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Your data matches 14 different statistics following compositions of up to 3 maps.
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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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000884: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => [1] => 0
[.,[.,.]]
=> [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
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: St000390
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00130: Permutations —descent tops⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Mp00130: Permutations —descent tops⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => => ? = 0
[.,[.,.]]
=> [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
[.,[.,[[[.,[.,.]],.],[[.,.],.]]]]
=> [7,8,4,3,5,6,2,1] => ? => ? = 4
[.,[.,[[.,[[[.,.],.],.]],[.,.]]]]
=> [8,4,5,6,3,7,2,1] => ? => ? = 2
[.,[.,[[[[.,[.,.]],.],[.,.]],.]]]
=> [7,4,3,5,6,8,2,1] => ? => ? = 3
[.,[.,[[[[.,[.,.]],[.,.]],.],.]]]
=> [6,4,3,5,7,8,2,1] => ? => ? = 4
[.,[[.,.],[[.,[.,.]],[[.,.],.]]]]
=> [7,8,5,4,6,2,3,1] => ? => ? = 3
[.,[[.,[[.,.],[.,[[.,.],.]]]],.]]
=> [6,7,5,3,4,2,8,1] => ? => ? = 2
[.,[[[[[.,.],[.,.]],[.,.]],.],.]]
=> [6,4,2,3,5,7,8,1] => ? => ? = 3
[[.,.],[.,[[.,.],[[.,[.,.]],.]]]]
=> [7,6,8,4,5,3,1,2] => ? => ? = 3
[[.,.],[.,[[.,[.,[[.,.],.]]],.]]]
=> [6,7,5,4,8,3,1,2] => ? => ? = 3
[[.,.],[.,[[[.,[.,.]],[.,.]],.]]]
=> [7,5,4,6,8,3,1,2] => ? => ? = 3
[[.,[.,.]],[[[.,[.,.]],[.,.]],.]]
=> [7,5,4,6,8,2,1,3] => ? => ? = 3
[[.,[.,[[.,.],.]]],[[.,.],[.,.]]]
=> [8,6,7,3,4,2,1,5] => ? => ? = 3
[[.,[[.,.],[.,.]]],[.,[[.,.],.]]]
=> [7,8,6,4,2,3,1,5] => ? => ? = 3
[[.,[[.,[.,.]],.]],[[.,[.,.]],.]]
=> [7,6,8,3,2,4,1,5] => ? => ? = 2
[[[.,[.,.]],[.,.]],[[.,[.,.]],.]]
=> [7,6,8,4,2,1,3,5] => ? => ? = 3
[[[.,.],[.,[[.,.],.]]],[.,[.,.]]]
=> [8,7,4,5,3,1,2,6] => ? => ? = 3
[[[[.,.],[.,[.,.]]],.],[.,[.,.]]]
=> [8,7,4,3,1,2,5,6] => ? => ? = 2
[[[[[.,.],[.,.]],.],.],[.,[.,.]]]
=> [8,7,3,1,2,4,5,6] => ? => ? = 2
[[[.,[.,.]],[[.,[.,.]],.]],[.,.]]
=> [8,5,4,6,2,1,3,7] => ? => ? = 3
[[[.,.],[.,[[.,[.,.]],[.,.]]]],.]
=> [7,5,4,6,3,1,2,8] => ? => ? = 2
[[[.,.],[[.,[[[.,.],.],.]],.]],.]
=> [4,5,6,3,7,1,2,8] => ? => ? = 1
[[[.,.],[[[[.,.],[.,.]],.],.]],.]
=> [5,3,4,6,7,1,2,8] => ? => ? = 2
[[[[.,.],[.,.]],[.,[.,[.,.]]]],.]
=> [7,6,5,3,1,2,4,8] => ? => ? = 2
[[[[.,.],[.,.]],[.,[[.,.],.]]],.]
=> [6,7,5,3,1,2,4,8] => ? => ? = 3
[[[[.,.],[.,[[.,.],.]]],[.,.]],.]
=> [7,4,5,3,1,2,6,8] => ? => ? = 3
[[[[[.,.],.],[[.,[.,.]],.]],.],.]
=> [5,4,6,1,2,3,7,8] => ? => ? = 1
[[[[[.,.],[.,.]],[.,[.,.]]],.],.]
=> [6,5,3,1,2,4,7,8] => ? => ? = 2
[[[[[[.,.],.],.],[.,[.,.]]],.],.]
=> [6,5,1,2,3,4,7,8] => ? => ? = 1
[[[[[.,.],[.,[[.,.],.]]],.],.],.]
=> [4,5,3,1,2,6,7,8] => ? => ? = 2
[.,[[.,[[.,[[.,[[.,[[.,[.,.]],.]],.]],.]],.]],.]]
=> [7,6,8,5,9,4,10,3,11,2,12,1] => 00000111111 => ? = 1
Description
The number of runs of ones in a binary word.
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: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 100%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000035: Permutations ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => [1] => 0
[.,[.,.]]
=> [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
[[.,.],[.,[[.,.],[.,[.,.]]]]]
=> [1,4,7,6,5,3,2] => [1,7,2,4,3,5,6] => [1,4,2,7,3,5,6] => ? = 2
[[.,.],[.,[[.,.],[[.,.],.]]]]
=> [1,4,6,7,5,3,2] => [1,7,2,4,3,6,5] => [1,4,2,6,3,7,5] => ? = 3
[[.,.],[.,[[[.,.],.],[.,.]]]]
=> [1,4,5,7,6,3,2] => [1,7,2,4,5,3,6] => [1,4,2,5,7,3,6] => ? = 2
[[.,.],[.,[[[.,.],[.,.]],.]]]
=> [1,4,6,5,7,3,2] => [1,7,2,4,6,5,3] => [1,4,2,6,7,3,5] => ? = 2
[[.,.],[.,[[[[.,.],.],.],.]]]
=> [1,4,5,6,7,3,2] => [1,7,2,4,5,6,3] => [1,4,2,5,6,7,3] => ? = 2
[[.,.],[[.,.],[.,[[.,.],.]]]]
=> [1,3,6,7,5,4,2] => [1,7,3,2,4,6,5] => [1,3,6,2,4,7,5] => ? = 2
[[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,3,5,7,6,4,2] => [1,7,3,2,5,4,6] => [1,3,5,2,7,4,6] => ? = 2
[[.,.],[[.,.],[[.,[.,.]],.]]]
=> [1,3,6,5,7,4,2] => [1,7,3,2,6,5,4] => [1,3,6,2,7,4,5] => ? = 2
[[.,.],[[.,.],[[[.,.],.],.]]]
=> [1,3,5,6,7,4,2] => [1,7,3,2,5,6,4] => [1,3,5,2,6,7,4] => ? = 2
[[.,.],[[.,[.,.]],[[.,.],.]]]
=> [1,4,3,6,7,5,2] => [1,7,4,3,2,6,5] => [1,4,6,2,3,7,5] => ? = 2
[[.,.],[[[.,.],.],[.,[.,.]]]]
=> [1,3,4,7,6,5,2] => [1,7,3,4,2,5,6] => [1,3,4,7,2,5,6] => ? = 1
[[.,.],[[[.,.],.],[[.,.],.]]]
=> [1,3,4,6,7,5,2] => [1,7,3,4,2,6,5] => [1,3,4,6,2,7,5] => ? = 2
[[.,.],[[.,[[.,.],.]],[.,.]]]
=> [1,4,5,3,7,6,2] => [1,7,5,4,3,2,6] => [1,4,5,7,2,3,6] => ? = 1
[[.,.],[[[.,.],[.,.]],[.,.]]]
=> [1,3,5,4,7,6,2] => [1,7,3,5,4,2,6] => [1,3,5,7,2,4,6] => ? = 1
[[.,.],[[[.,[.,.]],.],[.,.]]]
=> [1,4,3,5,7,6,2] => [1,7,4,3,5,2,6] => [1,4,5,2,7,3,6] => ? = 2
[[.,.],[[[[.,.],.],.],[.,.]]]
=> [1,3,4,5,7,6,2] => [1,7,3,4,5,2,6] => [1,3,4,5,7,2,6] => ? = 1
[[.,.],[[.,[[[.,.],.],.]],.]]
=> [1,4,5,6,3,7,2] => [1,7,6,4,5,3,2] => [1,4,5,6,7,2,3] => ? = 1
[[.,.],[[[.,.],[.,[.,.]]],.]]
=> [1,3,6,5,4,7,2] => [1,7,3,6,4,5,2] => [1,3,6,7,2,4,5] => ? = 1
[[.,.],[[[.,.],[[.,.],.]],.]]
