,4,7] => [7,4,3,6,5,2,1] => [7,4,3,6,5,2,1] => ? = 1
[1,2,5,6,4,7,3] => [1,2,5,6,3,4,7] => [7,4,3,6,5,2,1] => [7,4,3,6,5,2,1] => ? = 1
[1,2,5,6,7,3,4] => [1,2,5,6,7,3,4] => [4,3,7,6,5,2,1] => [4,3,7,6,5,2,1] => ? = 2
[1,2,5,6,7,4,3] => [1,2,5,6,7,3,4] => [4,3,7,6,5,2,1] => [4,3,7,6,5,2,1] => ? = 2
[1,2,5,7,3,4,6] => [1,2,5,7,3,4,6] => [6,4,3,7,5,2,1] => [6,4,3,7,5,2,1] => ? = 2
[1,2,5,7,3,6,4] => [1,2,5,7,3,6,4] => [4,6,3,7,5,2,1] => [4,7,3,6,5,2,1] => ? = 2
[1,2,5,7,4,3,6] => [1,2,5,7,3,6,4] => [4,6,3,7,5,2,1] => [4,7,3,6,5,2,1] => ? = 2
Description
The number of exclusive left-to-right maxima of a permutation.
This is the number of left-to-right maxima that are not right-to-left minima.
Matching statistic: St001004
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St001004: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St001004: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 1 = 0 + 1
[1,2] => [1,2] => [2,1] => [2,1] => 2 = 1 + 1
[2,1] => [1,2] => [2,1] => [2,1] => 2 = 1 + 1
[1,2,3] => [1,2,3] => [3,2,1] => [3,2,1] => 2 = 1 + 1
[2,3,1] => [1,2,3] => [3,2,1] => [3,2,1] => 2 = 1 + 1
[3,1,2] => [1,2,3] => [3,2,1] => [3,2,1] => 2 = 1 + 1
[3,2,1] => [1,2,3] => [3,2,1] => [3,2,1] => 2 = 1 + 1
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 2 = 1 + 1
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => 2 = 1 + 1
[1,4,2,3] => [1,4,2,3] => [3,2,4,1] => [3,2,4,1] => 3 = 2 + 1
[1,4,3,2] => [1,4,2,3] => [3,2,4,1] => [3,2,4,1] => 3 = 2 + 1
[2,1,4,3] => [1,4,2,3] => [3,2,4,1] => [3,2,4,1] => 3 = 2 + 1
[2,3,1,4] => [1,4,2,3] => [3,2,4,1] => [3,2,4,1] => 3 = 2 + 1
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 2 = 1 + 1
[2,4,1,3] => [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => 2 = 1 + 1
[3,1,4,2] => [1,4,2,3] => [3,2,4,1] => [3,2,4,1] => 3 = 2 + 1
[3,2,1,4] => [1,4,2,3] => [3,2,4,1] => [3,2,4,1] => 3 = 2 + 1
[3,4,1,2] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 2 = 1 + 1
[3,4,2,1] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 2 = 1 + 1
[4,1,2,3] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 2 = 1 + 1
[4,1,3,2] => [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => 2 = 1 + 1
[4,2,1,3] => [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => 2 = 1 + 1
[4,2,3,1] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 2 = 1 + 1
[4,3,1,2] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 2 = 1 + 1
[4,3,2,1] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 2 = 1 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => 2 = 1 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => [5,3,4,2,1] => 2 = 1 + 1
