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Your data matches 14 different statistics following compositions of up to 3 maps.
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Matching statistic: St000390
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Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00130: Permutations —descent tops⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00130: Permutations —descent tops⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => 1 => 1
[[.,.],.]
=> [1,2] => 0 => 0
[.,[.,[.,.]]]
=> [3,2,1] => 11 => 1
[.,[[.,.],.]]
=> [2,3,1] => 01 => 1
[[.,.],[.,.]]
=> [3,1,2] => 01 => 1
[[.,[.,.]],.]
=> [2,1,3] => 10 => 1
[[[.,.],.],.]
=> [1,2,3] => 00 => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 111 => 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => 101 => 2
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => 011 => 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => 011 => 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => 001 => 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => 011 => 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => 001 => 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => 101 => 2
[[[.,.],.],[.,.]]
=> [4,1,2,3] => 001 => 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => 110 => 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => 010 => 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => 010 => 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => 100 => 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => 000 => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 1111 => 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 1101 => 2
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => 1011 => 2
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 1011 => 2
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 1001 => 2
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => 0111 => 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => 0101 => 2
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => 0111 => 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => 0011 => 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 0111 => 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 0011 => 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => 0011 => 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 0101 => 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 0001 => 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 0111 => 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => 0101 => 2
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 0011 => 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => 0011 => 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => 0001 => 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 1011 => 2
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => 1001 => 2
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 0011 => 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => 0001 => 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => 1101 => 2
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => 0101 => 2
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 0101 => 2
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => 1001 => 2
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 0001 => 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 1110 => 1
Description
The number of runs of ones in a binary word.
Matching statistic: St000884
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000884: Permutations ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 86%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000884: Permutations ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 86%
Values
[.,[.,.]]
=> [2,1] => [2,1] => [2,1] => 1
[[.,.],.]
=> [1,2] => [1,2] => [1,2] => 0
[.,[.,[.,.]]]
=> [3,2,1] => [3,1,2] => [3,1,2] => 1
[.,[[.,.],.]]
=> [2,3,1] => [3,2,1] => [2,3,1] => 1
[[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => [1,3,2] => 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,1,2,3] => [4,1,2,3] => 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [4,1,3,2] => [3,1,4,2] => 2
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [4,2,1,3] => [2,4,1,3] => 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [4,3,2,1] => [3,4,1,2] => 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [4,2,3,1] => [2,3,4,1] => 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,2,3] => [1,4,2,3] => 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,4,3,2] => [1,3,4,2] => 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,1,2,4] => [3,1,2,4] => 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,2,1,4] => [2,3,1,4] => 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,1,2,3,4] => [5,1,2,3,4] => 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [5,1,2,4,3] => [4,1,2,5,3] => 2
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [5,1,3,2,4] => [3,1,5,2,4] => 2
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [5,1,4,3,2] => [4,1,5,2,3] => 2
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [5,1,3,4,2] => [3,1,4,5,2] => 2
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [5,2,1,3,4] => [2,5,1,3,4] => 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [5,2,1,4,3] => [2,4,1,5,3] => 2
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [5,3,2,1,4] => [3,5,1,2,4] => 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [5,2,3,1,4] => [2,3,5,1,4] => 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [5,4,2,3,1] => [4,5,1,2,3] => 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [5,4,3,2,1] => [3,4,5,1,2] => 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [5,2,4,3,1] => [2,4,5,1,3] => 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [5,3,2,4,1] => [3,4,1,5,2] => 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [5,2,3,4,1] => [2,3,4,5,1] => 1
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,5,2,3,4] => [1,5,2,3,4] => 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,5,2,4,3] => [1,4,2,5,3] => 2
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,5,3,2,4] => [1,3,5,2,4] => 1
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,5,4,3,2] => [1,4,5,2,3] => 1
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,5,3,4,2] => [1,3,4,5,2] => 1
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,5,3,4] => [2,1,5,3,4] => 2
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,5,4,3] => [2,1,4,5,3] => 2
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,5,3,4] => [1,2,5,3,4] => 1
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,5,4,3] => [1,2,4,5,3] => 1
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => 2
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [3,2,1,5,4] => [2,3,1,5,4] => 2
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 2
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,1,2,3,5] => [4,1,2,3,5] => 1
[.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [4,7,6,5,3,2,1] => [7,1,2,4,3,5,6] => [4,1,2,7,3,5,6] => ? = 2
[.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [4,6,7,5,3,2,1] => [7,1,2,4,3,6,5] => [4,1,2,6,3,7,5] => ? = 3
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [5,4,7,6,3,2,1] => [7,1,2,5,4,3,6] => [5,1,2,7,3,4,6] => ? = 2
[.,[.,[.,[[[.,.],.],[.,.]]]]]
=> [4,5,7,6,3,2,1] => [7,1,2,4,5,3,6] => [4,1,2,5,7,3,6] => ? = 2
[.,[.,[.,[[.,[[.,.],.]],.]]]]
=> [5,6,4,7,3,2,1] => [7,1,2,6,5,4,3] => [5,1,2,6,7,3,4] => ? = 2
[.,[.,[.,[[[.,.],[.,.]],.]]]]
=> [4,6,5,7,3,2,1] => [7,1,2,4,6,5,3] => [4,1,2,6,7,3,5] => ? = 2
[.,[.,[.,[[[.,[.,.]],.],.]]]]
=> [5,4,6,7,3,2,1] => [7,1,2,5,4,6,3] => [5,1,2,6,3,7,4] => ? = 3
[.,[.,[.,[[[[.,.],.],.],.]]]]
=> [4,5,6,7,3,2,1] => [7,1,2,4,5,6,3] => [4,1,2,5,6,7,3] => ? = 2
[.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [3,7,6,5,4,2,1] => [7,1,3,2,4,5,6] => [3,1,7,2,4,5,6] => ? = 2
[.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [3,6,7,5,4,2,1] => [7,1,3,2,4,6,5] => [3,1,6,2,4,7,5] => ? = 3
[.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [3,5,7,6,4,2,1] => [7,1,3,2,5,4,6] => [3,1,5,2,7,4,6] => ? = 3
[.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [3,6,5,7,4,2,1] => [7,1,3,2,6,5,4] => [3,1,6,2,7,4,5] => ? = 3
[.,[.,[[.,.],[[[.,.],.],.]]]]
=> [3,5,6,7,4,2,1] => [7,1,3,2,5,6,4] => [3,1,5,2,6,7,4] => ? = 3
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [4,3,7,6,5,2,1] => [7,1,4,3,2,5,6] => [4,1,7,2,3,5,6] => ? = 2
[.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [4,3,6,7,5,2,1] => [7,1,4,3,2,6,5] => [4,1,6,2,3,7,5] => ? = 3
[.,[.,[[[.,.],.],[.,[.,.]]]]]
=> [3,4,7,6,5,2,1] => [7,1,3,4,2,5,6] => [3,1,4,7,2,5,6] => ? = 2
[.,[.,[[[.,.],.],[[.,.],.]]]]
=> [3,4,6,7,5,2,1] => [7,1,3,4,2,6,5] => [3,1,4,6,2,7,5] => ? = 3
[.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [5,4,3,7,6,2,1] => [7,1,5,3,4,2,6] => [5,1,7,2,3,4,6] => ? = 2
[.,[.,[[.,[[.,.],.]],[.,.]]]]
=> [4,5,3,7,6,2,1] => [7,1,5,4,3,2,6] => [4,1,5,7,2,3,6] => ? = 2
[.,[.,[[[.,.],[.,.]],[.,.]]]]
=> [3,5,4,7,6,2,1] => [7,1,3,5,4,2,6] => [3,1,5,7,2,4,6] => ? = 2
[.,[.,[[[.,[.,.]],.],[.,.]]]]
=> [4,3,5,7,6,2,1] => [7,1,4,3,5,2,6] => [4,1,5,2,7,3,6] => ? = 3
[.,[.,[[[[.,.],.],.],[.,.]]]]
=> [3,4,5,7,6,2,1] => [7,1,3,4,5,2,6] => [3,1,4,5,7,2,6] => ? = 2
[.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> [6,5,4,3,7,2,1] => [7,1,6,3,4,5,2] => [6,1,7,2,3,4,5] => ? = 2
[.,[.,[[.,[.,[[.,.],.]]],.]]]
=> [5,6,4,3,7,2,1] => [7,1,6,3,5,4,2] => [5,1,6,2,7,3,4] => ? = 3
[.,[.,[[.,[[.,.],[.,.]]],.]]]
=> [4,6,5,3,7,2,1] => [7,1,6,4,3,5,2] => [4,1,6,7,2,3,5] => ? = 2
[.,[.,[[.,[[.,[.,.]],.]],.]]]
=> [5,4,6,3,7,2,1] => [7,1,6,5,4,3,2] => [5,1,6,7,2,3,4] => ? = 2
[.,[.,[[.,[[[.,.],.],.]],.]]]
=> [4,5,6,3,7,2,1] => [7,1,6,4,5,3,2] => [4,1,5,6,7,2,3] => ? = 2
[.,[.,[[[.,.],[.,[.,.]]],.]]]
=> [3,6,5,4,7,2,1] => [7,1,3,6,4,5,2] => [3,1,6,7,2,4,5] => ? = 2
[.,[.,[[[.,.],[[.,.],.]],.]]]
=> [3,5,6,4,7,2,1] => [7,1,3,6,5,4,2] => [3,1,5,6,7,2,4] => ? = 2
[.,[.,[[[.,[.,.]],[.,.]],.]]]
=> [4,3,6,5,7,2,1] => [7,1,4,3,6,5,2] => [4,1,6,2,7,3,5] => ? = 3
[.,[.,[[[[.,.],.],[.,.]],.]]]
=> [3,4,6,5,7,2,1] => [7,1,3,4,6,5,2] => [3,1,4,6,7,2,5] => ? = 2
[.,[.,[[[.,[[.,.],.]],.],.]]]
=> [4,5,3,6,7,2,1] => [7,1,5,4,3,6,2] => [4,1,5,6,2,7,3] => ? = 3
[.,[.,[[[[.,.],[.,.]],.],.]]]
=> [3,5,4,6,7,2,1] => [7,1,3,5,4,6,2] => [3,1,5,6,2,7,4] => ? = 3
[.,[.,[[[[.,[.,.]],.],.],.]]]
=> [4,3,5,6,7,2,1] => [7,1,4,3,5,6,2] => [4,1,5,2,6,7,3] => ? = 3
[.,[.,[[[[[.,.],.],.],.],.]]]
=> [3,4,5,6,7,2,1] => [7,1,3,4,5,6,2] => [3,1,4,5,6,7,2] => ? = 2
[.,[[.,.],[.,[.,[[.,.],.]]]]]
=> [2,6,7,5,4,3,1] => [7,2,1,3,4,6,5] => [2,6,1,3,4,7,5] => ? = 2
[.,[[.,.],[.,[[.,.],[.,.]]]]]
=> [2,5,7,6,4,3,1] => [7,2,1,3,5,4,6] => [2,5,1,3,7,4,6] => ? = 2
[.,[[.,.],[.,[[.,[.,.]],.]]]]
=> [2,6,5,7,4,3,1] => [7,2,1,3,6,5,4] => [2,6,1,3,7,4,5] => ? = 2
[.,[[.,.],[.,[[[.,.],.],.]]]]
=> [2,5,6,7,4,3,1] => [7,2,1,3,5,6,4] => [2,5,1,3,6,7,4] => ? = 2
[.,[[.,.],[[.,.],[.,[.,.]]]]]
=> [2,4,7,6,5,3,1] => [7,2,1,4,3,5,6] => [2,4,1,7,3,5,6] => ? = 2
[.,[[.,.],[[.,.],[[.,.],.]]]]
=> [2,4,6,7,5,3,1] => [7,2,1,4,3,6,5] => [2,4,1,6,3,7,5] => ? = 3
[.,[[.,.],[[.,[.,.]],[.,.]]]]
=> [2,5,4,7,6,3,1] => [7,2,1,5,4,3,6] => [2,5,1,7,3,4,6] => ? = 2
[.,[[.,.],[[[.,.],.],[.,.]]]]