=> [1,3,5,6,4,7,2] => [1,7,3,6,5,4,2] => [1,3,5,6,7,2,4] => ? = 1
[[.,.],[[[.,[.,.]],[.,.]],.]]
=> [1,4,3,6,5,7,2] => [1,7,4,3,6,5,2] => [1,4,6,2,7,3,5] => ? = 2
[[.,.],[[[[.,.],.],[.,.]],.]]
=> [1,3,4,6,5,7,2] => [1,7,3,4,6,5,2] => [1,3,4,6,7,2,5] => ? = 1
[[.,.],[[[.,[[.,.],.]],.],.]]
=> [1,4,5,3,6,7,2] => [1,7,5,4,3,6,2] => [1,4,5,6,2,7,3] => ? = 2
[[.,.],[[[[.,.],[.,.]],.],.]]
=> [1,3,5,4,6,7,2] => [1,7,3,5,4,6,2] => [1,3,5,6,2,7,4] => ? = 2
[[.,.],[[[[.,[.,.]],.],.],.]]
=> [1,4,3,5,6,7,2] => [1,7,4,3,5,6,2] => [1,4,5,2,6,7,3] => ? = 2
[[.,.],[[[[[.,.],.],.],.],.]]
=> [1,3,4,5,6,7,2] => [1,7,3,4,5,6,2] => [1,3,4,5,6,7,2] => ? = 1
[[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [1,2,7,6,5,4,3] => [1,2,7,3,4,5,6] => [1,2,7,3,4,5,6] => ? = 1
[[[.,.],.],[.,[[.,.],[.,.]]]]
=> [1,2,5,7,6,4,3] => [1,2,7,3,5,4,6] => [1,2,5,3,7,4,6] => ? = 2
[[[.,.],.],[.,[[.,[.,.]],.]]]
=> [1,2,6,5,7,4,3] => [1,2,7,3,6,5,4] => [1,2,6,3,7,4,5] => ? = 2
[[[.,.],.],[.,[[[.,.],.],.]]]
=> [1,2,5,6,7,4,3] => [1,2,7,3,5,6,4] => [1,2,5,3,6,7,4] => ? = 2
[[[.,.],.],[[.,.],[.,[.,.]]]]
=> [1,2,4,7,6,5,3] => [1,2,7,4,3,5,6] => [1,2,4,7,3,5,6] => ? = 1
[[[.,.],.],[[.,.],[[.,.],.]]]
=> [1,2,4,6,7,5,3] => [1,2,7,4,3,6,5] => [1,2,4,6,3,7,5] => ? = 2
[[[.,.],.],[[.,[.,.]],[.,.]]]
=> [1,2,5,4,7,6,3] => [1,2,7,5,4,3,6] => [1,2,5,7,3,4,6] => ? = 1
[[[.,.],.],[[[.,.],.],[.,.]]]
=> [1,2,4,5,7,6,3] => [1,2,7,4,5,3,6] => [1,2,4,5,7,3,6] => ? = 1
[[[.,.],.],[[.,[.,[.,.]]],.]]
=> [1,2,6,5,4,7,3] => [1,2,7,6,4,5,3] => [1,2,6,7,3,4,5] => ? = 1
[[[.,.],.],[[.,[[.,.],.]],.]]
=> [1,2,5,6,4,7,3] => [1,2,7,6,5,4,3] => [1,2,5,6,7,3,4] => ? = 1
[[[.,.],.],[[[.,.],[.,.]],.]]
=> [1,2,4,6,5,7,3] => [1,2,7,4,6,5,3] => [1,2,4,6,7,3,5] => ? = 1
[[[.,.],.],[[[.,[.,.]],.],.]]
=> [1,2,5,4,6,7,3] => [1,2,7,5,4,6,3] => [1,2,5,6,3,7,4] => ? = 2
[[[.,.],.],[[[[.,.],.],.],.]]
=> [1,2,4,5,6,7,3] => [1,2,7,4,5,6,3] => [1,2,4,5,6,7,3] => ? = 1
[[[.,.],[.,.]],[.,[.,[.,.]]]]
=> [1,3,2,7,6,5,4] => [1,3,2,7,4,5,6] => [1,3,2,7,4,5,6] => ? = 2
[[[.,.],[.,.]],[.,[[.,.],.]]]
=> [1,3,2,6,7,5,4] => [1,3,2,7,4,6,5] => [1,3,2,6,4,7,5] => ? = 3
[[[.,.],[.,.]],[[.,.],[.,.]]]
=> [1,3,2,5,7,6,4] => [1,3,2,7,5,4,6] => [1,3,2,5,7,4,6] => ? = 2
[[[.,.],[.,.]],[[.,[.,.]],.]]
=> [1,3,2,6,5,7,4] => [1,3,2,7,6,5,4] => [1,3,2,6,7,4,5] => ? = 2
[[[.,.],[.,.]],[[[.,.],.],.]]
=> [1,3,2,5,6,7,4] => [1,3,2,7,5,6,4] => [1,3,2,5,6,7,4] => ? = 2
[[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,2,3,7,6,5,4] => [1,2,3,7,4,5,6] => [1,2,3,7,4,5,6] => ? = 1
[[[[.,.],.],.],[[.,.],[.,.]]]
=> [1,2,3,5,7,6,4] => [1,2,3,7,5,4,6] => [1,2,3,5,7,4,6] => ? = 1
[[[[.,.],.],.],[[.,[.,.]],.]]
=> [1,2,3,6,5,7,4] => [1,2,3,7,6,5,4] => [1,2,3,6,7,4,5] => ? = 1
[[[[.,.],.],.],[[[.,.],.],.]]
=> [1,2,3,5,6,7,4] => [1,2,3,7,5,6,4] => [1,2,3,5,6,7,4] => ? = 1
[[[.,.],[.,[.,.]]],[.,[.,.]]]
=> [1,4,3,2,7,6,5] => [1,4,2,3,7,5,6] => [1,4,2,3,7,5,6] => ? = 2
[[[.,.],[.,[.,.]]],[[.,.],.]]
=> [1,4,3,2,6,7,5] => [1,4,2,3,7,6,5] => [1,4,2,3,6,7,5] => ? = 2
[[[.,.],[[.,.],.]],[.,[.,.]]]
=> [1,3,4,2,7,6,5] => [1,4,3,2,7,5,6] => [1,3,4,2,7,5,6] => ? = 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: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 67%
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St001737: Permutations ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 67%
Values
[.,.]
=> [1] => [1] => [1] => 0
[.,[.,.]]
=> [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
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [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
[.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [6,5,4,7,3,2,1] => [2,3,7,5,6,4,1] => [5,2,3,7,6,1,4] => ? = 2
[.,[.,[[[.,[.,[.,.]]],.],.]]]
=> [5,4,3,6,7,2,1] => [2,7,4,5,3,6,1] => [4,2,6,5,1,7,3] => ? = 3
[.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [2,7,6,5,4,3,1] => [3,2,4,5,6,7,1] => [2,7,3,4,5,6,1] => ? = 1
[.,[[.,.],[[.,[.,[.,.]]],.]]]
=> [2,6,5,4,7,3,1] => [3,2,7,5,6,4,1] => [2,5,3,7,6,1,4] => ? = 2
[.,[[.,[.,[.,.]]],[[.,.],.]]]
=> [4,3,2,6,7,5,1] => [5,3,4,2,7,6,1] => [3,6,4,1,5,7,2] => ? = 2
[.,[[[[.,.],.],.],[.,[.,.]]]]
=> [2,3,4,7,6,5,1] => [5,2,3,4,6,7,1] => [2,3,4,7,5,6,1] => ? = 1
[.,[[[[.,.],.],.],[[.,.],.]]]
=> [2,3,4,6,7,5,1] => [5,2,3,4,7,6,1] => [2,3,4,6,5,7,1] => ? = 2
[.,[[[.,.],[.,[.,.]]],[.,.]]]
=> [2,5,4,3,7,6,1] => [6,2,4,5,3,7,1] => [2,4,7,5,1,6,3] => ? = 1
[.,[[[[[.,.],.],.],.],[.,.]]]
=> [2,3,4,5,7,6,1] => [6,2,3,4,5,7,1] => [2,3,4,5,7,6,1] => ? = 1
[.,[[.,[.,[.,[.,[.,.]]]]],.]]
=> [6,5,4,3,2,7,1] => [7,3,4,5,6,2,1] => [3,7,4,5,6,1,2] => ? = 1
[.,[[.,[[.,[.,[.,.]]],.]],.]]
=> [5,4,3,6,2,7,1] => [7,6,4,5,3,2,1] => [4,5,7,6,1,2,3] => ? = 1
[.,[[.,[[[[.,.],.],.],.]],.]]