[1,2,5,3,4] => [1,2,5,3,4] => [4,3,5,2,1] => [4,3,5,2,1] => 3 = 2 + 1
[1,2,5,4,3] => [1,2,5,3,4] => [4,3,5,2,1] => [4,3,5,2,1] => 3 = 2 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => [5,4,2,3,1] => 2 = 1 + 1
[1,3,4,2,5] => [1,3,4,2,5] => [5,2,4,3,1] => [5,2,4,3,1] => 2 = 1 + 1
[1,3,5,2,4] => [1,3,5,2,4] => [4,2,5,3,1] => [4,2,5,3,1] => 3 = 2 + 1
[1,3,5,4,2] => [1,3,5,2,4] => [4,2,5,3,1] => [4,2,5,3,1] => 3 = 2 + 1
[1,4,2,3,5] => [1,4,2,3,5] => [5,3,2,4,1] => [5,3,2,4,1] => 2 = 1 + 1
[1,4,3,5,2] => [1,4,2,3,5] => [5,3,2,4,1] => [5,3,2,4,1] => 2 = 1 + 1
[1,4,5,2,3] => [1,4,5,2,3] => [3,2,5,4,1] => [3,2,5,4,1] => 3 = 2 + 1
[1,4,5,3,2] => [1,4,5,2,3] => [3,2,5,4,1] => [3,2,5,4,1] => 3 = 2 + 1
[1,5,2,3,4] => [1,5,2,3,4] => [4,3,2,5,1] => [4,3,2,5,1] => 3 = 2 + 1
[1,5,2,4,3] => [1,5,2,4,3] => [3,4,2,5,1] => [3,5,2,4,1] => 3 = 2 + 1
[1,5,3,2,4] => [1,5,2,4,3] => [3,4,2,5,1] => [3,5,2,4,1] => 3 = 2 + 1
[1,5,3,4,2] => [1,5,2,3,4] => [4,3,2,5,1] => [4,3,2,5,1] => 3 = 2 + 1
[1,5,4,2,3] => [1,5,2,3,4] => [4,3,2,5,1] => [4,3,2,5,1] => 3 = 2 + 1
[1,5,4,3,2] => [1,5,2,3,4] => [4,3,2,5,1] => [4,3,2,5,1] => 3 = 2 + 1
[2,1,3,5,4] => [1,3,5,2,4] => [4,2,5,3,1] => [4,2,5,3,1] => 3 = 2 + 1
[2,1,4,3,5] => [1,4,2,3,5] => [5,3,2,4,1] => [5,3,2,4,1] => 2 = 1 + 1
[2,1,4,5,3] => [1,4,5,2,3] => [3,2,5,4,1] => [3,2,5,4,1] => 3 = 2 + 1
[2,1,5,3,4] => [1,5,2,3,4] => [4,3,2,5,1] => [4,3,2,5,1] => 3 = 2 + 1
[2,1,5,4,3] => [1,5,2,3,4] => [4,3,2,5,1] => [4,3,2,5,1] => 3 = 2 + 1
[2,3,1,4,5] => [1,4,5,2,3] => [3,2,5,4,1] => [3,2,5,4,1] => 3 = 2 + 1
[2,3,1,5,4] => [1,5,2,3,4] => [4,3,2,5,1] => [4,3,2,5,1] => 3 = 2 + 1
[1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [7,5,6,4,3,2,1] => ? = 1 + 1
[1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => [6,5,7,4,3,2,1] => ? = 2 + 1
[1,2,3,4,7,6,5] => [1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => [6,5,7,4,3,2,1] => ? = 2 + 1
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => [7,6,4,5,3,2,1] => ? = 1 + 1
[1,2,3,5,6,4,7] => [1,2,3,5,6,4,7] => [7,4,6,5,3,2,1] => [7,4,6,5,3,2,1] => ? = 1 + 1
[1,2,3,5,7,4,6] => [1,2,3,5,7,4,6] => [6,4,7,5,3,2,1] => [6,4,7,5,3,2,1] => ? = 2 + 1
[1,2,3,5,7,6,4] => [1,2,3,5,7,4,6] => [6,4,7,5,3,2,1] => [6,4,7,5,3,2,1] => ? = 2 + 1
[1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [7,5,4,6,3,2,1] => [7,5,4,6,3,2,1] => ? = 1 + 1
[1,2,3,6,5,7,4] => [1,2,3,6,4,5,7] => [7,5,4,6,3,2,1] => [7,5,4,6,3,2,1] => ? = 1 + 1