=> [2,4,5,7,6,3,1] => [7,2,1,4,5,3,6] => [2,4,1,5,7,3,6] => ? = 2
[.,[[.,.],[[.,[[.,.],.]],.]]]
=> [2,5,6,4,7,3,1] => [7,2,1,6,5,4,3] => [2,5,1,6,7,3,4] => ? = 2
[.,[[.,.],[[[.,.],[.,.]],.]]]
=> [2,4,6,5,7,3,1] => [7,2,1,4,6,5,3] => [2,4,1,6,7,3,5] => ? = 2
[.,[[.,.],[[[.,[.,.]],.],.]]]
=> [2,5,4,6,7,3,1] => [7,2,1,5,4,6,3] => [2,5,1,6,3,7,4] => ? = 3
[.,[[.,.],[[[[.,.],.],.],.]]]
=> [2,4,5,6,7,3,1] => [7,2,1,4,5,6,3] => [2,4,1,5,6,7,3] => ? = 2
[.,[[.,[.,.]],[.,[.,[.,.]]]]]
=> [3,2,7,6,5,4,1] => [7,3,2,1,4,5,6] => [3,7,1,2,4,5,6] => ? = 1
[.,[[.,[.,.]],[.,[[.,.],.]]]]
=> [3,2,6,7,5,4,1] => [7,3,2,1,4,6,5] => [3,6,1,2,4,7,5] => ? = 2
[.,[[.,[.,.]],[[.,.],[.,.]]]]
=> [3,2,5,7,6,4,1] => [7,3,2,1,5,4,6] => [3,5,1,2,7,4,6] => ? = 2
Description
The number of isolated descents of a permutation.
A descent $i$ is isolated if neither $i+1$ nor $i-1$ are descents. If a permutation has only isolated descents, then it is called primitive in [1].
Matching statistic: St000035
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000035: Permutations ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 100%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000035: Permutations ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => [2,1] => [2,1] => 1
[[.,.],.]
=> [1,2] => [1,2] => [1,2] => 0
[.,[.,[.,.]]]
=> [3,2,1] => [3,1,2] => [3,1,2] => 1
[.,[[.,.],.]]
=> [2,3,1] => [3,2,1] => [2,3,1] => 1
[[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => [1,3,2] => 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,1,2,3] => [4,1,2,3] => 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [4,1,3,2] => [3,1,4,2] => 2
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [4,2,1,3] => [2,4,1,3] => 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [4,3,2,1] => [3,4,1,2] => 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [4,2,3,1] => [2,3,4,1] => 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,2,3] => [1,4,2,3] => 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,4,3,2] => [1,3,4,2] => 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,1,2,4] => [3,1,2,4] => 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,2,1,4] => [2,3,1,4] => 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,1,2,3,4] => [5,1,2,3,4] => 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [5,1,2,4,3] => [4,1,2,5,3] => 2
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [5,1,3,2,4] => [3,1,5,2,4] => 2
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [5,1,4,3,2] => [4,1,5,2,3] => 2
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [5,1,3,4,2] => [3,1,4,5,2] => 2
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [5,2,1,3,4] => [2,5,1,3,4] => 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [5,2,1,4,3] => [2,4,1,5,3] => 2
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [5,3,2,1,4] => [3,5,1,2,4] => 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [5,2,3,1,4] => [2,3,5,1,4] => 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [5,4,2,3,1] => [4,5,1,2,3] => 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [5,4,3,2,1] => [3,4,5,1,2] => 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [5,2,4,3,1] => [2,4,5,1,3] => 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [5,3,2,4,1] => [3,4,1,5,2] => 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [5,2,3,4,1] => [2,3,4,5,1] => 1
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,5,2,3,4] => [1,5,2,3,4] => 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,5,2,4,3] => [1,4,2,5,3] => 2
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,5,3,2,4] => [1,3,5,2,4] => 1
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,5,4,3,2] => [1,4,5,2,3] => 1
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,5,3,4,2] => [1,3,4,5,2] => 1
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,5,3,4] => [2,1,5,3,4] => 2
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,5,4,3] => [2,1,4,5,3] => 2
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,5,3,4] => [1,2,5,3,4] => 1
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,5,4,3] => [1,2,4,5,3] => 1
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => 2
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [3,2,1,5,4] => [2,3,1,5,4] => 2
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 2
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,1,2,3,5] => [4,1,2,3,5] => 1
[.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [4,7,6,5,3,2,1] => [7,1,2,4,3,5,6] => [4,1,2,7,3,5,6] => ? = 2
[.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [4,6,7,5,3,2,1] => [7,1,2,4,3,6,5] => [4,1,2,6,3,7,5] => ? = 3
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [5,4,7,6,3,2,1] => [7,1,2,5,4,3,6] => [5,1,2,7,3,4,6] => ? = 2
[.,[.,[.,[[[.,.],.],[.,.]]]]]
=> [4,5,7,6,3,2,1] => [7,1,2,4,5,3,6] => [4,1,2,5,7,3,6] => ? = 2
[.,[.,[.,[[.,[[.,.],.]],.]]]]
=> [5,6,4,7,3,2,1] => [7,1,2,6,5,4,3] => [5,1,2,6,7,3,4] => ? = 2
[.,[.,[.,[[[.,.],[.,.]],.]]]]
=> [4,6,5,7,3,2,1] => [7,1,2,4,6,5,3] => [4,1,2,6,7,3,5] => ? = 2
[.,[.,[.,[[[.,[.,.]],.],.]]]]
=> [5,4,6,7,3,2,1] => [7,1,2,5,4,6,3] => [5,1,2,6,3,7,4] => ? = 3
[.,[.,[.,[[[[.,.],.],.],.]]]]
=> [4,5,6,7,3,2,1] => [7,1,2,4,5,6,3] => [4,1,2,5,6,7,3] => ? = 2
[.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [3,7,6,5,4,2,1] => [7,1,3,2,4,5,6] => [3,1,7,2,4,5,6] => ? = 2
[.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [3,6,7,5,4,2,1] => [7,1,3,2,4,6,5] => [3,1,6,2,4,7,5] => ? = 3
[.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [3,5,7,6,4,2,1] => [7,1,3,2,5,4,6] => [3,1,5,2,7,4,6] => ? = 3
[.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [3,6,5,7,4,2,1] => [7,1,3,2,6,5,4] => [3,1,6,2,7,4,5] => ? = 3
[.,[.,[[.,.],[[[.,.],.],.]]]]
=> [3,5,6,7,4,2,1] => [7,1,3,2,5,6,4] => [3,1,5,2,6,7,4] => ? = 3
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [4,3,7,6,5,2,1] => [7,1,4,3,2,5,6] => [4,1,7,2,3,5,6] => ? = 2
[.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [4,3,6,7,5,2,1] => [7,1,4,3,2,6,5] => [4,1,6,2,3,7,5] => ? = 3
[.,[.,[[[.,.],.],[.,[.,.]]]]]
=> [3,4,7,6,5,2,1] => [7,1,3,4,2,5,6] => [3,1,4,7,2,5,6] => ? = 2
[.,[.,[[[.,.],.],[[.,.],.]]]]
=> [3,4,6,7,5,2,1] => [7,1,3,4,2,6,5] => [3,1,4,6,2,7,5] => ? = 3
[.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [5,4,3,7,6,2,1] => [7,1,5,3,4,2,6] => [5,1,7,2,3,4,6] => ? = 2
[.,[.,[[.,[[.,.],.]],[.,.]]]]
=> [4,5,3,7,6,2,1] => [7,1,5,4,3,2,6] => [4,1,5,7,2,3,6] => ? = 2
[.,[.,[[[.,.],[.,.]],[.,.]]]]
=> [3,5,4,7,6,2,1] => [7,1,3,5,4,2,6] => [3,1,5,7,2,4,6] => ? = 2
[.,[.,[[[.,[.,.]],.],[.,.]]]]
=> [4,3,5,7,6,2,1] => [7,1,4,3,5,2,6] => [4,1,5,2,7,3,6] => ? = 3
[.,[.,[[[[.,.],.],.],[.,.]]]]
=> [3,4,5,7,6,2,1] => [7,1,3,4,5,2,6] => [3,1,4,5,7,2,6] => ? = 2
[.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> [6,5,4,3,7,2,1] => [7,1,6,3,4,5,2] => [6,1,7,2,3,4,5] => ? = 2
[.,[.,[[.,[.,[[.,.],.]]],.]]]
=> [5,6,4,3,7,2,1] => [7,1,6,3,5,4,2] => [5,1,6,2,7,3,4] => ? = 3
[.,[.,[[.,[[.,.],[.,.]]],.]]]
=> [4,6,5,3,7,2,1] => [7,1,6,4,3,5,2] => [4,1,6,7,2,3,5] => ? = 2
[.,[.,[[.,[[.,[.,.]],.]],.]]]
=> [5,4,6,3,7,2,1] => [7,1,6,5,4,3,2] => [5,1,6,7,2,3,4] => ? = 2
[.,[.,[[.,[[[.,.],.],.]],.]]]
=> [4,5,6,3,7,2,1] => [7,1,6,4,5,3,2] => [4,1,5,6,7,2,3] => ? = 2
[.,[.,[[[.,.],[.,[.,.]]],.]]]
=> [3,6,5,4,7,2,1] => [7,1,3,6,4,5,2] => [3,1,6,7,2,4,5] => ? = 2
[.,[.,[[[.,.],[[.,.],.]],.]]]
=> [3,5,6,4,7,2,1] => [7,1,3,6,5,4,2] => [3,1,5,6,7,2,4] => ? = 2
[.,[.,[[[.,[.,.]],[.,.]],.]]]
=> [4,3,6,5,7,2,1] => [7,1,4,3,6,5,2] => [4,1,6,2,7,3,5] => ? = 3
[.,[.,[[[[.,.],.],[.,.]],.]]]
=> [3,4,6,5,7,2,1] => [7,1,3,4,6,5,2] => [3,1,4,6,7,2,5] => ? = 2
[.,[.,[[[.,[[.,.],.]],.],.]]]
=> [4,5,3,6,7,2,1] => [7,1,5,4,3,6,2] => [4,1,5,6,2,7,3] => ? = 3
[.,[.,[[[[.,.],[.,.]],.],.]]]
=> [3,5,4,6,7,2,1] => [7,1,3,5,4,6,2] => [3,1,5,6,2,7,4] => ? = 3
[.,[.,[[[[.,[.,.]],.],.],.]]]
=> [4,3,5,6,7,2,1] => [7,1,4,3,5,6,2] => [4,1,5,2,6,7,3] => ? = 3
[.,[.,[[[[[.,.],.],.],.],.]]]
=> [3,4,5,6,7,2,1] => [7,1,3,4,5,6,2] => [3,1,4,5,6,7,2] => ? = 2
[.,[[.,.],[.,[.,[[.,.],.]]]]]
=> [2,6,7,5,4,3,1] => [7,2,1,3,4,6,5] => [2,6,1,3,4,7,5] => ? = 2
[.,[[.,.],[.,[[.,.],[.,.]]]]]
=> [2,5,7,6,4,3,1] => [7,2,1,3,5,4,6] => [2,5,1,3,7,4,6] => ? = 2
[.,[[.,.],[.,[[.,[.,.]],.]]]]
=> [2,6,5,7,4,3,1] => [7,2,1,3,6,5,4] => [2,6,1,3,7,4,5] => ? = 2
[.,[[.,.],[.,[[[.,.],.],.]]]]
=> [2,5,6,7,4,3,1] => [7,2,1,3,5,6,4] => [2,5,1,3,6,7,4] => ? = 2
[.,[[.,.],[[.,.],[.,[.,.]]]]]
=> [2,4,7,6,5,3,1] => [7,2,1,4,3,5,6] => [2,4,1,7,3,5,6] => ? = 2
[.,[[.,.],[[.,.],[[.,.],.]]]]
=> [2,4,6,7,5,3,1] => [7,2,1,4,3,6,5] => [2,4,1,6,3,7,5] => ? = 3
[.,[[.,.],[[.,[.,.]],[.,.]]]]
=> [2,5,4,7,6,3,1] => [7,2,1,5,4,3,6] => [2,5,1,7,3,4,6] => ? = 2
[.,[[.,.],[[[.,.],.],[.,.]]]]