=> [3,4,5,6,2,7,1] => [7,6,3,4,5,2,1] => [3,4,5,6,7,1,2] => ? = 1
[.,[[[.,.],[.,[.,[.,.]]]],.]]
=> [2,6,5,4,3,7,1] => [7,2,4,5,6,3,1] => [2,4,7,5,6,1,3] => ? = 1
[.,[[[[.,.],.],[[.,.],.]],.]]
=> [2,3,5,6,4,7,1] => [7,2,3,6,5,4,1] => [2,3,5,6,7,1,4] => ? = 1
[.,[[[[[.,.],.],.],[.,.]],.]]
=> [2,3,4,6,5,7,1] => [7,2,3,4,6,5,1] => [2,3,4,6,7,1,5] => ? = 1
[.,[[[.,[.,[.,[.,.]]]],.],.]]
=> [5,4,3,2,6,7,1] => [7,3,4,5,2,6,1] => [3,6,4,5,1,7,2] => ? = 2
[.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => [7,2,3,4,5,6,1] => [2,3,4,5,6,7,1] => ? = 1
[[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,7,6,5,4,3,2] => [1,3,4,5,6,7,2] => [1,7,3,4,5,6,2] => ? = 1
[[.,.],[.,[.,[.,[[.,.],.]]]]]
=> [1,6,7,5,4,3,2] => [1,3,4,5,7,6,2] => [1,6,3,4,5,7,2] => ? = 2
[[.,.],[.,[.,[[.,[.,.]],.]]]]
=> [1,6,5,7,4,3,2] => [1,3,4,7,6,5,2] => [1,6,3,4,7,2,5] => ? = 2
[[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [1,5,4,7,6,3,2] => [1,3,6,5,4,7,2] => [1,5,3,7,2,6,4] => ? = 2
[[.,.],[.,[[.,[.,[.,.]]],.]]]
=> [1,6,5,4,7,3,2] => [1,3,7,5,6,4,2] => [1,5,3,7,6,2,4] => ? = 2
[[.,.],[[.,[.,.]],[.,[.,.]]]]
=> [1,4,3,7,6,5,2] => [1,5,4,3,6,7,2] => [1,4,7,2,5,6,3] => ? = 1
[[.,.],[[.,[.,[.,.]]],[.,.]]]
=> [1,5,4,3,7,6,2] => [1,6,4,5,3,7,2] => [1,4,7,5,2,6,3] => ? = 1
[[.,.],[[.,[.,[.,[.,.]]]],.]]
=> [1,6,5,4,3,7,2] => [1,7,4,5,6,3,2] => [1,4,7,5,6,2,3] => ? = 1
[[.,.],[[.,[.,[[.,.],.]]],.]]
=> [1,5,6,4,3,7,2] => [1,7,4,6,5,3,2] => [1,4,6,5,7,2,3] => ? = 2
[[.,.],[[.,[[.,[.,.]],.]],.]]
=> [1,5,4,6,3,7,2] => [1,7,6,5,4,3,2] => [1,5,6,7,2,3,4] => ? = 1
[[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => [2,1,4,5,6,7,3] => [2,1,7,4,5,6,3] => ? = 2
[[[.,[.,.]],.],[.,[.,[.,.]]]]
=> [2,1,3,7,6,5,4] => [2,1,3,5,6,7,4] => [2,1,3,7,5,6,4] => ? = 2
[[[[.,[.,.]],.],.],[.,[.,.]]]
=> [2,1,3,4,7,6,5] => [2,1,3,4,6,7,5] => [2,1,3,4,7,6,5] => ? = 2
[[.,[.,[.,[.,[.,.]]]]],[.,.]]
=> [5,4,3,2,1,7,6] => [2,3,4,5,1,7,6] => [5,2,3,4,1,7,6] => ? = 2
[[.,[[[[.,.],.],.],.]],[.,.]]
=> [2,3,4,5,1,7,6] => [5,2,3,4,1,7,6] => [2,3,4,5,1,7,6] => ? = 2
[[[.,.],[.,[.,[.,.]]]],[.,.]]
=> [1,5,4,3,2,7,6] => [1,3,4,5,2,7,6] => [1,5,3,4,2,7,6] => ? = 2
[[[.,[.,[.,[.,.]]]],.],[.,.]]
=> [4,3,2,1,5,7,6] => [2,3,4,1,5,7,6] => [4,2,3,1,5,7,6] => ? = 2
[[[.,[[[.,.],.],.]],.],[.,.]]
=> [2,3,4,1,5,7,6] => [4,2,3,1,5,7,6] => [2,3,4,1,5,7,6] => ? = 2
[[[[.,[.,[.,.]]],.],.],[.,.]]
=> [3,2,1,4,5,7,6] => [2,3,1,4,5,7,6] => [3,2,1,4,5,7,6] => ? = 2
[[.,[.,[.,[.,[.,[.,.]]]]]],.]
=> [6,5,4,3,2,1,7] => [2,3,4,5,6,1,7] => [6,2,3,4,5,1,7] => ? = 1
[[.,[.,[.,[.,[[.,.],.]]]]],.]
=> [5,6,4,3,2,1,7] => [2,3,4,6,5,1,7] => [5,2,3,4,6,1,7] => ? = 2
[[.,[.,[.,[[.,[.,.]],.]]]],.]
=> [5,4,6,3,2,1,7] => [2,3,6,5,4,1,7] => [5,2,3,6,1,4,7] => ? = 2
[[.,[.,[[.,[.,.]],[.,.]]]],.]
=> [4,3,6,5,2,1,7] => [2,5,4,3,6,1,7] => [4,2,6,1,5,3,7] => ? = 2
[[.,[.,[[.,[.,[.,.]]],.]]],.]
=> [5,4,3,6,2,1,7] => [2,6,4,5,3,1,7] => [4,2,6,5,1,3,7] => ? = 2
[[.,[[.,.],[.,[.,[.,.]]]]],.]
=> [2,6,5,4,3,1,7] => [3,2,4,5,6,1,7] => [2,6,3,4,5,1,7] => ? = 1
[[.,[[.,[.,.]],[.,[.,.]]]],.]
=> [3,2,6,5,4,1,7] => [4,3,2,5,6,1,7] => [3,6,1,4,5,2,7] => ? = 1
[[.,[[.,[.,[.,.]]],[.,.]]],.]
=> [4,3,2,6,5,1,7] => [5,3,4,2,6,1,7] => [3,6,4,1,5,2,7] => ? = 1
[[.,[[.,[.,[.,[.,.]]]],.]],.]
=> [5,4,3,2,6,1,7] => [6,3,4,5,2,1,7] => [3,6,4,5,1,2,7] => ? = 1
Description
The number of descents of type 2 in a permutation.
A position $i\in[1,n-1]$ is a descent of type 2 of a permutation $\pi$ of $n$ letters, if it is a descent and if $\pi(j) < \pi(i)$ for all $j < i$.
Matching statistic: St001729
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St001729: Permutations ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 67%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St001729: Permutations ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 67%
Values
[.,.]
=> [1] => [1] => [1] => 0
[.,[.,.]]
=> [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
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [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
[.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [6,5,4,7,3,2,1] => [7,1,2,6,4,5,3] => [6,1,2,7,3,4,5] => ? = 2
[.,[.,[[[.,[.,[.,.]]],.],.]]]
=> [5,4,3,6,7,2,1] => [7,1,5,3,4,6,2] => [5,1,6,2,3,7,4] => ? = 3
[.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [2,7,6,5,4,3,1] => [7,2,1,3,4,5,6] => [2,7,1,3,4,5,6] => ? = 1
[.,[[.,.],[[.,[.,[.,.]]],.]]]
=> [2,6,5,4,7,3,1] => [7,2,1,6,4,5,3] => [2,6,1,7,3,4,5] => ? = 2
[.,[[.,[.,[.,.]]],[[.,.],.]]]
=> [4,3,2,6,7,5,1] => [7,4,2,3,1,6,5] => [4,6,1,2,3,7,5] => ? = 2
[.,[[[[.,.],.],.],[.,[.,.]]]]
=> [2,3,4,7,6,5,1] => [7,2,3,4,1,5,6] => [2,3,4,7,1,5,6] => ? = 1
[.,[[[[.,.],.],.],[[.,.],.]]]
=> [2,3,4,6,7,5,1] => [7,2,3,4,1,6,5] => [2,3,4,6,1,7,5] => ? = 2
[.,[[[.,.],[.,[.,.]]],[.,.]]]
=> [2,5,4,3,7,6,1] => [7,2,5,3,4,1,6] => [2,5,7,1,3,4,6] => ? = 1
[.,[[[[[.,.],.],.],.],[.,.]]]
=> [2,3,4,5,7,6,1] => [7,2,3,4,5,1,6] => [2,3,4,5,7,1,6] => ? = 1
[.,[[.,[.,[.,[.,[.,.]]]]],.]]