[1,2,3,6,7,4,5] => [1,2,3,6,7,4,5] => [5,4,7,6,3,2,1] => [5,4,7,6,3,2,1] => ? = 2 + 1
[1,2,3,6,7,5,4] => [1,2,3,6,7,4,5] => [5,4,7,6,3,2,1] => [5,4,7,6,3,2,1] => ? = 2 + 1
[1,2,3,7,4,5,6] => [1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => [6,5,4,7,3,2,1] => ? = 2 + 1
[1,2,3,7,4,6,5] => [1,2,3,7,4,6,5] => [5,6,4,7,3,2,1] => [5,7,4,6,3,2,1] => ? = 2 + 1
[1,2,3,7,5,4,6] => [1,2,3,7,4,6,5] => [5,6,4,7,3,2,1] => [5,7,4,6,3,2,1] => ? = 2 + 1
[1,2,3,7,5,6,4] => [1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => [6,5,4,7,3,2,1] => ? = 2 + 1
[1,2,3,7,6,4,5] => [1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => [6,5,4,7,3,2,1] => ? = 2 + 1
[1,2,3,7,6,5,4] => [1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => [6,5,4,7,3,2,1] => ? = 2 + 1
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [7,6,5,3,4,2,1] => [7,6,5,3,4,2,1] => ? = 1 + 1
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [7,5,6,3,4,2,1] => [7,5,6,3,4,2,1] => ? = 1 + 1
[1,2,4,3,7,5,6] => [1,2,4,3,7,5,6] => [6,5,7,3,4,2,1] => [6,5,7,3,4,2,1] => ? = 2 + 1
[1,2,4,3,7,6,5] => [1,2,4,3,7,5,6] => [6,5,7,3,4,2,1] => [6,5,7,3,4,2,1] => ? = 2 + 1
[1,2,4,5,3,6,7] => [1,2,4,5,3,6,7] => [7,6,3,5,4,2,1] => [7,6,3,5,4,2,1] => ? = 1 + 1
[1,2,4,5,6,3,7] => [1,2,4,5,6,3,7] => [7,3,6,5,4,2,1] => [7,3,6,5,4,2,1] => ? = 1 + 1
[1,2,4,5,7,3,6] => [1,2,4,5,7,3,6] => [6,3,7,5,4,2,1] => [6,3,7,5,4,2,1] => ? = 2 + 1
[1,2,4,5,7,6,3] => [1,2,4,5,7,3,6] => [6,3,7,5,4,2,1] => [6,3,7,5,4,2,1] => ? = 2 + 1
[1,2,4,6,3,5,7] => [1,2,4,6,3,5,7] => [7,5,3,6,4,2,1] => [7,5,3,6,4,2,1] => ? = 1 + 1
[1,2,4,6,5,7,3] => [1,2,4,6,3,5,7] => [7,5,3,6,4,2,1] => [7,5,3,6,4,2,1] => ? = 1 + 1
[1,2,4,6,7,3,5] => [1,2,4,6,7,3,5] => [5,3,7,6,4,2,1] => [5,3,7,6,4,2,1] => ? = 2 + 1
[1,2,4,6,7,5,3] => [1,2,4,6,7,3,5] => [5,3,7,6,4,2,1] => [5,3,7,6,4,2,1] => ? = 2 + 1
[1,2,4,7,3,5,6] => [1,2,4,7,3,5,6] => [6,5,3,7,4,2,1] => [6,5,3,7,4,2,1] => ? = 2 + 1
[1,2,4,7,3,6,5] => [1,2,4,7,3,6,5] => [5,6,3,7,4,2,1] => [5,7,3,6,4,2,1] => ? = 2 + 1
[1,2,4,7,5,3,6] => [1,2,4,7,3,6,5] => [5,6,3,7,4,2,1] => [5,7,3,6,4,2,1] => ? = 2 + 1
[1,2,4,7,5,6,3] => [1,2,4,7,3,5,6] => [6,5,3,7,4,2,1] => [6,5,3,7,4,2,1] => ? = 2 + 1
[1,2,4,7,6,3,5] => [1,2,4,7,3,5,6] => [6,5,3,7,4,2,1] => [6,5,3,7,4,2,1] => ? = 2 + 1
[1,2,4,7,6,5,3] => [1,2,4,7,3,5,6] => [6,5,3,7,4,2,1] => [6,5,3,7,4,2,1] => ? = 2 + 1
[1,2,5,3,4,6,7] => [1,2,5,3,4,6,7] => [7,6,4,3,5,2,1] => [7,6,4,3,5,2,1] => ? = 1 + 1
[1,2,5,3,6,4,7] => [1,2,5,3,6,4,7] => [7,4,6,3,5,2,1] => [7,4,6,3,5,2,1] => ? = 1 + 1