=> [2,4,5,7,6,3,1] => [7,2,1,4,5,3,6] => [2,4,1,5,7,3,6] => ? = 2
[.,[[.,.],[[.,[[.,.],.]],.]]]
=> [2,5,6,4,7,3,1] => [7,2,1,6,5,4,3] => [2,5,1,6,7,3,4] => ? = 2
[.,[[.,.],[[[.,.],[.,.]],.]]]
=> [2,4,6,5,7,3,1] => [7,2,1,4,6,5,3] => [2,4,1,6,7,3,5] => ? = 2
[.,[[.,.],[[[.,[.,.]],.],.]]]
=> [2,5,4,6,7,3,1] => [7,2,1,5,4,6,3] => [2,5,1,6,3,7,4] => ? = 3
[.,[[.,.],[[[[.,.],.],.],.]]]
=> [2,4,5,6,7,3,1] => [7,2,1,4,5,6,3] => [2,4,1,5,6,7,3] => ? = 2
[.,[[.,[.,.]],[.,[.,[.,.]]]]]
=> [3,2,7,6,5,4,1] => [7,3,2,1,4,5,6] => [3,7,1,2,4,5,6] => ? = 1
[.,[[.,[.,.]],[.,[[.,.],.]]]]
=> [3,2,6,7,5,4,1] => [7,3,2,1,4,6,5] => [3,6,1,2,4,7,5] => ? = 2
[.,[[.,[.,.]],[[.,.],[.,.]]]]
=> [3,2,5,7,6,4,1] => [7,3,2,1,5,4,6] => [3,5,1,2,7,4,6] => ? = 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: St001737
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St001737: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 57%
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St001737: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 57%
Values
[.,[.,.]]
=> [2,1] => [2,1] => [2,1] => 1
[[.,.],.]
=> [1,2] => [1,2] => [1,2] => 0
[.,[.,[.,.]]]
=> [3,2,1] => [2,3,1] => [3,2,1] => 1
[.,[[.,.],.]]
=> [2,3,1] => [3,2,1] => [2,3,1] => 1
[[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => [1,3,2] => 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [2,3,4,1] => [4,2,3,1] => 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [2,4,3,1] => [3,2,4,1] => 2
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [3,2,4,1] => [2,4,3,1] => 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [4,3,2,1] => [3,4,1,2] => 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [4,2,3,1] => [2,3,4,1] => 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,3,4,2] => [1,4,3,2] => 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,4,3,2] => [1,3,4,2] => 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [2,3,1,4] => [3,2,1,4] => 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,2,1,4] => [2,3,1,4] => 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [2,3,4,5,1] => [5,2,3,4,1] => 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [2,3,5,4,1] => [4,2,3,5,1] => 2
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [2,4,3,5,1] => [3,2,5,4,1] => 2
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [2,5,4,3,1] => [4,2,5,1,3] => 2
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [2,5,3,4,1] => [3,2,4,5,1] => 2
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [3,2,4,5,1] => [2,5,3,4,1] => 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [3,2,5,4,1] => [2,4,3,5,1] => 2
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [4,3,2,5,1] => [3,5,1,4,2] => 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [4,2,3,5,1] => [2,3,5,4,1] => 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [5,3,4,2,1] => [3,5,4,1,2] => 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [5,4,3,2,1] => [3,4,5,1,2] => 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [5,2,4,3,1] => [2,4,5,1,3] => 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [5,3,2,4,1] => [3,4,1,5,2] => 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [5,2,3,4,1] => [2,3,4,5,1] => 1
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,3,4,5,2] => [1,5,3,4,2] => 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,3,5,4,2] => [1,4,3,5,2] => 2
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,4,3,5,2] => [1,3,5,4,2] => 1
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,5,4,3,2] => [1,4,5,2,3] => 1
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,5,3,4,2] => [1,3,4,5,2] => 1
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,4,5,3] => [2,1,5,4,3] => 2
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,5,4,3] => [2,1,4,5,3] => 2
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,4,5,3] => [1,2,5,4,3] => 1
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,5,4,3] => [1,2,4,5,3] => 1
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [2,3,1,5,4] => [3,2,1,5,4] => 2
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [3,2,1,5,4] => [2,3,1,5,4] => 2
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 2
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [2,3,4,1,5] => [4,2,3,1,5] => 1
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [7,6,5,4,3,2,1] => [2,3,4,5,6,7,1] => [7,2,3,4,5,6,1] => ? = 1
[.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [6,7,5,4,3,2,1] => [2,3,4,5,7,6,1] => [6,2,3,4,5,7,1] => ? = 2
[.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [5,7,6,4,3,2,1] => [2,3,4,6,5,7,1] => [5,2,3,4,7,6,1] => ? = 2
[.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [6,5,7,4,3,2,1] => [2,3,4,7,6,5,1] => [6,2,3,4,7,1,5] => ? = 2
[.,[.,[.,[.,[[[.,.],.],.]]]]]
=> [5,6,7,4,3,2,1] => [2,3,4,7,5,6,1] => [5,2,3,4,6,7,1] => ? = 2
[.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [4,7,6,5,3,2,1] => [2,3,5,4,6,7,1] => [4,2,3,7,5,6,1] => ? = 2
[.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [4,6,7,5,3,2,1] => [2,3,5,4,7,6,1] => [4,2,3,6,5,7,1] => ? = 3
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [5,4,7,6,3,2,1] => [2,3,6,5,4,7,1] => [5,2,3,7,1,6,4] => ? = 2
[.,[.,[.,[[[.,.],.],[.,.]]]]]
=> [4,5,7,6,3,2,1] => [2,3,6,4,5,7,1] => [4,2,3,5,7,6,1] => ? = 2
[.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [6,5,4,7,3,2,1] => [2,3,7,5,6,4,1] => [5,2,3,7,6,1,4] => ? = 2
[.,[.,[.,[[.,[[.,.],.]],.]]]]
=> [5,6,4,7,3,2,1] => [2,3,7,6,5,4,1] => [5,2,3,6,7,1,4] => ? = 2
[.,[.,[.,[[[.,.],[.,.]],.]]]]
=> [4,6,5,7,3,2,1] => [2,3,7,4,6,5,1] => [4,2,3,6,7,1,5] => ? = 2
[.,[.,[.,[[[.,[.,.]],.],.]]]]
=> [5,4,6,7,3,2,1] => [2,3,7,5,4,6,1] => [5,2,3,6,1,7,4] => ? = 3
[.,[.,[.,[[[[.,.],.],.],.]]]]
=> [4,5,6,7,3,2,1] => [2,3,7,4,5,6,1] => [4,2,3,5,6,7,1] => ? = 2
[.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [3,7,6,5,4,2,1] => [2,4,3,5,6,7,1] => [3,2,7,4,5,6,1] => ? = 2
[.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [3,6,7,5,4,2,1] => [2,4,3,5,7,6,1] => [3,2,6,4,5,7,1] => ? = 3
[.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [3,5,7,6,4,2,1] => [2,4,3,6,5,7,1] => [3,2,5,4,7,6,1] => ? = 3
[.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [3,6,5,7,4,2,1] => [2,4,3,7,6,5,1] => [3,2,6,4,7,1,5] => ? = 3
[.,[.,[[.,.],[[[.,.],.],.]]]]
=> [3,5,6,7,4,2,1] => [2,4,3,7,5,6,1] => [3,2,5,4,6,7,1] => ? = 3
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [4,3,7,6,5,2,1] => [2,5,4,3,6,7,1] => [4,2,7,1,5,6,3] => ? = 2
[.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [4,3,6,7,5,2,1] => [2,5,4,3,7,6,1] => [4,2,6,1,5,7,3] => ? = 3
[.,[.,[[[.,.],.],[.,[.,.]]]]]
=> [3,4,7,6,5,2,1] => [2,5,3,4,6,7,1] => [3,2,4,7,5,6,1] => ? = 2
[.,[.,[[[.,.],.],[[.,.],.]]]]
=> [3,4,6,7,5,2,1] => [2,5,3,4,7,6,1] => [3,2,4,6,5,7,1] => ? = 3
[.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [5,4,3,7,6,2,1] => [2,6,4,5,3,7,1] => [4,2,7,5,1,6,3] => ? = 2
[.,[.,[[.,[[.,.],.]],[.,.]]]]
=> [4,5,3,7,6,2,1] => [2,6,5,4,3,7,1] => [4,2,5,7,1,6,3] => ? = 2
[.,[.,[[[.,.],[.,.]],[.,.]]]]
=> [3,5,4,7,6,2,1] => [2,6,3,5,4,7,1] => [3,2,5,7,1,6,4] => ? = 2
[.,[.,[[[.,[.,.]],.],[.,.]]]]
=> [4,3,5,7,6,2,1] => [2,6,4,3,5,7,1] => [4,2,5,1,7,6,3] => ? = 3
[.,[.,[[[[.,.],.],.],[.,.]]]]
=> [3,4,5,7,6,2,1] => [2,6,3,4,5,7,1] => [3,2,4,5,7,6,1] => ? = 2
[.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> [6,5,4,3,7,2,1] => [2,7,4,5,6,3,1] => [4,2,7,5,6,1,3] => ? = 2
[.,[.,[[.,[.,[[.,.],.]]],.]]]
=> [5,6,4,3,7,2,1] => [2,7,4,6,5,3,1] => [4,2,6,5,7,1,3] => ? = 3
[.,[.,[[.,[[.,.],[.,.]]],.]]]
=> [4,6,5,3,7,2,1] => [2,7,5,4,6,3,1] => [4,2,5,7,6,1,3] => ? = 2
[.,[.,[[.,[[.,[.,.]],.]],.]]]
=> [5,4,6,3,7,2,1] => [2,7,6,5,4,3,1] => [5,2,6,7,1,3,4] => ? = 2
[.,[.,[[.,[[[.,.],.],.]],.]]]
=> [4,5,6,3,7,2,1] => [2,7,6,4,5,3,1] => [4,2,5,6,7,1,3] => ? = 2
[.,[.,[[[.,.],[.,[.,.]]],.]]]
=> [3,6,5,4,7,2,1] => [2,7,3,5,6,4,1] => [3,2,5,7,6,1,4] => ? = 2
[.,[.,[[[.,.],[[.,.],.]],.]]]
=> [3,5,6,4,7,2,1] => [2,7,3,6,5,4,1] => [3,2,5,6,7,1,4] => ? = 2
[.,[.,[[[.,[.,.]],[.,.]],.]]]
=> [4,3,6,5,7,2,1] => [2,7,4,3,6,5,1] => [4,2,6,1,7,3,5] => ? = 3
[.,[.,[[[[.,.],.],[.,.]],.]]]
=> [3,4,6,5,7,2,1] => [2,7,3,4,6,5,1] => [3,2,4,6,7,1,5] => ? = 2
[.,[.,[[[.,[.,[.,.]]],.],.]]]
=> [5,4,3,6,7,2,1] => [2,7,4,5,3,6,1] => [4,2,6,5,1,7,3] => ? = 3
[.,[.,[[[.,[[.,.],.]],.],.]]]
=> [4,5,3,6,7,2,1] => [2,7,5,4,3,6,1] => [4,2,5,6,1,7,3] => ? = 3
[.,[.,[[[[.,.],[.,.]],.],.]]]
=> [3,5,4,6,7,2,1] => [2,7,3,5,4,6,1] => [3,2,5,6,1,7,4] => ? = 3
[.,[.,[[[[.,[.,.]],.],.],.]]]
=> [4,3,5,6,7,2,1] => [2,7,4,3,5,6,1] => [4,2,5,1,6,7,3] => ? = 3
[.,[.,[[[[[.,.],.],.],.],.]]]
=> [3,4,5,6,7,2,1] => [2,7,3,4,5,6,1] => [3,2,4,5,6,7,1] => ? = 2
[.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [2,7,6,5,4,3,1] => [3,2,4,5,6,7,1] => [2,7,3,4,5,6,1] => ? = 1
[.,[[.,.],[.,[.,[[.,.],.]]]]]
=> [2,6,7,5,4,3,1] => [3,2,4,5,7,6,1] => [2,6,3,4,5,7,1] => ? = 2
[.,[[.,.],[.,[[.,.],[.,.]]]]]