=> [6,5,4,3,2,7,1] => [7,6,2,3,4,5,1] => [6,7,1,2,3,4,5] => ? = 1
[.,[[.,[[.,[.,[.,.]]],.]],.]]
=> [5,4,3,6,2,7,1] => [7,6,5,3,4,2,1] => [5,6,7,1,2,3,4] => ? = 1
[.,[[.,[[[[.,.],.],.],.]],.]]
=> [3,4,5,6,2,7,1] => [7,6,3,4,5,2,1] => [3,4,5,6,7,1,2] => ? = 1
[.,[[[.,.],[.,[.,[.,.]]]],.]]
=> [2,6,5,4,3,7,1] => [7,2,6,3,4,5,1] => [2,6,7,1,3,4,5] => ? = 1
[.,[[[[.,.],.],[[.,.],.]],.]]
=> [2,3,5,6,4,7,1] => [7,2,3,6,5,4,1] => [2,3,5,6,7,1,4] => ? = 1
[.,[[[[[.,.],.],.],[.,.]],.]]
=> [2,3,4,6,5,7,1] => [7,2,3,4,6,5,1] => [2,3,4,6,7,1,5] => ? = 1
[.,[[[.,[.,[.,[.,.]]]],.],.]]
=> [5,4,3,2,6,7,1] => [7,5,2,3,4,6,1] => [5,6,1,2,3,7,4] => ? = 2
[.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => [7,2,3,4,5,6,1] => [2,3,4,5,6,7,1] => ? = 1
[[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 1
[[.,.],[.,[.,[.,[[.,.],.]]]]]
=> [1,6,7,5,4,3,2] => [1,7,2,3,4,6,5] => [1,6,2,3,4,7,5] => ? = 2
[[.,.],[.,[.,[[.,[.,.]],.]]]]
=> [1,6,5,7,4,3,2] => [1,7,2,3,6,5,4] => [1,6,2,3,7,4,5] => ? = 2
[[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [1,5,4,7,6,3,2] => [1,7,2,5,4,3,6] => [1,5,2,7,3,4,6] => ? = 2
[[.,.],[.,[[.,[.,[.,.]]],.]]]
=> [1,6,5,4,7,3,2] => [1,7,2,6,4,5,3] => [1,6,2,7,3,4,5] => ? = 2
[[.,.],[[.,[.,.]],[.,[.,.]]]]
=> [1,4,3,7,6,5,2] => [1,7,4,3,2,5,6] => [1,4,7,2,3,5,6] => ? = 1
[[.,.],[[.,[.,[.,.]]],[.,.]]]
=> [1,5,4,3,7,6,2] => [1,7,5,3,4,2,6] => [1,5,7,2,3,4,6] => ? = 1
[[.,.],[[.,[.,[.,[.,.]]]],.]]
=> [1,6,5,4,3,7,2] => [1,7,6,3,4,5,2] => [1,6,7,2,3,4,5] => ? = 1
[[.,.],[[.,[.,[[.,.],.]]],.]]
=> [1,5,6,4,3,7,2] => [1,7,6,3,5,4,2] => [1,5,6,2,7,3,4] => ? = 2
[[.,.],[[.,[[.,.],[.,.]]],.]]
=> [1,4,6,5,3,7,2] => [1,7,6,4,3,5,2] => [1,4,6,7,2,3,5] => ? = 1
[[.,.],[[.,[[.,[.,.]],.]],.]]
=> [1,5,4,6,3,7,2] => [1,7,6,5,4,3,2] => [1,5,6,7,2,3,4] => ? = 1
[[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => [2,1,7,3,4,5,6] => [2,1,7,3,4,5,6] => ? = 2
[[[.,[.,.]],.],[.,[.,[.,.]]]]
=> [2,1,3,7,6,5,4] => [2,1,3,7,4,5,6] => [2,1,3,7,4,5,6] => ? = 2
[[[[.,[.,.]],.],.],[.,[.,.]]]
=> [2,1,3,4,7,6,5] => [2,1,3,4,7,5,6] => [2,1,3,4,7,5,6] => ? = 2
[[.,[.,[.,[.,[.,.]]]]],[.,.]]
=> [5,4,3,2,1,7,6] => [5,1,2,3,4,7,6] => [5,1,2,3,4,7,6] => ? = 2
[[.,[[[[.,.],.],.],.]],[.,.]]
=> [2,3,4,5,1,7,6] => [5,2,3,4,1,7,6] => [2,3,4,5,1,7,6] => ? = 2
[[[.,.],[.,[.,[.,.]]]],[.,.]]
=> [1,5,4,3,2,7,6] => [1,5,2,3,4,7,6] => [1,5,2,3,4,7,6] => ? = 2
[[[.,[.,[.,[.,.]]]],.],[.,.]]
=> [4,3,2,1,5,7,6] => [4,1,2,3,5,7,6] => [4,1,2,3,5,7,6] => ? = 2
[[[.,[[[.,.],.],.]],.],[.,.]]
=> [2,3,4,1,5,7,6] => [4,2,3,1,5,7,6] => [2,3,4,1,5,7,6] => ? = 2
[[[[.,[.,[.,.]]],.],.],[.,.]]
=> [3,2,1,4,5,7,6] => [3,1,2,4,5,7,6] => [3,1,2,4,5,7,6] => ? = 2
[[.,[.,[.,[.,[.,[.,.]]]]]],.]
=> [6,5,4,3,2,1,7] => [6,1,2,3,4,5,7] => [6,1,2,3,4,5,7] => ? = 1
[[.,[.,[.,[.,[[.,.],.]]]]],.]
=> [5,6,4,3,2,1,7] => [6,1,2,3,5,4,7] => [5,1,2,3,6,4,7] => ? = 2
[[.,[.,[.,[[.,[.,.]],.]]]],.]
=> [5,4,6,3,2,1,7] => [6,1,2,5,4,3,7] => [5,1,2,6,3,4,7] => ? = 2
[[.,[.,[[.,[.,.]],[.,.]]]],.]
=> [4,3,6,5,2,1,7] => [6,1,4,3,2,5,7] => [4,1,6,2,3,5,7] => ? = 2
[[.,[.,[[.,[.,[.,.]]],.]]],.]
=> [5,4,3,6,2,1,7] => [6,1,5,3,4,2,7] => [5,1,6,2,3,4,7] => ? = 2
[[.,[[.,.],[.,[.,[.,.]]]]],.]
=> [2,6,5,4,3,1,7] => [6,2,1,3,4,5,7] => [2,6,1,3,4,5,7] => ? = 1
[[.,[[.,[.,.]],[.,[.,.]]]],.]
=> [3,2,6,5,4,1,7] => [6,3,2,1,4,5,7] => [3,6,1,2,4,5,7] => ? = 1
[[.,[[.,[.,[.,.]]],[.,.]]],.]
=> [4,3,2,6,5,1,7] => [6,4,2,3,1,5,7] => [4,6,1,2,3,5,7] => ? = 1
Description
The number of visible descents of a permutation.
A visible descent of a permutation $\pi$ is a position $i$ such that $\pi(i+1) \leq \min(i, \pi(i))$.
Matching statistic: St001928
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St001928: Permutations ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 67%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St001928: Permutations ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 67%
Values
[.,.]
=> [1] => [1] => [1] => 0
[.,[.,.]]
=> [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
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [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
[.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [6,5,4,7,3,2,1] => [7,1,2,6,4,5,3] => [6,1,2,7,3,4,5] => ? = 2
[.,[.,[[[.,[.,[.,.]]],.],.]]]
=> [5,4,3,6,7,2,1] => [7,1,5,3,4,6,2] => [5,1,6,2,3,7,4] => ? = 3
[.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [2,7,6,5,4,3,1] => [7,2,1,3,4,5,6] => [2,7,1,3,4,5,6] => ? = 1
[.,[[.,.],[[.,[.,[.,.]]],.]]]
=> [2,6,5,4,7,3,1] => [7,2,1,6,4,5,3] => [2,6,1,7,3,4,5] => ? = 2
[.,[[.,[.,[.,.]]],[[.,.],.]]]
=> [4,3,2,6,7,5,1] => [7,4,2,3,1,6,5] => [4,6,1,2,3,7,5] => ? = 2
[.,[[[[.,.],.],.],[.,[.,.]]]]
=> [2,3,4,7,6,5,1] => [7,2,3,4,1,5,6] => [2,3,4,7,1,5,6] => ? = 1
[.,[[[[.,.],.],.],[[.,.],.]]]
=> [2,3,4,6,7,5,1] => [7,2,3,4,1,6,5] => [2,3,4,6,1,7,5] => ? = 2
[.,[[[.,.],[.,[.,.]]],[.,.]]]