[1,2,5,3,7,4,6] => [1,2,5,3,7,4,6] => [6,4,7,3,5,2,1] => [6,4,7,3,5,2,1] => ? = 2 + 1
[1,2,5,3,7,6,4] => [1,2,5,3,7,4,6] => [6,4,7,3,5,2,1] => [6,4,7,3,5,2,1] => ? = 2 + 1
[1,2,5,4,3,7,6] => [1,2,5,3,7,4,6] => [6,4,7,3,5,2,1] => [6,4,7,3,5,2,1] => ? = 2 + 1
[1,2,5,4,6,3,7] => [1,2,5,3,7,4,6] => [6,4,7,3,5,2,1] => [6,4,7,3,5,2,1] => ? = 2 + 1
[1,2,5,4,6,7,3] => [1,2,5,3,4,6,7] => [7,6,4,3,5,2,1] => [7,6,4,3,5,2,1] => ? = 1 + 1
[1,2,5,4,7,3,6] => [1,2,5,3,6,4,7] => [7,4,6,3,5,2,1] => [7,4,6,3,5,2,1] => ? = 1 + 1
[1,2,5,6,3,4,7] => [1,2,5,6,3,4,7] => [7,4,3,6,5,2,1] => [7,4,3,6,5,2,1] => ? = 1 + 1
[1,2,5,6,4,7,3] => [1,2,5,6,3,4,7] => [7,4,3,6,5,2,1] => [7,4,3,6,5,2,1] => ? = 1 + 1
[1,2,5,6,7,3,4] => [1,2,5,6,7,3,4] => [4,3,7,6,5,2,1] => [4,3,7,6,5,2,1] => ? = 2 + 1
[1,2,5,6,7,4,3] => [1,2,5,6,7,3,4] => [4,3,7,6,5,2,1] => [4,3,7,6,5,2,1] => ? = 2 + 1
[1,2,5,7,3,4,6] => [1,2,5,7,3,4,6] => [6,4,3,7,5,2,1] => [6,4,3,7,5,2,1] => ? = 2 + 1
[1,2,5,7,3,6,4] => [1,2,5,7,3,6,4] => [4,6,3,7,5,2,1] => [4,7,3,6,5,2,1] => ? = 2 + 1
[1,2,5,7,4,3,6] => [1,2,5,7,3,6,4] => [4,6,3,7,5,2,1] => [4,7,3,6,5,2,1] => ? = 2 + 1
Description
The number of indices that are either left-to-right maxima or right-to-left minima.
The (bivariate) generating function for this statistic is (essentially) given in [1], the mid points of a $321$ pattern in the permutation are those elements which are neither left-to-right maxima nor a right-to-left minima, see [[St000371]] and [[St000372]].
The following 24 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001737The number of descents of type 2 in a permutation. St000451The length of the longest pattern of the form k 1 2. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000353The number of inner valleys of a permutation. St000872The number of very big descents of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St000259The diameter of a connected graph. St001330The hat guessing number of a graph. St000100The number of linear extensions 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. St001520The number of strict 3-descents. St001557The number of inversions of the second entry of a permutation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000635The number of strictly order preserving maps of a poset into itself. St001890The maximum magnitude of the Möbius function of a poset. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001860The number of factors of the Stanley symmetric function associated with a signed permutation.
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