=> [2,5,7,6,4,3,1] => [3,2,4,6,5,7,1] => [2,5,3,4,7,6,1] => ? = 2
[.,[[.,.],[.,[[.,[.,.]],.]]]]
=> [2,6,5,7,4,3,1] => [3,2,4,7,6,5,1] => [2,6,3,4,7,1,5] => ? = 2
[.,[[.,.],[.,[[[.,.],.],.]]]]
=> [2,5,6,7,4,3,1] => [3,2,4,7,5,6,1] => [2,5,3,4,6,7,1] => ? = 2
[.,[[.,.],[[.,.],[.,[.,.]]]]]
=> [2,4,7,6,5,3,1] => [3,2,5,4,6,7,1] => [2,4,3,7,5,6,1] => ? = 2
[.,[[.,.],[[.,.],[[.,.],.]]]]
=> [2,4,6,7,5,3,1] => [3,2,5,4,7,6,1] => [2,4,3,6,5,7,1] => ? = 3
[.,[[.,.],[[.,[.,.]],[.,.]]]]
=> [2,5,4,7,6,3,1] => [3,2,6,5,4,7,1] => [2,5,3,7,1,6,4] => ? = 2
Description
The number of descents of type 2 in a permutation.
A position $i\in[1,n-1]$ is a descent of type 2 of a permutation $\pi$ of $n$ letters, if it is a descent and if $\pi(j) < \pi(i)$ for all $j < i$.
Matching statistic: St000354
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000354: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 57%
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000354: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 57%
Values
[.,[.,.]]
=> [1,1,0,0]
=> [1,0,1,0]
=> [2,1] => 1
[[.,.],.]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,2] => 0
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [2,3,1] => 1
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,3,2] => 1
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [3,1,2] => 1
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [2,1,3] => 1
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 1
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 1
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 1
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 1
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 1
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 2
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 1
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => 1
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 1
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => 2
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => 2
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => 2
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => 2
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => 1
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 2
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => 1
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 1
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => 1
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => 1
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => 1
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => 2
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 1
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => 1
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => 2
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => 1
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => 1
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => 2
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => 2
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => 1
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => 2
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => 2
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => 2
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => 2
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => 1
[[.,[.,[.,[.,.]]]],.]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => 1
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,7,1] => ? = 1
[.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,3,4,5,1,7,6] => ? = 2
[.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,4,1,6,7,5] => ? = 2
[.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,3,4,6,1,7,5] => ? = 2
[.,[.,[.,[.,[[[.,.],.],.]]]]]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,3,4,1,5,7,6] => ? = 2
[.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,1,5,6,7,4] => ? = 2
[.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,3,1,5,4,7,6] => ? = 3
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,3,5,1,6,7,4] => ? = 2
[.,[.,[.,[[[.,.],.],[.,.]]]]]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,3,1,4,6,7,5] => ? = 2
[.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,3,5,6,1,7,4] => ? = 2
[.,[.,[.,[[.,[[.,.],.]],.]]]]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [2,3,1,6,4,7,5] => ? = 2
[.,[.,[.,[[[.,.],[.,.]],.]]]]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [2,3,6,1,4,7,5] => ? = 2
[.,[.,[.,[[[.,[.,.]],.],.]]]]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,3,5,1,4,7,6] => ? = 3
[.,[.,[.,[[[[.,.],.],.],.]]]]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,3,1,4,5,7,6] => ? = 2
[.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,4,5,6,7,3] => ? = 2
[.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3,7,6] => ? = 3
[.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,6,7,5] => ? = 3
[.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,1,4,6,3,7,5] => ? = 3
[.,[.,[[.,.],[[[.,.],.],.]]]]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,4,3,5,7,6] => ? = 3
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [2,4,1,5,6,7,3] => ? = 2
[.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [2,4,1,5,3,7,6] => ? = 3
[.,[.,[[[.,.],.],[.,[.,.]]]]]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,1,3,5,6,7,4] => ? = 2
[.,[.,[[[.,.],.],[[.,.],.]]]]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => ? = 3
[.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [2,4,5,1,6,7,3] => ? = 2
[.,[.,[[.,[[.,.],.]],[.,.]]]]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [2,1,5,3,6,7,4] => ? = 2
[.,[.,[[[.,.],[.,.]],[.,.]]]]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [2,5,1,3,6,7,4] => ? = 2
[.,[.,[[[.,[.,.]],.],[.,.]]]]
=> [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [2,4,1,3,6,7,5] => ? = 3
[.,[.,[[[[.,.],.],.],[.,.]]]]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,1,3,4,6,7,5] => ? = 2
[.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [2,4,5,6,1,7,3] => ? = 2
[.,[.,[[.,[.,[[.,.],.]]],.]]]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [2,4,1,6,3,7,5] => ? = 3
[.,[.,[[.,[[.,.],[.,.]]],.]]]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [2,1,5,6,3,7,4] => ? = 2
[.,[.,[[.,[[.,[.,.]],.]],.]]]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [2,5,1,6,3,7,4] => ? = 2
[.,[.,[[.,[[[.,.],.],.]],.]]]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,1,3,6,4,7,5] => ? = 2
[.,[.,[[[.,.],[.,[.,.]]],.]]]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [2,5,6,1,3,7,4] => ? = 2
[.,[.,[[[.,.],[[.,.],.]],.]]]
=> [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,1,6,3,4,7,5] => ? = 2
[.,[.,[[[.,[.,.]],[.,.]],.]]]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [2,4,6,1,3,7,5] => ? = 3
[.,[.,[[[[.,.],.],[.,.]],.]]]
=> [1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [2,6,1,3,4,7,5] => ? = 2
[.,[.,[[[.,[.,[.,.]]],.],.]]]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [2,4,5,1,3,7,6] => ? = 3
[.,[.,[[[.,[[.,.],.]],.],.]]]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [2,1,5,3,4,7,6] => ? = 3
[.,[.,[[[[.,.],[.,.]],.],.]]]
=> [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [2,5,1,3,4,7,6] => ? = 3
[.,[.,[[[[.,[.,.]],.],.],.]]]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,4,1,3,5,7,6] => ? = 3
[.,[.,[[[[[.,.],.],.],.],.]]]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,1,3,4,5,7,6] => ? = 2
[.,[[.,[.,.]],[.,[.,[.,.]]]]]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [3,1,4,5,6,7,2] => ? = 1
[.,[[.,[.,.]],[.,[[.,.],.]]]]
=> [1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [3,1,4,5,2,7,6] => ? = 2
[.,[[.,[.,.]],[[.,.],[.,.]]]]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [3,1,4,2,6,7,5] => ? = 2
[.,[[.,[.,.]],[[.,[.,.]],.]]]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [3,1,4,6,2,7,5] => ? = 2
[.,[[.,[.,.]],[[[.,.],.],.]]]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> [3,1,4,2,5,7,6] => ? = 2
[.,[[.,[.,[.,.]]],[.,[.,.]]]]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [3,4,1,5,6,7,2] => ? = 1
[.,[[.,[.,[.,.]]],[[.,.],.]]]
=> [1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [3,4,1,5,2,7,6] => ? = 2
[.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [4,1,2,5,6,7,3] => ? = 1
Description
The number of recoils of a permutation.
A '''recoil''', or '''inverse descent''' of a permutation $\pi$ is a value $i$ such that $i+1$ appears to the left of $i$ in $\pi_1,\pi_2,\dots,\pi_n$.
In other words, this is the number of descents of the inverse permutation. It can be also be described as the number of occurrences of the mesh pattern $([2,1], {(0,1),(1,1),(2,1)})$, i.e., the middle row is shaded.
Matching statistic: St001729
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St001729: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 57%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St001729: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 57%
Values
[.,[.,.]]
=> [2,1] => [2,1] => [2,1] => 1
[[.,.],.]
=> [1,2] => [1,2] => [1,2] => 0
[.,[.,[.,.]]]
=> [3,2,1] => [3,1,2] => [3,1,2] => 1
[.,[[.,.],.]]
=> [2,3,1] => [3,2,1] => [2,3,1] => 1
[[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => [1,3,2] => 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,1,2,3] => [4,1,2,3] => 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [4,1,3,2] => [3,1,4,2] => 2
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [4,2,1,3] => [2,4,1,3] => 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [4,3,2,1] => [3,4,1,2] => 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [4,2,3,1] => [2,3,4,1] => 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,2,3] => [1,4,2,3] => 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,4,3,2] => [1,3,4,2] => 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,1,2,4] => [3,1,2,4] => 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,2,1,4] => [2,3,1,4] => 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,1,2,3,4] => [5,1,2,3,4] => 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [5,1,2,4,3] => [4,1,2,5,3] => 2
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [5,1,3,2,4] => [3,1,5,2,4] => 2
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [5,1,4,3,2] => [4,1,5,2,3] => 2
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [5,1,3,4,2] => [3,1,4,5,2] => 2
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [5,2,1,3,4] => [2,5,1,3,4] => 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [5,2,1,4,3] => [2,4,1,5,3] => 2
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [5,3,2,1,4] => [3,5,1,2,4] => 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [5,2,3,1,4] => [2,3,5,1,4] => 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [5,4,2,3,1] => [4,5,1,2,3] => 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [5,4,3,2,1] => [3,4,5,1,2] => 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [5,2,4,3,1] => [2,4,5,1,3] => 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [5,3,2,4,1] => [3,4,1,5,2] => 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [5,2,3,4,1] => [2,3,4,5,1] => 1
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,5,2,3,4] => [1,5,2,3,4] => 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,5,2,4,3] => [1,4,2,5,3] => 2
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,5,3,2,4] => [1,3,5,2,4] => 1
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,5,4,3,2] => [1,4,5,2,3] => 1
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,5,3,4,2] => [1,3,4,5,2] => 1
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,5,3,4] => [2,1,5,3,4] => 2
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,5,4,3] => [2,1,4,5,3] => 2
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,5,3,4] => [1,2,5,3,4] => 1
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,5,4,3] => [1,2,4,5,3] => 1
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => 2
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [3,2,1,5,4] => [2,3,1,5,4] => 2
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 2
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,1,2,3,5] => [4,1,2,3,5] => 1
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [7,6,5,4,3,2,1] => [7,1,2,3,4,5,6] => [7,1,2,3,4,5,6] => ? = 1
[.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [6,7,5,4,3,2,1] => [7,1,2,3,4,6,5] => [6,1,2,3,4,7,5] => ? = 2
[.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [5,7,6,4,3,2,1] => [7,1,2,3,5,4,6] => [5,1,2,3,7,4,6] => ? = 2
[.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [6,5,7,4,3,2,1] => [7,1,2,3,6,5,4] => [6,1,2,3,7,4,5] => ? = 2
[.,[.,[.,[.,[[[.,.],.],.]]]]]
=> [5,6,7,4,3,2,1] => [7,1,2,3,5,6,4] => [5,1,2,3,6,7,4] => ? = 2
[.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [4,7,6,5,3,2,1] => [7,1,2,4,3,5,6] => [4,1,2,7,3,5,6] => ? = 2
[.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [4,6,7,5,3,2,1] => [7,1,2,4,3,6,5] => [4,1,2,6,3,7,5] => ? = 3
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [5,4,7,6,3,2,1] => [7,1,2,5,4,3,6] => [5,1,2,7,3,4,6] => ? = 2
[.,[.,[.,[[[.,.],.],[.,.]]]]]
=> [4,5,7,6,3,2,1] => [7,1,2,4,5,3,6] => [4,1,2,5,7,3,6] => ? = 2
[.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [6,5,4,7,3,2,1] => [7,1,2,6,4,5,3] => [6,1,2,7,3,4,5] => ? = 2
[.,[.,[.,[[.,[[.,.],.]],.]]]]
=> [5,6,4,7,3,2,1] => [7,1,2,6,5,4,3] => [5,1,2,6,7,3,4] => ? = 2
[.,[.,[.,[[[.,.],[.,.]],.]]]]
=> [4,6,5,7,3,2,1] => [7,1,2,4,6,5,3] => [4,1,2,6,7,3,5] => ? = 2
[.,[.,[.,[[[.,[.,.]],.],.]]]]