=> [2,5,4,3,7,6,1] => [7,2,5,3,4,1,6] => [2,5,7,1,3,4,6] => ? = 1
[.,[[[[[.,.],.],.],.],[.,.]]]
=> [2,3,4,5,7,6,1] => [7,2,3,4,5,1,6] => [2,3,4,5,7,1,6] => ? = 1
[.,[[.,[.,[.,[.,[.,.]]]]],.]]
=> [6,5,4,3,2,7,1] => [7,6,2,3,4,5,1] => [6,7,1,2,3,4,5] => ? = 1
[.,[[.,[[.,[.,[.,.]]],.]],.]]
=> [5,4,3,6,2,7,1] => [7,6,5,3,4,2,1] => [5,6,7,1,2,3,4] => ? = 1
[.,[[.,[[[[.,.],.],.],.]],.]]
=> [3,4,5,6,2,7,1] => [7,6,3,4,5,2,1] => [3,4,5,6,7,1,2] => ? = 1
[.,[[[.,.],[.,[.,[.,.]]]],.]]
=> [2,6,5,4,3,7,1] => [7,2,6,3,4,5,1] => [2,6,7,1,3,4,5] => ? = 1
[.,[[[[.,.],.],[[.,.],.]],.]]
=> [2,3,5,6,4,7,1] => [7,2,3,6,5,4,1] => [2,3,5,6,7,1,4] => ? = 1
[.,[[[[[.,.],.],.],[.,.]],.]]
=> [2,3,4,6,5,7,1] => [7,2,3,4,6,5,1] => [2,3,4,6,7,1,5] => ? = 1
[.,[[[.,[.,[.,[.,.]]]],.],.]]
=> [5,4,3,2,6,7,1] => [7,5,2,3,4,6,1] => [5,6,1,2,3,7,4] => ? = 2
[.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => [7,2,3,4,5,6,1] => [2,3,4,5,6,7,1] => ? = 1
[[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 1
[[.,.],[.,[.,[.,[[.,.],.]]]]]
=> [1,6,7,5,4,3,2] => [1,7,2,3,4,6,5] => [1,6,2,3,4,7,5] => ? = 2
[[.,.],[.,[.,[[.,[.,.]],.]]]]
=> [1,6,5,7,4,3,2] => [1,7,2,3,6,5,4] => [1,6,2,3,7,4,5] => ? = 2
[[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [1,5,4,7,6,3,2] => [1,7,2,5,4,3,6] => [1,5,2,7,3,4,6] => ? = 2
[[.,.],[.,[[.,[.,[.,.]]],.]]]
=> [1,6,5,4,7,3,2] => [1,7,2,6,4,5,3] => [1,6,2,7,3,4,5] => ? = 2
[[.,.],[[.,[.,.]],[.,[.,.]]]]
=> [1,4,3,7,6,5,2] => [1,7,4,3,2,5,6] => [1,4,7,2,3,5,6] => ? = 1
[[.,.],[[.,[.,[.,.]]],[.,.]]]
=> [1,5,4,3,7,6,2] => [1,7,5,3,4,2,6] => [1,5,7,2,3,4,6] => ? = 1
[[.,.],[[.,[.,[.,[.,.]]]],.]]
=> [1,6,5,4,3,7,2] => [1,7,6,3,4,5,2] => [1,6,7,2,3,4,5] => ? = 1
[[.,.],[[.,[.,[[.,.],.]]],.]]
=> [1,5,6,4,3,7,2] => [1,7,6,3,5,4,2] => [1,5,6,2,7,3,4] => ? = 2
[[.,.],[[.,[[.,.],[.,.]]],.]]
=> [1,4,6,5,3,7,2] => [1,7,6,4,3,5,2] => [1,4,6,7,2,3,5] => ? = 1
[[.,.],[[.,[[.,[.,.]],.]],.]]
=> [1,5,4,6,3,7,2] => [1,7,6,5,4,3,2] => [1,5,6,7,2,3,4] => ? = 1
[[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => [2,1,7,3,4,5,6] => [2,1,7,3,4,5,6] => ? = 2
[[[.,[.,.]],.],[.,[.,[.,.]]]]
=> [2,1,3,7,6,5,4] => [2,1,3,7,4,5,6] => [2,1,3,7,4,5,6] => ? = 2
[[[[.,[.,.]],.],.],[.,[.,.]]]
=> [2,1,3,4,7,6,5] => [2,1,3,4,7,5,6] => [2,1,3,4,7,5,6] => ? = 2
[[.,[.,[.,[.,[.,.]]]]],[.,.]]
=> [5,4,3,2,1,7,6] => [5,1,2,3,4,7,6] => [5,1,2,3,4,7,6] => ? = 2
[[.,[[[[.,.],.],.],.]],[.,.]]
=> [2,3,4,5,1,7,6] => [5,2,3,4,1,7,6] => [2,3,4,5,1,7,6] => ? = 2
[[[.,.],[.,[.,[.,.]]]],[.,.]]
=> [1,5,4,3,2,7,6] => [1,5,2,3,4,7,6] => [1,5,2,3,4,7,6] => ? = 2
[[[.,[.,[.,[.,.]]]],.],[.,.]]
=> [4,3,2,1,5,7,6] => [4,1,2,3,5,7,6] => [4,1,2,3,5,7,6] => ? = 2
[[[.,[[[.,.],.],.]],.],[.,.]]
=> [2,3,4,1,5,7,6] => [4,2,3,1,5,7,6] => [2,3,4,1,5,7,6] => ? = 2
[[[[.,[.,[.,.]]],.],.],[.,.]]
=> [3,2,1,4,5,7,6] => [3,1,2,4,5,7,6] => [3,1,2,4,5,7,6] => ? = 2
[[.,[.,[.,[.,[.,[.,.]]]]]],.]
=> [6,5,4,3,2,1,7] => [6,1,2,3,4,5,7] => [6,1,2,3,4,5,7] => ? = 1
[[.,[.,[.,[.,[[.,.],.]]]]],.]
=> [5,6,4,3,2,1,7] => [6,1,2,3,5,4,7] => [5,1,2,3,6,4,7] => ? = 2
[[.,[.,[.,[[.,[.,.]],.]]]],.]
=> [5,4,6,3,2,1,7] => [6,1,2,5,4,3,7] => [5,1,2,6,3,4,7] => ? = 2
[[.,[.,[[.,[.,.]],[.,.]]]],.]
=> [4,3,6,5,2,1,7] => [6,1,4,3,2,5,7] => [4,1,6,2,3,5,7] => ? = 2
[[.,[.,[[.,[.,[.,.]]],.]]],.]
=> [5,4,3,6,2,1,7] => [6,1,5,3,4,2,7] => [5,1,6,2,3,4,7] => ? = 2
[[.,[[.,.],[.,[.,[.,.]]]]],.]
=> [2,6,5,4,3,1,7] => [6,2,1,3,4,5,7] => [2,6,1,3,4,5,7] => ? = 1
[[.,[[.,[.,.]],[.,[.,.]]]],.]
=> [3,2,6,5,4,1,7] => [6,3,2,1,4,5,7] => [3,6,1,2,4,5,7] => ? = 1
[[.,[[.,[.,[.,.]]],[.,.]]],.]
=> [4,3,2,6,5,1,7] => [6,4,2,3,1,5,7] => [4,6,1,2,3,5,7] => ? = 1
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: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 67%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 67%
Values
[.,.]
=> [1] => [1] => [1] => 1 = 0 + 1
[.,[.,.]]
=> [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
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [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
[.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [6,5,4,7,3,2,1] => [7,1,2,6,4,5,3] => [6,1,2,7,3,4,5] => ? = 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
[.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [2,7,6,5,4,3,1] => [7,2,1,3,4,5,6] => [2,7,1,3,4,5,6] => ? = 1 + 1
[.,[[.,.],[[.,[.,[.,.]]],.]]]
=> [2,6,5,4,7,3,1] => [7,2,1,6,4,5,3] => [2,6,1,7,3,4,5] => ? = 2 + 1
[.,[[.,[.,[.,.]]],[[.,.],.]]]
=> [4,3,2,6,7,5,1] => [7,4,2,3,1,6,5] => [4,6,1,2,3,7,5] => ? = 2 + 1
[.,[[[[.,.],.],.],[.,[.,.]]]]
=> [2,3,4,7,6,5,1] => [7,2,3,4,1,5,6] => [2,3,4,7,1,5,6] => ? = 1 + 1
[.,[[[[.,.],.],.],[[.,.],.]]]
=> [2,3,4,6,7,5,1] => [7,2,3,4,1,6,5] => [2,3,4,6,1,7,5] => ? = 2 + 1
[.,[[[.,.],[.,[.,.]]],[.,.]]]