=> [5,4,6,7,3,2,1] => [7,1,2,5,4,6,3] => [5,1,2,6,3,7,4] => ? = 3
[.,[.,[.,[[[[.,.],.],.],.]]]]
=> [4,5,6,7,3,2,1] => [7,1,2,4,5,6,3] => [4,1,2,5,6,7,3] => ? = 2
[.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [3,7,6,5,4,2,1] => [7,1,3,2,4,5,6] => [3,1,7,2,4,5,6] => ? = 2
[.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [3,6,7,5,4,2,1] => [7,1,3,2,4,6,5] => [3,1,6,2,4,7,5] => ? = 3
[.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [3,5,7,6,4,2,1] => [7,1,3,2,5,4,6] => [3,1,5,2,7,4,6] => ? = 3
[.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [3,6,5,7,4,2,1] => [7,1,3,2,6,5,4] => [3,1,6,2,7,4,5] => ? = 3
[.,[.,[[.,.],[[[.,.],.],.]]]]
=> [3,5,6,7,4,2,1] => [7,1,3,2,5,6,4] => [3,1,5,2,6,7,4] => ? = 3
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [4,3,7,6,5,2,1] => [7,1,4,3,2,5,6] => [4,1,7,2,3,5,6] => ? = 2
[.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [4,3,6,7,5,2,1] => [7,1,4,3,2,6,5] => [4,1,6,2,3,7,5] => ? = 3
[.,[.,[[[.,.],.],[.,[.,.]]]]]
=> [3,4,7,6,5,2,1] => [7,1,3,4,2,5,6] => [3,1,4,7,2,5,6] => ? = 2
[.,[.,[[[.,.],.],[[.,.],.]]]]
=> [3,4,6,7,5,2,1] => [7,1,3,4,2,6,5] => [3,1,4,6,2,7,5] => ? = 3
[.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [5,4,3,7,6,2,1] => [7,1,5,3,4,2,6] => [5,1,7,2,3,4,6] => ? = 2
[.,[.,[[.,[[.,.],.]],[.,.]]]]
=> [4,5,3,7,6,2,1] => [7,1,5,4,3,2,6] => [4,1,5,7,2,3,6] => ? = 2
[.,[.,[[[.,.],[.,.]],[.,.]]]]
=> [3,5,4,7,6,2,1] => [7,1,3,5,4,2,6] => [3,1,5,7,2,4,6] => ? = 2
[.,[.,[[[.,[.,.]],.],[.,.]]]]
=> [4,3,5,7,6,2,1] => [7,1,4,3,5,2,6] => [4,1,5,2,7,3,6] => ? = 3
[.,[.,[[[[.,.],.],.],[.,.]]]]
=> [3,4,5,7,6,2,1] => [7,1,3,4,5,2,6] => [3,1,4,5,7,2,6] => ? = 2
[.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> [6,5,4,3,7,2,1] => [7,1,6,3,4,5,2] => [6,1,7,2,3,4,5] => ? = 2
[.,[.,[[.,[.,[[.,.],.]]],.]]]
=> [5,6,4,3,7,2,1] => [7,1,6,3,5,4,2] => [5,1,6,2,7,3,4] => ? = 3
[.,[.,[[.,[[.,.],[.,.]]],.]]]
=> [4,6,5,3,7,2,1] => [7,1,6,4,3,5,2] => [4,1,6,7,2,3,5] => ? = 2
[.,[.,[[.,[[.,[.,.]],.]],.]]]
=> [5,4,6,3,7,2,1] => [7,1,6,5,4,3,2] => [5,1,6,7,2,3,4] => ? = 2
[.,[.,[[.,[[[.,.],.],.]],.]]]
=> [4,5,6,3,7,2,1] => [7,1,6,4,5,3,2] => [4,1,5,6,7,2,3] => ? = 2
[.,[.,[[[.,.],[.,[.,.]]],.]]]
=> [3,6,5,4,7,2,1] => [7,1,3,6,4,5,2] => [3,1,6,7,2,4,5] => ? = 2
[.,[.,[[[.,.],[[.,.],.]],.]]]
=> [3,5,6,4,7,2,1] => [7,1,3,6,5,4,2] => [3,1,5,6,7,2,4] => ? = 2
[.,[.,[[[.,[.,.]],[.,.]],.]]]
=> [4,3,6,5,7,2,1] => [7,1,4,3,6,5,2] => [4,1,6,2,7,3,5] => ? = 3
[.,[.,[[[[.,.],.],[.,.]],.]]]
=> [3,4,6,5,7,2,1] => [7,1,3,4,6,5,2] => [3,1,4,6,7,2,5] => ? = 2
[.,[.,[[[.,[.,[.,.]]],.],.]]]
=> [5,4,3,6,7,2,1] => [7,1,5,3,4,6,2] => [5,1,6,2,3,7,4] => ? = 3
[.,[.,[[[.,[[.,.],.]],.],.]]]
=> [4,5,3,6,7,2,1] => [7,1,5,4,3,6,2] => [4,1,5,6,2,7,3] => ? = 3
[.,[.,[[[[.,.],[.,.]],.],.]]]
=> [3,5,4,6,7,2,1] => [7,1,3,5,4,6,2] => [3,1,5,6,2,7,4] => ? = 3
[.,[.,[[[[.,[.,.]],.],.],.]]]
=> [4,3,5,6,7,2,1] => [7,1,4,3,5,6,2] => [4,1,5,2,6,7,3] => ? = 3
[.,[.,[[[[[.,.],.],.],.],.]]]
=> [3,4,5,6,7,2,1] => [7,1,3,4,5,6,2] => [3,1,4,5,6,7,2] => ? = 2
[.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [2,7,6,5,4,3,1] => [7,2,1,3,4,5,6] => [2,7,1,3,4,5,6] => ? = 1
[.,[[.,.],[.,[.,[[.,.],.]]]]]
=> [2,6,7,5,4,3,1] => [7,2,1,3,4,6,5] => [2,6,1,3,4,7,5] => ? = 2
[.,[[.,.],[.,[[.,.],[.,.]]]]]
=> [2,5,7,6,4,3,1] => [7,2,1,3,5,4,6] => [2,5,1,3,7,4,6] => ? = 2
[.,[[.,.],[.,[[.,[.,.]],.]]]]
=> [2,6,5,7,4,3,1] => [7,2,1,3,6,5,4] => [2,6,1,3,7,4,5] => ? = 2
[.,[[.,.],[.,[[[.,.],.],.]]]]
=> [2,5,6,7,4,3,1] => [7,2,1,3,5,6,4] => [2,5,1,3,6,7,4] => ? = 2
[.,[[.,.],[[.,.],[.,[.,.]]]]]
=> [2,4,7,6,5,3,1] => [7,2,1,4,3,5,6] => [2,4,1,7,3,5,6] => ? = 2
[.,[[.,.],[[.,.],[[.,.],.]]]]
=> [2,4,6,7,5,3,1] => [7,2,1,4,3,6,5] => [2,4,1,6,3,7,5] => ? = 3
[.,[[.,.],[[.,[.,.]],[.,.]]]]
=> [2,5,4,7,6,3,1] => [7,2,1,5,4,3,6] => [2,5,1,7,3,4,6] => ? = 2
Description
The number of visible descents of a permutation.
A visible descent of a permutation $\pi$ is a position $i$ such that $\pi(i+1) \leq \min(i, \pi(i))$.
Matching statistic: St001928
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St001928: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 57%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St001928: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 57%
Values
[.,[.,.]]
=> [2,1] => [2,1] => [2,1] => 1
[[.,.],.]
=> [1,2] => [1,2] => [1,2] => 0
[.,[.,[.,.]]]
=> [3,2,1] => [3,1,2] => [3,1,2] => 1
[.,[[.,.],.]]
=> [2,3,1] => [3,2,1] => [2,3,1] => 1
[[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => [1,3,2] => 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,1,2,3] => [4,1,2,3] => 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [4,1,3,2] => [3,1,4,2] => 2
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [4,2,1,3] => [2,4,1,3] => 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [4,3,2,1] => [3,4,1,2] => 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [4,2,3,1] => [2,3,4,1] => 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,2,3] => [1,4,2,3] => 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,4,3,2] => [1,3,4,2] => 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,1,2,4] => [3,1,2,4] => 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,2,1,4] => [2,3,1,4] => 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,1,2,3,4] => [5,1,2,3,4] => 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [5,1,2,4,3] => [4,1,2,5,3] => 2
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [5,1,3,2,4] => [3,1,5,2,4] => 2
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [5,1,4,3,2] => [4,1,5,2,3] => 2
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [5,1,3,4,2] => [3,1,4,5,2] => 2
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [5,2,1,3,4] => [2,5,1,3,4] => 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [5,2,1,4,3] => [2,4,1,5,3] => 2
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [5,3,2,1,4] => [3,5,1,2,4] => 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [5,2,3,1,4] => [2,3,5,1,4] => 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [5,4,2,3,1] => [4,5,1,2,3] => 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [5,4,3,2,1] => [3,4,5,1,2] => 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [5,2,4,3,1] => [2,4,5,1,3] => 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [5,3,2,4,1] => [3,4,1,5,2] => 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [5,2,3,4,1] => [2,3,4,5,1] => 1
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,5,2,3,4] => [1,5,2,3,4] => 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,5,2,4,3] => [1,4,2,5,3] => 2
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,5,3,2,4] => [1,3,5,2,4] => 1
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,5,4,3,2] => [1,4,5,2,3] => 1
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,5,3,4,2] => [1,3,4,5,2] => 1
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,5,3,4] => [2,1,5,3,4] => 2
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,5,4,3] => [2,1,4,5,3] => 2
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,5,3,4] => [1,2,5,3,4] => 1
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,5,4,3] => [1,2,4,5,3] => 1
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => 2
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [3,2,1,5,4] => [2,3,1,5,4] => 2
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 2
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,1,2,3,5] => [4,1,2,3,5] => 1
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [7,6,5,4,3,2,1] => [7,1,2,3,4,5,6] => [7,1,2,3,4,5,6] => ? = 1
[.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [6,7,5,4,3,2,1] => [7,1,2,3,4,6,5] => [6,1,2,3,4,7,5] => ? = 2
[.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [5,7,6,4,3,2,1] => [7,1,2,3,5,4,6] => [5,1,2,3,7,4,6] => ? = 2
[.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [6,5,7,4,3,2,1] => [7,1,2,3,6,5,4] => [6,1,2,3,7,4,5] => ? = 2
[.,[.,[.,[.,[[[.,.],.],.]]]]]
=> [5,6,7,4,3,2,1] => [7,1,2,3,5,6,4] => [5,1,2,3,6,7,4] => ? = 2
[.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [4,7,6,5,3,2,1] => [7,1,2,4,3,5,6] => [4,1,2,7,3,5,6] => ? = 2
[.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [4,6,7,5,3,2,1] => [7,1,2,4,3,6,5] => [4,1,2,6,3,7,5] => ? = 3
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [5,4,7,6,3,2,1] => [7,1,2,5,4,3,6] => [5,1,2,7,3,4,6] => ? = 2
[.,[.,[.,[[[.,.],.],[.,.]]]]]
=> [4,5,7,6,3,2,1] => [7,1,2,4,5,3,6] => [4,1,2,5,7,3,6] => ? = 2
[.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [6,5,4,7,3,2,1] => [7,1,2,6,4,5,3] => [6,1,2,7,3,4,5] => ? = 2
[.,[.,[.,[[.,[[.,.],.]],.]]]]
=> [5,6,4,7,3,2,1] => [7,1,2,6,5,4,3] => [5,1,2,6,7,3,4] => ? = 2
[.,[.,[.,[[[.,.],[.,.]],.]]]]
=> [4,6,5,7,3,2,1] => [7,1,2,4,6,5,3] => [4,1,2,6,7,3,5] => ? = 2
[.,[.,[.,[[[.,[.,.]],.],.]]]]
=> [5,4,6,7,3,2,1] => [7,1,2,5,4,6,3] => [5,1,2,6,3,7,4] => ? = 3
[.,[.,[.,[[[[.,.],.],.],.]]]]
=> [4,5,6,7,3,2,1] => [7,1,2,4,5,6,3] => [4,1,2,5,6,7,3] => ? = 2
[.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [3,7,6,5,4,2,1] => [7,1,3,2,4,5,6] => [3,1,7,2,4,5,6] => ? = 2
[.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [3,6,7,5,4,2,1] => [7,1,3,2,4,6,5] => [3,1,6,2,4,7,5] => ? = 3
[.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [3,5,7,6,4,2,1] => [7,1,3,2,5,4,6] => [3,1,5,2,7,4,6] => ? = 3
[.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [3,6,5,7,4,2,1] => [7,1,3,2,6,5,4] => [3,1,6,2,7,4,5] => ? = 3
[.,[.,[[.,.],[[[.,.],.],.]]]]
=> [3,5,6,7,4,2,1] => [7,1,3,2,5,6,4] => [3,1,5,2,6,7,4] => ? = 3
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [4,3,7,6,5,2,1] => [7,1,4,3,2,5,6] => [4,1,7,2,3,5,6] => ? = 2
[.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [4,3,6,7,5,2,1] => [7,1,4,3,2,6,5] => [4,1,6,2,3,7,5] => ? = 3
[.,[.,[[[.,.],.],[.,[.,.]]]]]
=> [3,4,7,6,5,2,1] => [7,1,3,4,2,5,6] => [3,1,4,7,2,5,6] => ? = 2
[.,[.,[[[.,.],.],[[.,.],.]]]]
=> [3,4,6,7,5,2,1] => [7,1,3,4,2,6,5] => [3,1,4,6,2,7,5] => ? = 3
[.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [5,4,3,7,6,2,1] => [7,1,5,3,4,2,6] => [5,1,7,2,3,4,6] => ? = 2
[.,[.,[[.,[[.,.],.]],[.,.]]]]
=> [4,5,3,7,6,2,1] => [7,1,5,4,3,2,6] => [4,1,5,7,2,3,6] => ? = 2
[.,[.,[[[.,.],[.,.]],[.,.]]]]
=> [3,5,4,7,6,2,1] => [7,1,3,5,4,2,6] => [3,1,5,7,2,4,6] => ? = 2
[.,[.,[[[.,[.,.]],.],[.,.]]]]
=> [4,3,5,7,6,2,1] => [7,1,4,3,5,2,6] => [4,1,5,2,7,3,6] => ? = 3
[.,[.,[[[[.,.],.],.],[.,.]]]]
=> [3,4,5,7,6,2,1] => [7,1,3,4,5,2,6] => [3,1,4,5,7,2,6] => ? = 2
[.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> [6,5,4,3,7,2,1] => [7,1,6,3,4,5,2] => [6,1,7,2,3,4,5] => ? = 2
[.,[.,[[.,[.,[[.,.],.]]],.]]]
=> [5,6,4,3,7,2,1] => [7,1,6,3,5,4,2] => [5,1,6,2,7,3,4] => ? = 3
[.,[.,[[.,[[.,.],[.,.]]],.]]]
=> [4,6,5,3,7,2,1] => [7,1,6,4,3,5,2] => [4,1,6,7,2,3,5] => ? = 2
[.,[.,[[.,[[.,[.,.]],.]],.]]]
=> [5,4,6,3,7,2,1] => [7,1,6,5,4,3,2] => [5,1,6,7,2,3,4] => ? = 2
[.,[.,[[.,[[[.,.],.],.]],.]]]
=> [4,5,6,3,7,2,1] => [7,1,6,4,5,3,2] => [4,1,5,6,7,2,3] => ? = 2
[.,[.,[[[.,.],[.,[.,.]]],.]]]
=> [3,6,5,4,7,2,1] => [7,1,3,6,4,5,2] => [3,1,6,7,2,4,5] => ? = 2
[.,[.,[[[.,.],[[.,.],.]],.]]]
=> [3,5,6,4,7,2,1] => [7,1,3,6,5,4,2] => [3,1,5,6,7,2,4] => ? = 2
[.,[.,[[[.,[.,.]],[.,.]],.]]]
=> [4,3,6,5,7,2,1] => [7,1,4,3,6,5,2] => [4,1,6,2,7,3,5] => ? = 3
[.,[.,[[[[.,.],.],[.,.]],.]]]
=> [3,4,6,5,7,2,1] => [7,1,3,4,6,5,2] => [3,1,4,6,7,2,5] => ? = 2
[.,[.,[[[.,[.,[.,.]]],.],.]]]
=> [5,4,3,6,7,2,1] => [7,1,5,3,4,6,2] => [5,1,6,2,3,7,4] => ? = 3
[.,[.,[[[.,[[.,.],.]],.],.]]]
=> [4,5,3,6,7,2,1] => [7,1,5,4,3,6,2] => [4,1,5,6,2,7,3] => ? = 3
[.,[.,[[[[.,.],[.,.]],.],.]]]
=> [3,5,4,6,7,2,1] => [7,1,3,5,4,6,2] => [3,1,5,6,2,7,4] => ? = 3
[.,[.,[[[[.,[.,.]],.],.],.]]]
=> [4,3,5,6,7,2,1] => [7,1,4,3,5,6,2] => [4,1,5,2,6,7,3] => ? = 3
[.,[.,[[[[[.,.],.],.],.],.]]]
=> [3,4,5,6,7,2,1] => [7,1,3,4,5,6,2] => [3,1,4,5,6,7,2] => ? = 2
[.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [2,7,6,5,4,3,1] => [7,2,1,3,4,5,6] => [2,7,1,3,4,5,6] => ? = 1
[.,[[.,.],[.,[.,[[.,.],.]]]]]
=> [2,6,7,5,4,3,1] => [7,2,1,3,4,6,5] => [2,6,1,3,4,7,5] => ? = 2
[.,[[.,.],[.,[[.,.],[.,.]]]]]
=> [2,5,7,6,4,3,1] => [7,2,1,3,5,4,6] => [2,5,1,3,7,4,6] => ? = 2
[.,[[.,.],[.,[[.,[.,.]],.]]]]
=> [2,6,5,7,4,3,1] => [7,2,1,3,6,5,4] => [2,6,1,3,7,4,5] => ? = 2
[.,[[.,.],[.,[[[.,.],.],.]]]]
=> [2,5,6,7,4,3,1] => [7,2,1,3,5,6,4] => [2,5,1,3,6,7,4] => ? = 2
[.,[[.,.],[[.,.],[.,[.,.]]]]]
=> [2,4,7,6,5,3,1] => [7,2,1,4,3,5,6] => [2,4,1,7,3,5,6] => ? = 2
[.,[[.,.],[[.,.],[[.,.],.]]]]
=> [2,4,6,7,5,3,1] => [7,2,1,4,3,6,5] => [2,4,1,6,3,7,5] => ? = 3
[.,[[.,.],[[.,[.,.]],[.,.]]]]
=> [2,5,4,7,6,3,1] => [7,2,1,5,4,3,6] => [2,5,1,7,3,4,6] => ? = 2
Description
The number of non-overlapping descents in a permutation.
In other words, any maximal descending subsequence $\pi_i,\pi_{i+1},\dots,\pi_k$ contributes $\lfloor\frac{k-i+1}{2}\rfloor$ to the total count.
Matching statistic: St000470
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 57%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 57%
Values
[.,[.,.]]
=> [2,1] => [2,1] => [2,1] => 2 = 1 + 1
[[.,.],.]
=> [1,2] => [1,2] => [1,2] => 1 = 0 + 1
[.,[.,[.,.]]]
=> [3,2,1] => [3,1,2] => [3,1,2] => 2 = 1 + 1
[.,[[.,.],.]]
=> [2,3,1] => [3,2,1] => [2,3,1] => 2 = 1 + 1
[[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => [2,1,3] => 2 = 1 + 1
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,1,2,3] => [4,1,2,3] => 2 = 1 + 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [4,1,3,2] => [3,1,4,2] => 3 = 2 + 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [4,2,1,3] => [2,4,1,3] => 2 = 1 + 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [4,3,2,1] => [3,4,1,2] => 2 = 1 + 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [4,2,3,1] => [2,3,4,1] => 2 = 1 + 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,2,3] => [1,4,2,3] => 2 = 1 + 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,4,3,2] => [1,3,4,2] => 2 = 1 + 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 3 = 2 + 1
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,1,2,4] => [3,1,2,4] => 2 = 1 + 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,2,1,4] => [2,3,1,4] => 2 = 1 + 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 2 = 1 + 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,1,2,3,4] => [5,1,2,3,4] => 2 = 1 + 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [5,1,2,4,3] => [4,1,2,5,3] => 3 = 2 + 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [5,1,3,2,4] => [3,1,5,2,4] => 3 = 2 + 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [5,1,4,3,2] => [4,1,5,2,3] => 3 = 2 + 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [5,1,3,4,2] => [3,1,4,5,2] => 3 = 2 + 1
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [5,2,1,3,4] => [2,5,1,3,4] => 2 = 1 + 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [5,2,1,4,3] => [2,4,1,5,3] => 3 = 2 + 1
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [5,3,2,1,4] => [3,5,1,2,4] => 2 = 1 + 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [5,2,3,1,4] => [2,3,5,1,4] => 2 = 1 + 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [5,4,2,3,1] => [4,5,1,2,3] => 2 = 1 + 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [5,4,3,2,1] => [3,4,5,1,2] => 2 = 1 + 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [5,2,4,3,1] => [2,4,5,1,3] => 2 = 1 + 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [5,3,2,4,1] => [3,4,1,5,2] => 3 = 2 + 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [5,2,3,4,1] => [2,3,4,5,1] => 2 = 1 + 1
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,5,2,3,4] => [1,5,2,3,4] => 2 = 1 + 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,5,2,4,3] => [1,4,2,5,3] => 3 = 2 + 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,5,3,2,4] => [1,3,5,2,4] => 2 = 1 + 1
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,5,4,3,2] => [1,4,5,2,3] => 2 = 1 + 1
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,5,3,4,2] => [1,3,4,5,2] => 2 = 1 + 1
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,5,3,4] => [2,1,5,3,4] => 3 = 2 + 1
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,5,4,3] => [2,1,4,5,3] => 3 = 2 + 1
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,5,3,4] => [1,2,5,3,4] => 2 = 1 + 1
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,5,4,3] => [1,2,4,5,3] => 2 = 1 + 1
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => 3 = 2 + 1
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [3,2,1,5,4] => [2,3,1,5,4] => 3 = 2 + 1
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 3 = 2 + 1
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 3 = 2 + 1
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 2 = 1 + 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,1,2,3,5] => [4,1,2,3,5] => 2 = 1 + 1
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [7,6,5,4,3,2,1] => [7,1,2,3,4,5,6] => [7,1,2,3,4,5,6] => ? = 1 + 1
[.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [6,7,5,4,3,2,1] => [7,1,2,3,4,6,5] => [6,1,2,3,4,7,5] => ? = 2 + 1
[.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [5,7,6,4,3,2,1] => [7,1,2,3,5,4,6] => [5,1,2,3,7,4,6] => ? = 2 + 1
[.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [6,5,7,4,3,2,1] => [7,1,2,3,6,5,4] => [6,1,2,3,7,4,5] => ? = 2 + 1
[.,[.,[.,[.,[[[.,.],.],.]]]]]
=> [5,6,7,4,3,2,1] => [7,1,2,3,5,6,4] => [5,1,2,3,6,7,4] => ? = 2 + 1
[.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [4,7,6,5,3,2,1] => [7,1,2,4,3,5,6] => [4,1,2,7,3,5,6] => ? = 2 + 1
[.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [4,6,7,5,3,2,1] => [7,1,2,4,3,6,5] => [4,1,2,6,3,7,5] => ? = 3 + 1
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [5,4,7,6,3,2,1] => [7,1,2,5,4,3,6] => [5,1,2,7,3,4,6] => ? = 2 + 1
[.,[.,[.,[[[.,.],.],[.,.]]]]]
=> [4,5,7,6,3,2,1] => [7,1,2,4,5,3,6] => [4,1,2,5,7,3,6] => ? = 2 + 1
[.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [6,5,4,7,3,2,1] => [7,1,2,6,4,5,3] => [6,1,2,7,3,4,5] => ? = 2 + 1
[.,[.,[.,[[.,[[.,.],.]],.]]]]