=> [2,5,4,3,7,6,1] => [7,2,5,3,4,1,6] => [2,5,7,1,3,4,6] => ? = 1 + 1
[.,[[[[[.,.],.],.],.],[.,.]]]
=> [2,3,4,5,7,6,1] => [7,2,3,4,5,1,6] => [2,3,4,5,7,1,6] => ? = 1 + 1
[.,[[.,[.,[.,[.,[.,.]]]]],.]]
=> [6,5,4,3,2,7,1] => [7,6,2,3,4,5,1] => [6,7,1,2,3,4,5] => ? = 1 + 1
[.,[[.,[[.,[.,[.,.]]],.]],.]]
=> [5,4,3,6,2,7,1] => [7,6,5,3,4,2,1] => [5,6,7,1,2,3,4] => ? = 1 + 1
[.,[[.,[[[[.,.],.],.],.]],.]]
=> [3,4,5,6,2,7,1] => [7,6,3,4,5,2,1] => [3,4,5,6,7,1,2] => ? = 1 + 1
[.,[[[.,.],[.,[.,[.,.]]]],.]]
=> [2,6,5,4,3,7,1] => [7,2,6,3,4,5,1] => [2,6,7,1,3,4,5] => ? = 1 + 1
[.,[[[[.,.],.],[[.,.],.]],.]]
=> [2,3,5,6,4,7,1] => [7,2,3,6,5,4,1] => [2,3,5,6,7,1,4] => ? = 1 + 1
[.,[[[[[.,.],.],.],[.,.]],.]]
=> [2,3,4,6,5,7,1] => [7,2,3,4,6,5,1] => [2,3,4,6,7,1,5] => ? = 1 + 1
[.,[[[.,[.,[.,[.,.]]]],.],.]]
=> [5,4,3,2,6,7,1] => [7,5,2,3,4,6,1] => [5,6,1,2,3,7,4] => ? = 2 + 1
[.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => [7,2,3,4,5,6,1] => [2,3,4,5,6,7,1] => ? = 1 + 1
[[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 1 + 1
[[.,.],[.,[.,[.,[[.,.],.]]]]]
=> [1,6,7,5,4,3,2] => [1,7,2,3,4,6,5] => [1,6,2,3,4,7,5] => ? = 2 + 1
[[.,.],[.,[.,[[.,[.,.]],.]]]]
=> [1,6,5,7,4,3,2] => [1,7,2,3,6,5,4] => [1,6,2,3,7,4,5] => ? = 2 + 1
[[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [1,5,4,7,6,3,2] => [1,7,2,5,4,3,6] => [1,5,2,7,3,4,6] => ? = 2 + 1
[[.,.],[.,[[.,[.,[.,.]]],.]]]
=> [1,6,5,4,7,3,2] => [1,7,2,6,4,5,3] => [1,6,2,7,3,4,5] => ? = 2 + 1
[[.,.],[[.,[.,.]],[.,[.,.]]]]
=> [1,4,3,7,6,5,2] => [1,7,4,3,2,5,6] => [1,4,7,2,3,5,6] => ? = 1 + 1
[[.,.],[[.,[.,[.,.]]],[.,.]]]
=> [1,5,4,3,7,6,2] => [1,7,5,3,4,2,6] => [1,5,7,2,3,4,6] => ? = 1 + 1
[[.,.],[[.,[.,[.,[.,.]]]],.]]
=> [1,6,5,4,3,7,2] => [1,7,6,3,4,5,2] => [1,6,7,2,3,4,5] => ? = 1 + 1
[[.,.],[[.,[.,[[.,.],.]]],.]]
=> [1,5,6,4,3,7,2] => [1,7,6,3,5,4,2] => [1,5,6,2,7,3,4] => ? = 2 + 1
[[.,.],[[.,[[.,.],[.,.]]],.]]
=> [1,4,6,5,3,7,2] => [1,7,6,4,3,5,2] => [1,4,6,7,2,3,5] => ? = 1 + 1
[[.,.],[[.,[[.,[.,.]],.]],.]]
=> [1,5,4,6,3,7,2] => [1,7,6,5,4,3,2] => [1,5,6,7,2,3,4] => ? = 1 + 1
[[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => [2,1,7,3,4,5,6] => [2,1,7,3,4,5,6] => ? = 2 + 1
[[[.,[.,.]],.],[.,[.,[.,.]]]]
=> [2,1,3,7,6,5,4] => [2,1,3,7,4,5,6] => [2,1,3,7,4,5,6] => ? = 2 + 1
[[[[.,[.,.]],.],.],[.,[.,.]]]
=> [2,1,3,4,7,6,5] => [2,1,3,4,7,5,6] => [2,1,3,4,7,5,6] => ? = 2 + 1
[[.,[.,[.,[.,[.,.]]]]],[.,.]]
=> [5,4,3,2,1,7,6] => [5,1,2,3,4,7,6] => [5,1,2,3,4,7,6] => ? = 2 + 1
[[.,[[[[.,.],.],.],.]],[.,.]]
=> [2,3,4,5,1,7,6] => [5,2,3,4,1,7,6] => [2,3,4,5,1,7,6] => ? = 2 + 1
[[[.,.],[.,[.,[.,.]]]],[.,.]]
=> [1,5,4,3,2,7,6] => [1,5,2,3,4,7,6] => [1,5,2,3,4,7,6] => ? = 2 + 1
[[[.,[.,[.,[.,.]]]],.],[.,.]]
=> [4,3,2,1,5,7,6] => [4,1,2,3,5,7,6] => [4,1,2,3,5,7,6] => ? = 2 + 1
[[[.,[[[.,.],.],.]],.],[.,.]]
=> [2,3,4,1,5,7,6] => [4,2,3,1,5,7,6] => [2,3,4,1,5,7,6] => ? = 2 + 1
[[[[.,[.,[.,.]]],.],.],[.,.]]
=> [3,2,1,4,5,7,6] => [3,1,2,4,5,7,6] => [3,1,2,4,5,7,6] => ? = 2 + 1
[[.,[.,[.,[.,[.,[.,.]]]]]],.]
=> [6,5,4,3,2,1,7] => [6,1,2,3,4,5,7] => [6,1,2,3,4,5,7] => ? = 1 + 1
[[.,[.,[.,[.,[[.,.],.]]]]],.]
=> [5,6,4,3,2,1,7] => [6,1,2,3,5,4,7] => [5,1,2,3,6,4,7] => ? = 2 + 1
[[.,[.,[.,[[.,[.,.]],.]]]],.]
=> [5,4,6,3,2,1,7] => [6,1,2,5,4,3,7] => [5,1,2,6,3,4,7] => ? = 2 + 1
[[.,[.,[[.,[.,.]],[.,.]]]],.]
=> [4,3,6,5,2,1,7] => [6,1,4,3,2,5,7] => [4,1,6,2,3,5,7] => ? = 2 + 1
[[.,[.,[[.,[.,[.,.]]],.]]],.]
=> [5,4,3,6,2,1,7] => [6,1,5,3,4,2,7] => [5,1,6,2,3,4,7] => ? = 2 + 1
[[.,[[.,.],[.,[.,[.,.]]]]],.]
=> [2,6,5,4,3,1,7] => [6,2,1,3,4,5,7] => [2,6,1,3,4,5,7] => ? = 1 + 1
[[.,[[.,[.,.]],[.,[.,.]]]],.]
=> [3,2,6,5,4,1,7] => [6,3,2,1,4,5,7] => [3,6,1,2,4,5,7] => ? = 1 + 1
[[.,[[.,[.,[.,.]]],[.,.]]],.]
=> [4,3,2,6,5,1,7] => [6,4,2,3,1,5,7] => [4,6,1,2,3,5,7] => ? = 1 + 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: 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: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 67%
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000354: Permutations ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 67%
Values
[.,.]
=> [1,0]
=> [1,0]
=> [1] => ? = 0
[.,[.,.]]
=> [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
[.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,3,5,6,1,7,4] => ? = 2
[.,[.,[[[.,[.,[.,.]]],.],.]]]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [2,4,5,1,3,7,6] => ? = 3
[.,[[.,[.,[.,.]]],[[.,.],.]]]
=> [1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [3,4,1,5,2,7,6] => ? = 2
[.,[[[.,.],[.,[.,.]]],[.,.]]]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [4,5,1,2,6,7,3] => ? = 1
[.,[[.,[.,[.,[.,[.,.]]]]],.]]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [3,4,5,6,1,7,2] => ? = 1
[.,[[.,[[.,[.,[.,.]]],.]],.]]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [4,5,1,6,2,7,3] => ? = 1
[.,[[[.,.],[.,[.,[.,.]]]],.]]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [4,5,6,1,2,7,3] => ? = 1
[.,[[[[.,.],.],[[.,.],.]],.]]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,6,2,3,4,7,5] => ? = 1
[.,[[[[[.,.],.],.],[.,.]],.]]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,7,5] => ? = 1
[.,[[[.,[.,[.,[.,.]]]],.],.]]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [3,4,5,1,2,7,6] => ? = 2
[[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,7,1,2] => ? = 1
[[.,.],[.,[.,[.,[[.,.],.]]]]]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [3,4,5,1,7,2,6] => ? = 2
[[.,.],[.,[.,[[.,[.,.]],.]]]]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [3,4,6,1,7,2,5] => ? = 2
[[.,.],[.,[[.,.],[.,[.,.]]]]]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,5,6,7,2,4] => ? = 2
[[.,.],[.,[[.,.],[[.,.],.]]]]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [3,1,5,2,7,4,6] => ? = 3
[[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [3,5,1,6,7,2,4] => ? = 2
[[.,.],[.,[[[.,.],.],[.,.]]]]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [3,1,2,6,7,4,5] => ? = 2
[[.,.],[.,[[.,[.,[.,.]]],.]]]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [3,5,6,1,7,2,4] => ? = 2
[[.,.],[.,[[[.,.],[.,.]],.]]]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [3,6,1,2,7,4,5] => ? = 2
[[.,.],[.,[[[[.,.],.],.],.]]]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [3,1,2,4,7,5,6] => ? = 2
[[.,.],[[.,[.,.]],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [4,1,5,6,7,2,3] => ? = 1
[[.,.],[[.,[.,.]],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> [4,1,5,2,7,3,6] => ? = 2
[[.,.],[[.,[.,[.,.]]],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [4,5,1,6,7,2,3] => ? = 1
[[.,.],[[.,[[.,.],.]],[.,.]]]
=> [1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> [1,5,2,6,7,3,4] => ? = 1
[[.,.],[[[.,.],[.,.]],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [5,1,2,6,7,3,4] => ? = 1
[[.,.],[[[.,[.,.]],.],[.,.]]]
=> [1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [4,1,2,6,7,3,5] => ? = 2
[[.,.],[[.,[.,[.,[.,.]]]],.]]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [4,5,6,1,7,2,3] => ? = 1
[[.,.],[[.,[.,[[.,.],.]]],.]]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> [4,1,6,2,7,3,5] => ? = 2
[[.,.],[[.,[[.,.],[.,.]]],.]]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [1,5,6,2,7,3,4] => ? = 1
[[.,.],[[.,[[.,[.,.]],.]],.]]
=> [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [5,1,6,2,7,3,4] => ? = 1
[[.,.],[[[.,.],[.,[.,.]]],.]]
=> [1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [5,6,1,2,7,3,4] => ? = 1
[[.,.],[[[.,.],[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,6,2,3,7,4,5] => ? = 1
[[.,.],[[[.,[.,.]],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [4,6,1,2,7,3,5] => ? = 2
[[.,.],[[[[.,.],.],[.,.]],.]]
=> [1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,1,2,3,7,4,5] => ? = 1
[[.,.],[[[.,[[.,.],.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> [1,5,2,3,7,4,6] => ? = 2
[[.,.],[[[[.,.],[.,.]],.],.]]
=> [1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [5,1,2,3,7,4,6] => ? = 2
[[.,.],[[[[.,[.,.]],.],.],.]]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [4,1,2,3,7,5,6] => ? = 2
[[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,4,5,6,7,1,3] => ? = 2
[[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [4,5,6,7,1,2,3] => ? = 1
[[[.,.],.],[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [4,5,1,7,2,3,6] => ? = 2
[[[.,.],.],[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [4,1,6,7,2,3,5] => ? = 2
[[[.,.],.],[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [4,6,1,7,2,3,5] => ? = 2
[[[.,.],.],[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [4,1,2,7,3,5,6] => ? = 2
[[[.,.],.],[[.,.],[.,[.,.]]]]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,5,6,7,2,3,4] => ? = 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: St000834
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00326: Permutations —weak order rowmotion⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 67%
Mp00326: Permutations —weak order rowmotion⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 67%
Values
[.,.]
=> [1] => [1] => [1] => 0
[.,[.,.]]
=> [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
[.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [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
[.,[.,[[[.,[.,[.,.]]],.],.]]]
=> [5,4,3,6,7,2,1] => [6,3,4,5,1,2,7] => [5,6,2,3,4,1,7] => ? = 3
[.,[[.,.],[[.,[.,[.,.]]],.]]]
=> [2,6,5,4,7,3,1] => [4,5,6,2,1,3,7] => [5,4,6,1,2,3,7] => ? = 2
[.,[[.,[.,[.,.]]],[[.,.],.]]]
=> [4,3,2,6,7,5,1] => [6,2,3,4,1,5,7] => [5,2,3,4,6,1,7] => ? = 2
[.,[[[.,.],[.,[.,.]]],[.,.]]]
=> [2,5,4,3,7,6,1] => [3,4,5,2,1,6,7] => [5,4,1,2,3,6,7] => ? = 1
[.,[[.,[[.,[.,[.,.]]],.]],.]]
=> [5,4,3,6,2,7,1] => [3,4,5,2,6,1,7] => [6,4,1,2,3,5,7] => ? = 1
[.,[[.,[[[[.,.],.],.],.]],.]]
=> [3,4,5,6,2,7,1] => [5,4,3,2,6,1,7] => [6,4,3,2,1,5,7] => ? = 1
[.,[[[.,.],[.,[.,[.,.]]]],.]]
=> [2,6,5,4,3,7,1] => [3,4,5,6,2,1,7] => [6,5,1,2,3,4,7] => ? = 1
[.,[[[[.,.],.],[[.,.],.]],.]]
=> [2,3,5,6,4,7,1] => [5,4,6,3,2,1,7] => [6,5,4,2,1,3,7] => ? = 1
[.,[[[[[.,.],.],.],[.,.]],.]]
=> [2,3,4,6,5,7,1] => [5,6,4,3,2,1,7] => [6,5,4,3,1,2,7] => ? = 1
[.,[[[.,[.,[.,[.,.]]]],.],.]]
=> [5,4,3,2,6,7,1] => [6,2,3,4,5,1,7] => [6,2,3,4,5,1,7] => ? = 2
[[.,.],[.,[.,[[.,[.,.]],.]]]]
=> [1,6,5,7,4,3,2] => [5,6,2,3,4,7,1] => [7,3,4,5,1,2,6] => ? = 2
[[.,.],[.,[[.,.],[[.,.],.]]]]
=> [1,4,6,7,5,3,2] => [6,4,2,3,5,7,1] => [7,3,4,2,5,1,6] => ? = 3
[[.,.],[.,[[[.,.],.],[.,.]]]]
=> [1,4,5,7,6,3,2] => [5,4,2,3,6,7,1] => [7,3,4,2,1,5,6] => ? = 2
[[.,.],[.,[[.,[.,[.,.]]],.]]]
=> [1,6,5,4,7,3,2] => [4,5,6,2,3,7,1] => [7,4,5,1,2,3,6] => ? = 2
[[.,.],[.,[[[.,.],[.,.]],.]]]
=> [1,4,6,5,7,3,2] => [5,6,4,2,3,7,1] => [7,4,5,3,1,2,6] => ? = 2
[[.,.],[.,[[[[.,.],.],.],.]]]
=> [1,4,5,6,7,3,2] => [6,5,4,2,3,7,1] => [7,4,5,3,2,1,6] => ? = 2
[[.,.],[[.,.],[.,[[.,.],.]]]]
=> [1,3,6,7,5,4,2] => [6,3,2,4,5,7,1] => [7,3,2,4,5,1,6] => ? = 2
[[.,.],[[.,.],[[.,[.,.]],.]]]
=> [1,3,6,5,7,4,2] => [5,6,3,2,4,7,1] => [7,4,3,5,1,2,6] => ? = 2
[[.,.],[[.,.],[[[.,.],.],.]]]
=> [1,3,5,6,7,4,2] => [6,5,3,2,4,7,1] => [7,4,3,5,2,1,6] => ? = 2
[[.,.],[[.,[.,.]],[[.,.],.]]]
=> [1,4,3,6,7,5,2] => [6,3,4,2,5,7,1] => [7,4,2,3,5,1,6] => ? = 2
[[.,.],[[[.,.],.],[[.,.],.]]]
=> [1,3,4,6,7,5,2] => [6,4,3,2,5,7,1] => [7,4,3,2,5,1,6] => ? = 2
[[.,.],[[[.,.],[.,.]],[.,.]]]