=> [5,6,4,7,3,2,1] => [7,1,2,6,5,4,3] => [5,1,2,6,7,3,4] => ? = 2 + 1
[.,[.,[.,[[[.,.],[.,.]],.]]]]
=> [4,6,5,7,3,2,1] => [7,1,2,4,6,5,3] => [4,1,2,6,7,3,5] => ? = 2 + 1
[.,[.,[.,[[[.,[.,.]],.],.]]]]
=> [5,4,6,7,3,2,1] => [7,1,2,5,4,6,3] => [5,1,2,6,3,7,4] => ? = 3 + 1
[.,[.,[.,[[[[.,.],.],.],.]]]]
=> [4,5,6,7,3,2,1] => [7,1,2,4,5,6,3] => [4,1,2,5,6,7,3] => ? = 2 + 1
[.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [3,7,6,5,4,2,1] => [7,1,3,2,4,5,6] => [3,1,7,2,4,5,6] => ? = 2 + 1
[.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [3,6,7,5,4,2,1] => [7,1,3,2,4,6,5] => [3,1,6,2,4,7,5] => ? = 3 + 1
[.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [3,5,7,6,4,2,1] => [7,1,3,2,5,4,6] => [3,1,5,2,7,4,6] => ? = 3 + 1
[.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [3,6,5,7,4,2,1] => [7,1,3,2,6,5,4] => [3,1,6,2,7,4,5] => ? = 3 + 1
[.,[.,[[.,.],[[[.,.],.],.]]]]
=> [3,5,6,7,4,2,1] => [7,1,3,2,5,6,4] => [3,1,5,2,6,7,4] => ? = 3 + 1
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [4,3,7,6,5,2,1] => [7,1,4,3,2,5,6] => [4,1,7,2,3,5,6] => ? = 2 + 1
[.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [4,3,6,7,5,2,1] => [7,1,4,3,2,6,5] => [4,1,6,2,3,7,5] => ? = 3 + 1
[.,[.,[[[.,.],.],[.,[.,.]]]]]
=> [3,4,7,6,5,2,1] => [7,1,3,4,2,5,6] => [3,1,4,7,2,5,6] => ? = 2 + 1
[.,[.,[[[.,.],.],[[.,.],.]]]]
=> [3,4,6,7,5,2,1] => [7,1,3,4,2,6,5] => [3,1,4,6,2,7,5] => ? = 3 + 1
[.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [5,4,3,7,6,2,1] => [7,1,5,3,4,2,6] => [5,1,7,2,3,4,6] => ? = 2 + 1
[.,[.,[[.,[[.,.],.]],[.,.]]]]
=> [4,5,3,7,6,2,1] => [7,1,5,4,3,2,6] => [4,1,5,7,2,3,6] => ? = 2 + 1
[.,[.,[[[.,.],[.,.]],[.,.]]]]
=> [3,5,4,7,6,2,1] => [7,1,3,5,4,2,6] => [3,1,5,7,2,4,6] => ? = 2 + 1
[.,[.,[[[.,[.,.]],.],[.,.]]]]
=> [4,3,5,7,6,2,1] => [7,1,4,3,5,2,6] => [4,1,5,2,7,3,6] => ? = 3 + 1
[.,[.,[[[[.,.],.],.],[.,.]]]]
=> [3,4,5,7,6,2,1] => [7,1,3,4,5,2,6] => [3,1,4,5,7,2,6] => ? = 2 + 1
[.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> [6,5,4,3,7,2,1] => [7,1,6,3,4,5,2] => [6,1,7,2,3,4,5] => ? = 2 + 1
[.,[.,[[.,[.,[[.,.],.]]],.]]]
=> [5,6,4,3,7,2,1] => [7,1,6,3,5,4,2] => [5,1,6,2,7,3,4] => ? = 3 + 1
[.,[.,[[.,[[.,.],[.,.]]],.]]]
=> [4,6,5,3,7,2,1] => [7,1,6,4,3,5,2] => [4,1,6,7,2,3,5] => ? = 2 + 1
[.,[.,[[.,[[.,[.,.]],.]],.]]]
=> [5,4,6,3,7,2,1] => [7,1,6,5,4,3,2] => [5,1,6,7,2,3,4] => ? = 2 + 1
[.,[.,[[.,[[[.,.],.],.]],.]]]
=> [4,5,6,3,7,2,1] => [7,1,6,4,5,3,2] => [4,1,5,6,7,2,3] => ? = 2 + 1
[.,[.,[[[.,.],[.,[.,.]]],.]]]
=> [3,6,5,4,7,2,1] => [7,1,3,6,4,5,2] => [3,1,6,7,2,4,5] => ? = 2 + 1
[.,[.,[[[.,.],[[.,.],.]],.]]]
=> [3,5,6,4,7,2,1] => [7,1,3,6,5,4,2] => [3,1,5,6,7,2,4] => ? = 2 + 1
[.,[.,[[[.,[.,.]],[.,.]],.]]]
=> [4,3,6,5,7,2,1] => [7,1,4,3,6,5,2] => [4,1,6,2,7,3,5] => ? = 3 + 1
[.,[.,[[[[.,.],.],[.,.]],.]]]
=> [3,4,6,5,7,2,1] => [7,1,3,4,6,5,2] => [3,1,4,6,7,2,5] => ? = 2 + 1
[.,[.,[[[.,[.,[.,.]]],.],.]]]
=> [5,4,3,6,7,2,1] => [7,1,5,3,4,6,2] => [5,1,6,2,3,7,4] => ? = 3 + 1
[.,[.,[[[.,[[.,.],.]],.],.]]]
=> [4,5,3,6,7,2,1] => [7,1,5,4,3,6,2] => [4,1,5,6,2,7,3] => ? = 3 + 1
[.,[.,[[[[.,.],[.,.]],.],.]]]
=> [3,5,4,6,7,2,1] => [7,1,3,5,4,6,2] => [3,1,5,6,2,7,4] => ? = 3 + 1
[.,[.,[[[[.,[.,.]],.],.],.]]]
=> [4,3,5,6,7,2,1] => [7,1,4,3,5,6,2] => [4,1,5,2,6,7,3] => ? = 3 + 1
[.,[.,[[[[[.,.],.],.],.],.]]]
=> [3,4,5,6,7,2,1] => [7,1,3,4,5,6,2] => [3,1,4,5,6,7,2] => ? = 2 + 1
[.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [2,7,6,5,4,3,1] => [7,2,1,3,4,5,6] => [2,7,1,3,4,5,6] => ? = 1 + 1
[.,[[.,.],[.,[.,[[.,.],.]]]]]
=> [2,6,7,5,4,3,1] => [7,2,1,3,4,6,5] => [2,6,1,3,4,7,5] => ? = 2 + 1
[.,[[.,.],[.,[[.,.],[.,.]]]]]
=> [2,5,7,6,4,3,1] => [7,2,1,3,5,4,6] => [2,5,1,3,7,4,6] => ? = 2 + 1
[.,[[.,.],[.,[[.,[.,.]],.]]]]
=> [2,6,5,7,4,3,1] => [7,2,1,3,6,5,4] => [2,6,1,3,7,4,5] => ? = 2 + 1
[.,[[.,.],[.,[[[.,.],.],.]]]]
=> [2,5,6,7,4,3,1] => [7,2,1,3,5,6,4] => [2,5,1,3,6,7,4] => ? = 2 + 1
[.,[[.,.],[[.,.],[.,[.,.]]]]]
=> [2,4,7,6,5,3,1] => [7,2,1,4,3,5,6] => [2,4,1,7,3,5,6] => ? = 2 + 1
[.,[[.,.],[[.,.],[[.,.],.]]]]
=> [2,4,6,7,5,3,1] => [7,2,1,4,3,6,5] => [2,4,1,6,3,7,5] => ? = 3 + 1
[.,[[.,.],[[.,[.,.]],[.,.]]]]
=> [2,5,4,7,6,3,1] => [7,2,1,5,4,3,6] => [2,5,1,7,3,4,6] => ? = 2 + 1
Description
The number of runs in a permutation.
A run in a permutation is an inclusion-wise maximal increasing substring, i.e., a contiguous subsequence.
This is the same as the number of descents plus 1.
Matching statistic: St000834
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00326: Permutations —weak order rowmotion⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 57%
Mp00326: Permutations —weak order rowmotion⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 57%
Values
[.,[.,.]]
=> [2,1] => [1,2] => [1,2] => 1
[[.,.],.]
=> [1,2] => [2,1] => [2,1] => 0
[.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => [1,2,3] => 1
[.,[[.,.],.]]
=> [2,3,1] => [2,1,3] => [2,1,3] => 1
[[.,.],[.,.]]
=> [1,3,2] => [2,3,1] => [3,1,2] => 1
[[.,[.,.]],.]
=> [2,1,3] => [3,1,2] => [2,3,1] => 1
[[[.,.],.],.]
=> [1,2,3] => [3,2,1] => [3,2,1] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,1,2,4] => [2,3,1,4] => 2
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [2,1,3,4] => [2,1,3,4] => 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [2,3,1,4] => [3,1,2,4] => 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [3,2,1,4] => [3,2,1,4] => 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [2,3,4,1] => [4,1,2,3] => 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [3,2,4,1] => [4,2,1,3] => 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [3,4,1,2] => [3,4,1,2] => 2
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [3,4,2,1] => [4,3,1,2] => 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [4,1,2,3] => [2,3,4,1] => 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [4,2,1,3] => [3,2,4,1] => 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [4,3,1,2] => [3,4,2,1] => 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,1,2,3,5] => [2,3,4,1,5] => 2
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [3,1,2,4,5] => [2,3,1,4,5] => 2
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [3,4,1,2,5] => [3,4,1,2,5] => 2
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [4,3,1,2,5] => [3,4,2,1,5] => 2
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [4,2,1,3,5] => [3,2,4,1,5] => 2
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [2,3,1,4,5] => [3,1,2,4,5] => 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [3,2,1,4,5] => [3,2,1,4,5] => 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [2,3,4,1,5] => [4,1,2,3,5] => 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,2,4,1,5] => [4,2,1,3,5] => 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [3,4,2,1,5] => [4,3,1,2,5] => 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [4,2,3,1,5] => [4,2,3,1,5] => 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [4,3,2,1,5] => [4,3,2,1,5] => 1
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [2,3,4,5,1] => [5,1,2,3,4] => 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [4,2,3,5,1] => [5,2,3,1,4] => 2
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [3,2,4,5,1] => [5,2,1,3,4] => 1
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [3,4,2,5,1] => [5,3,1,2,4] => 1
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [4,3,2,5,1] => [5,3,2,1,4] => 1
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [3,4,5,1,2] => [4,5,1,2,3] => 2
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [4,3,5,1,2] => [4,5,2,1,3] => 2
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [3,4,5,2,1] => [5,4,1,2,3] => 1
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [4,3,5,2,1] => [5,4,2,1,3] => 1
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [4,5,1,2,3] => [3,4,5,1,2] => 2
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [4,5,2,1,3] => [4,3,5,1,2] => 2
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [4,5,2,3,1] => [5,3,4,1,2] => 2
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [4,5,3,1,2] => [4,5,3,1,2] => 2
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [4,5,3,2,1] => [5,4,3,1,2] => 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [5,1,2,3,4] => [2,3,4,5,1] => 1
[.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [6,5,7,4,3,2,1] => [5,6,1,2,3,4,7] => [3,4,5,6,1,2,7] => ? = 2
[.,[.,[.,[.,[[[.,.],.],.]]]]]
=> [5,6,7,4,3,2,1] => [6,5,1,2,3,4,7] => [3,4,5,6,2,1,7] => ? = 2
[.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [4,6,7,5,3,2,1] => [6,4,1,2,3,5,7] => [3,4,5,2,6,1,7] => ? = 3
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [5,4,7,6,3,2,1] => [4,5,1,2,3,6,7] => [3,4,5,1,2,6,7] => ? = 2
[.,[.,[.,[[[.,.],.],[.,.]]]]]
=> [4,5,7,6,3,2,1] => [5,4,1,2,3,6,7] => [3,4,5,2,1,6,7] => ? = 2
[.,[.,[.,[[.,[[.,.],.]],.]]]]
=> [5,6,4,7,3,2,1] => [5,4,6,1,2,3,7] => [4,5,6,2,1,3,7] => ? = 2
[.,[.,[.,[[[.,.],[.,.]],.]]]]
=> [4,6,5,7,3,2,1] => [5,6,4,1,2,3,7] => [4,5,6,3,1,2,7] => ? = 2
[.,[.,[.,[[[.,[.,.]],.],.]]]]
=> [5,4,6,7,3,2,1] => [6,4,5,1,2,3,7] => [4,5,6,2,3,1,7] => ? = 3
[.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [3,6,7,5,4,2,1] => [6,3,1,2,4,5,7] => [3,4,2,5,6,1,7] => ? = 3
[.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [3,5,7,6,4,2,1] => [5,3,1,2,4,6,7] => [3,4,2,5,1,6,7] => ? = 3
[.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [3,6,5,7,4,2,1] => [5,6,3,1,2,4,7] => [4,5,3,6,1,2,7] => ? = 3
[.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [4,3,6,7,5,2,1] => [6,3,4,1,2,5,7] => [4,5,2,3,6,1,7] => ? = 3
[.,[.,[[[.,.],.],[.,[.,.]]]]]
=> [3,4,7,6,5,2,1] => [4,3,1,2,5,6,7] => [3,4,2,1,5,6,7] => ? = 2
[.,[.,[[[.,.],.],[[.,.],.]]]]
=> [3,4,6,7,5,2,1] => [6,4,3,1,2,5,7] => [4,5,3,2,6,1,7] => ? = 3
[.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [5,4,3,7,6,2,1] => [3,4,5,1,2,6,7] => [4,5,1,2,3,6,7] => ? = 2
[.,[.,[[.,[[.,.],.]],[.,.]]]]
=> [4,5,3,7,6,2,1] => [4,3,5,1,2,6,7] => [4,5,2,1,3,6,7] => ? = 2
[.,[.,[[[.,.],[.,.]],[.,.]]]]
=> [3,5,4,7,6,2,1] => [4,5,3,1,2,6,7] => [4,5,3,1,2,6,7] => ? = 2
[.,[.,[[[.,[.,.]],.],[.,.]]]]
=> [4,3,5,7,6,2,1] => [5,3,4,1,2,6,7] => [4,5,2,3,1,6,7] => ? = 3
[.,[.,[[[[.,.],.],.],[.,.]]]]
=> [3,4,5,7,6,2,1] => [5,4,3,1,2,6,7] => [4,5,3,2,1,6,7] => ? = 2
[.,[.,[[.,[.,[[.,.],.]]],.]]]
=> [5,6,4,3,7,2,1] => [5,3,4,6,1,2,7] => [5,6,2,3,1,4,7] => ? = 3
[.,[.,[[.,[[.,.],[.,.]]],.]]]
=> [4,6,5,3,7,2,1] => [4,3,5,6,1,2,7] => [5,6,2,1,3,4,7] => ? = 2
[.,[.,[[.,[[.,[.,.]],.]],.]]]
=> [5,4,6,3,7,2,1] => [4,5,3,6,1,2,7] => [5,6,3,1,2,4,7] => ? = 2
[.,[.,[[.,[[[.,.],.],.]],.]]]
=> [4,5,6,3,7,2,1] => [5,4,3,6,1,2,7] => [5,6,3,2,1,4,7] => ? = 2
[.,[.,[[[.,.],[.,[.,.]]],.]]]
=> [3,6,5,4,7,2,1] => [4,5,6,3,1,2,7] => [5,6,4,1,2,3,7] => ? = 2
[.,[.,[[[.,.],[[.,.],.]],.]]]
=> [3,5,6,4,7,2,1] => [5,4,6,3,1,2,7] => [5,6,4,2,1,3,7] => ? = 2
[.,[.,[[[[.,.],.],[.,.]],.]]]
=> [3,4,6,5,7,2,1] => [5,6,4,3,1,2,7] => [5,6,4,3,1,2,7] => ? = 2
[.,[.,[[[.,[.,[.,.]]],.],.]]]
=> [5,4,3,6,7,2,1] => [6,3,4,5,1,2,7] => [5,6,2,3,4,1,7] => ? = 3
[.,[.,[[[.,[[.,.],.]],.],.]]]
=> [4,5,3,6,7,2,1] => [6,4,3,5,1,2,7] => [5,6,3,2,4,1,7] => ? = 3
[.,[.,[[[[.,.],[.,.]],.],.]]]
=> [3,5,4,6,7,2,1] => [6,4,5,3,1,2,7] => [5,6,4,2,3,1,7] => ? = 3
[.,[.,[[[[.,[.,.]],.],.],.]]]
=> [4,3,5,6,7,2,1] => [6,5,3,4,1,2,7] => [5,6,3,4,2,1,7] => ? = 3
[.,[[.,.],[.,[.,[[.,.],.]]]]]
=> [2,6,7,5,4,3,1] => [6,2,1,3,4,5,7] => [3,2,4,5,6,1,7] => ? = 2
[.,[[.,.],[.,[[.,.],[.,.]]]]]
=> [2,5,7,6,4,3,1] => [5,2,1,3,4,6,7] => [3,2,4,5,1,6,7] => ? = 2
[.,[[.,.],[.,[[.,[.,.]],.]]]]
=> [2,6,5,7,4,3,1] => [5,6,2,1,3,4,7] => [4,3,5,6,1,2,7] => ? = 2
[.,[[.,.],[.,[[[.,.],.],.]]]]
=> [2,5,6,7,4,3,1] => [6,5,2,1,3,4,7] => [4,3,5,6,2,1,7] => ? = 2
[.,[[.,.],[[.,.],[.,[.,.]]]]]
=> [2,4,7,6,5,3,1] => [4,2,1,3,5,6,7] => [3,2,4,1,5,6,7] => ? = 2
[.,[[.,.],[[.,.],[[.,.],.]]]]
=> [2,4,6,7,5,3,1] => [6,4,2,1,3,5,7] => [4,3,5,2,6,1,7] => ? = 3
[.,[[.,.],[[.,[.,.]],[.,.]]]]
=> [2,5,4,7,6,3,1] => [4,5,2,1,3,6,7] => [4,3,5,1,2,6,7] => ? = 2
[.,[[.,.],[[[.,.],.],[.,.]]]]
=> [2,4,5,7,6,3,1] => [5,4,2,1,3,6,7] => [4,3,5,2,1,6,7] => ? = 2
[.,[[.,.],[[.,[.,[.,.]]],.]]]
=> [2,6,5,4,7,3,1] => [4,5,6,2,1,3,7] => [5,4,6,1,2,3,7] => ? = 2
[.,[[.,.],[[.,[[.,.],.]],.]]]
=> [2,5,6,4,7,3,1] => [5,4,6,2,1,3,7] => [5,4,6,2,1,3,7] => ? = 2
[.,[[.,.],[[[.,.],[.,.]],.]]]
=> [2,4,6,5,7,3,1] => [5,6,4,2,1,3,7] => [5,4,6,3,1,2,7] => ? = 2
[.,[[.,.],[[[.,[.,.]],.],.]]]
=> [2,5,4,6,7,3,1] => [6,4,5,2,1,3,7] => [5,4,6,2,3,1,7] => ? = 3
[.,[[.,[.,.]],[.,[[.,.],.]]]]
=> [3,2,6,7,5,4,1] => [6,2,3,1,4,5,7] => [4,2,3,5,6,1,7] => ? = 2
[.,[[.,[.,.]],[[.,.],[.,.]]]]
=> [3,2,5,7,6,4,1] => [5,2,3,1,4,6,7] => [4,2,3,5,1,6,7] => ? = 2
[.,[[.,[.,.]],[[.,[.,.]],.]]]
=> [3,2,6,5,7,4,1] => [5,6,2,3,1,4,7] => [5,3,4,6,1,2,7] => ? = 2
[.,[[.,[.,.]],[[[.,.],.],.]]]
=> [3,2,5,6,7,4,1] => [6,5,2,3,1,4,7] => [5,3,4,6,2,1,7] => ? = 2
[.,[[[.,.],.],[.,[[.,.],.]]]]
=> [2,3,6,7,5,4,1] => [6,3,2,1,4,5,7] => [4,3,2,5,6,1,7] => ? = 2
[.,[[[.,.],.],[[.,.],[.,.]]]]
=> [2,3,5,7,6,4,1] => [5,3,2,1,4,6,7] => [4,3,2,5,1,6,7] => ? = 2
[.,[[[.,.],.],[[.,[.,.]],.]]]
=> [2,3,6,5,7,4,1] => [5,6,3,2,1,4,7] => [5,4,3,6,1,2,7] => ? = 2
[.,[[.,[.,[.,.]]],[[.,.],.]]]
=> [4,3,2,6,7,5,1] => [6,2,3,4,1,5,7] => [5,2,3,4,6,1,7] => ? = 2
Description
The number of right outer peaks of a permutation.
A right 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 $n$ if $w_n > w_{n-1}$.
In other words, it is a peak in the word $[w_1,..., w_n,0]$.
Matching statistic: St001212
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
St001212: Dyck paths ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 57%
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
St001212: Dyck paths ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 57%
Values
[.,[.,.]]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[[.,.],.]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[[.,[.,[.,[.,.]]]],.]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
[.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 2
[.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> ? = 2
[.,[.,[.,[.,[[[.,.],.],.]]]]]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 2
[.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2
[.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 2
[.,[.,[.,[[[.,.],.],[.,.]]]]]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 2
[.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 2
[.,[.,[.,[[.,[[.,.],.]],.]]]]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> ? = 2
[.,[.,[.,[[[.,.],[.,.]],.]]]]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> ? = 2
[.,[.,[.,[[[.,[.,.]],.],.]]]]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 3
[.,[.,[.,[[[[.,.],.],.],.]]]]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 2
[.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 2
[.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 3
[.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 3
[.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 3
[.,[.,[[.,.],[[[.,.],.],.]]]]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 3
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 2
[.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 3
[.,[.,[[[.,.],.],[.,[.,.]]]]]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 2
[.,[.,[[[.,.],.],[[.,.],.]]]]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 3
[.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 2
[.,[.,[[.,[[.,.],.]],[.,.]]]]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> ? = 2
[.,[.,[[[.,.],[.,.]],[.,.]]]]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> ? = 2
[.,[.,[[[.,[.,.]],.],[.,.]]]]
=> [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> ? = 3
[.,[.,[[[[.,.],.],.],[.,.]]]]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 2
[.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 2
[.,[.,[[.,[.,[[.,.],.]]],.]]]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 3
[.,[.,[[.,[[.,.],[.,.]]],.]]]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> ? = 2
[.,[.,[[.,[[.,[.,.]],.]],.]]]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> ? = 2
[.,[.,[[.,[[[.,.],.],.]],.]]]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> ? = 2
[.,[.,[[[.,.],[.,[.,.]]],.]]]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> ? = 2
[.,[.,[[[.,.],[[.,.],.]],.]]]
=> [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 2
[.,[.,[[[.,[.,.]],[.,.]],.]]]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> ? = 3
[.,[.,[[[[.,.],.],[.,.]],.]]]
=> [1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 2
[.,[.,[[[.,[.,[.,.]]],.],.]]]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> ? = 3
[.,[.,[[[.,[[.,.],.]],.],.]]]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> ? = 3
[.,[.,[[[[.,.],[.,.]],.],.]]]
=> [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> ? = 3
[.,[.,[[[[.,[.,.]],.],.],.]]]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> ? = 3
[.,[.,[[[[[.,.],.],.],.],.]]]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 2
[.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[.,[[.,.],[.,[.,[[.,.],.]]]]]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
[.,[[.,.],[.,[[.,.],[.,.]]]]]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 2
[.,[[.,.],[.,[[.,[.,.]],.]]]]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> ? = 2
[.,[[.,.],[.,[[[.,.],.],.]]]]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 2
[.,[[.,.],[[.,.],[.,[.,.]]]]]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2
[.,[[.,.],[[.,.],[[.,.],.]]]]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3
[.,[[.,.],[[.,[.,.]],[.,.]]]]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 2
Description
The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module.
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