=> [1,3,5,4,7,6,2] => [4,5,3,2,6,7,1] => [7,4,3,1,2,5,6] => ? = 1
[[.,.],[[[.,[.,.]],.],[.,.]]]
=> [1,4,3,5,7,6,2] => [5,3,4,2,6,7,1] => [7,4,2,3,1,5,6] => ? = 2
[[.,.],[[[[.,.],.],.],[.,.]]]
=> [1,3,4,5,7,6,2] => [5,4,3,2,6,7,1] => [7,4,3,2,1,5,6] => ? = 1
[[.,.],[[.,[.,[[.,.],.]]],.]]
=> [1,5,6,4,3,7,2] => [5,3,4,6,2,7,1] => [7,5,2,3,1,4,6] => ? = 2
[[.,.],[[.,[[.,.],[.,.]]],.]]
=> [1,4,6,5,3,7,2] => [4,3,5,6,2,7,1] => [7,5,2,1,3,4,6] => ? = 1
[[.,.],[[.,[[.,[.,.]],.]],.]]
=> [1,5,4,6,3,7,2] => [4,5,3,6,2,7,1] => [7,5,3,1,2,4,6] => ? = 1
[[.,.],[[.,[[[.,.],.],.]],.]]
=> [1,4,5,6,3,7,2] => [5,4,3,6,2,7,1] => [7,5,3,2,1,4,6] => ? = 1
[[.,.],[[[.,.],[.,[.,.]]],.]]
=> [1,3,6,5,4,7,2] => [4,5,6,3,2,7,1] => [7,5,4,1,2,3,6] => ? = 1
[[.,.],[[[.,.],[[.,.],.]],.]]
=> [1,3,5,6,4,7,2] => [5,4,6,3,2,7,1] => [7,5,4,2,1,3,6] => ? = 1
[[.,.],[[[.,[.,.]],[.,.]],.]]
=> [1,4,3,6,5,7,2] => [5,6,3,4,2,7,1] => [7,5,3,4,1,2,6] => ? = 2
[[.,.],[[[[.,.],.],[.,.]],.]]
=> [1,3,4,6,5,7,2] => [5,6,4,3,2,7,1] => [7,5,4,3,1,2,6] => ? = 1
[[.,.],[[[.,[[.,.],.]],.],.]]
=> [1,4,5,3,6,7,2] => [6,4,3,5,2,7,1] => [7,5,3,2,4,1,6] => ? = 2
[[.,.],[[[[.,.],[.,.]],.],.]]
=> [1,3,5,4,6,7,2] => [6,4,5,3,2,7,1] => [7,5,4,2,3,1,6] => ? = 2
[[.,.],[[[[.,[.,.]],.],.],.]]
=> [1,4,3,5,6,7,2] => [6,5,3,4,2,7,1] => [7,5,3,4,2,1,6] => ? = 2
[[.,.],[[[[[.,.],.],.],.],.]]
=> [1,3,4,5,6,7,2] => [6,5,4,3,2,7,1] => [7,5,4,3,2,1,6] => ? = 1
[[[.,.],.],[.,[.,[[.,.],.]]]]
=> [1,2,6,7,5,4,3] => [6,3,4,5,7,2,1] => [7,6,2,3,4,1,5] => ? = 2
[[[.,.],.],[.,[[.,.],[.,.]]]]
=> [1,2,5,7,6,4,3] => [5,3,4,6,7,2,1] => [7,6,2,3,1,4,5] => ? = 2
[[[.,.],.],[.,[[.,[.,.]],.]]]
=> [1,2,6,5,7,4,3] => [5,6,3,4,7,2,1] => [7,6,3,4,1,2,5] => ? = 2
[[[.,.],.],[.,[[[.,.],.],.]]]
=> [1,2,5,6,7,4,3] => [6,5,3,4,7,2,1] => [7,6,3,4,2,1,5] => ? = 2
[[[.,.],.],[[.,.],[.,[.,.]]]]
=> [1,2,4,7,6,5,3] => [4,3,5,6,7,2,1] => [7,6,2,1,3,4,5] => ? = 1
[[[.,.],.],[[.,.],[[.,.],.]]]
=> [1,2,4,6,7,5,3] => [6,4,3,5,7,2,1] => [7,6,3,2,4,1,5] => ? = 2
[[[.,.],.],[[.,[.,.]],[.,.]]]
=> [1,2,5,4,7,6,3] => [4,5,3,6,7,2,1] => [7,6,3,1,2,4,5] => ? = 1
[[[.,.],.],[[[.,.],.],[.,.]]]
=> [1,2,4,5,7,6,3] => [5,4,3,6,7,2,1] => [7,6,3,2,1,4,5] => ? = 1
[[[.,.],.],[[.,[.,[.,.]]],.]]
=> [1,2,6,5,4,7,3] => [4,5,6,3,7,2,1] => [7,6,4,1,2,3,5] => ? = 1
[[[.,.],.],[[.,[[.,.],.]],.]]
=> [1,2,5,6,4,7,3] => [5,4,6,3,7,2,1] => [7,6,4,2,1,3,5] => ? = 1
[[[.,.],.],[[[.,.],[.,.]],.]]
=> [1,2,4,6,5,7,3] => [5,6,4,3,7,2,1] => [7,6,4,3,1,2,5] => ? = 1
[[[.,.],.],[[[.,[.,.]],.],.]]
=> [1,2,5,4,6,7,3] => [6,4,5,3,7,2,1] => [7,6,4,2,3,1,5] => ? = 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: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 67%
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
St001212: Dyck paths ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 67%
Values
[.,.]
=> [1,0]
=> [1,0]
=> 0
[.,[.,.]]
=> [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
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [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
[.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 2
[.,[.,[[[.,[.,[.,.]]],.],.]]]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> ? = 3
[.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[.,[[.,.],[[.,[.,[.,.]]],.]]]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 2
[.,[[.,[.,[.,.]]],[[.,.],.]]]
=> [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]
=> ? = 2
[.,[[[[.,.],.],.],[.,[.,.]]]]
=> [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
[.,[[[[.,.],.],.],[[.,.],.]]]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 2
[.,[[[.,.],[.,[.,.]]],[.,.]]]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> ? = 1
[.,[[[[[.,.],.],.],.],[.,.]]]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 1
[.,[[.,[.,[.,[.,[.,.]]]]],.]]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 1
[.,[[.,[[.,[.,[.,.]]],.]],.]]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 1
[.,[[.,[[[[.,.],.],.],.]],.]]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> ? = 1
[.,[[[.,.],[.,[.,[.,.]]]],.]]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 1
[.,[[[[.,.],.],[[.,.],.]],.]]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> ? = 1
[.,[[[[[.,.],.],.],[.,.]],.]]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 1
[.,[[[.,[.,[.,[.,.]]]],.],.]]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> ? = 2
[.,[[[[[[.,.],.],.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[[.,.],[.,[.,[.,[[.,.],.]]]]]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 2
[[.,.],[.,[.,[[.,[.,.]],.]]]]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> ? = 2
[[.,.],[.,[[.,.],[.,[.,.]]]]]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 2
[[.,.],[.,[[.,.],[[.,.],.]]]]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 3
[[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 2
[[.,.],[.,[[[.,.],.],[.,.]]]]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 2
[[.,.],[.,[[.,[.,[.,.]]],.]]]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> ? = 2
[[.,.],[.,[[[.,.],[.,.]],.]]]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 2
[[.,.],[.,[[[[.,.],.],.],.]]]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 2
[[.,.],[[.,.],[.,[.,[.,.]]]]]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[[.,.],[[.,.],[.,[[.,.],.]]]]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> ? = 2
[[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> ? = 2
[[.,.],[[.,.],[[.,[.,.]],.]]]
=> [1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> ? = 2
[[.,.],[[.,.],[[[.,.],.],.]]]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> ? = 2
[[.,.],[[.,[.,.]],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 1
[[.,.],[[.,[.,.]],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 2
[[.,.],[[[.,.],.],[.,[.,.]]]]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 1
[[.,.],[[[.,.],.],[[.,.],.]]]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> ? = 2
[[.,.],[[.,[.,[.,.]]],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 1
[[.,.],[[.,[[.,.],.]],[.,.]]]
=> [1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> ? = 1
[[.,.],[[[.,.],[.,.]],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 1
[[.,.],[[[.,[.,.]],.],[.,.]]]
=> [1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 2
[[.,.],[[[[.,.],.],.],[.,.]]]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 1
[[.,.],[[.,[.,[.,[.,.]]]],.]]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 1
[[.,.],[[.,[.,[[.,.],.]]],.]]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> ? = 2
[[.,.],[[.,[[.,.],[.,.]]],.]]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> ? = 1
[[.,.],[[.,[[.,[.,.]],.]],.]]
=> [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> ? = 1
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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