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Your data matches 53 different statistics following compositions of up to 3 maps.
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Matching statistic: St000390
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(load all 3 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => [1,2] => 0 => 0
[[.,.],.]
=> [1,2] => [2,1] => 1 => 1
[.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => 00 => 0
[.,[[.,.],.]]
=> [2,3,1] => [1,3,2] => 01 => 1
[[.,.],[.,.]]
=> [3,1,2] => [2,1,3] => 10 => 1
[[.,[.,.]],.]
=> [2,1,3] => [3,1,2] => 10 => 1
[[[.,.],.],.]
=> [1,2,3] => [3,2,1] => 11 => 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => 000 => 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,2,4,3] => 001 => 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,3,2,4] => 010 => 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,4,2,3] => 010 => 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,4,3,2] => 011 => 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [2,1,3,4] => 100 => 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [2,1,4,3] => 101 => 2
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [3,1,2,4] => 100 => 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [3,2,1,4] => 110 => 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [4,1,2,3] => 100 => 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [4,1,3,2] => 101 => 2
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [4,2,1,3] => 110 => 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [4,3,1,2] => 110 => 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [4,3,2,1] => 111 => 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,2,3,4,5] => 0000 => 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,2,3,5,4] => 0001 => 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,2,4,3,5] => 0010 => 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,5,3,4] => 0010 => 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,5,4,3] => 0011 => 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,3,2,4,5] => 0100 => 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,3,2,5,4] => 0101 => 2
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,4,2,3,5] => 0100 => 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,4,3,2,5] => 0110 => 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,5,2,3,4] => 0100 => 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,5,2,4,3] => 0101 => 2
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,5,3,2,4] => 0110 => 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,5,4,2,3] => 0110 => 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,5,4,3,2] => 0111 => 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [2,1,3,4,5] => 1000 => 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [2,1,3,5,4] => 1001 => 2
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [2,1,4,3,5] => 1010 => 2
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [2,1,5,3,4] => 1010 => 2
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [2,1,5,4,3] => 1011 => 2
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [3,1,2,4,5] => 1000 => 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [3,1,2,5,4] => 1001 => 2
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [3,2,1,4,5] => 1100 => 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [3,2,1,5,4] => 1101 => 2
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [4,1,2,3,5] => 1000 => 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [4,1,3,2,5] => 1010 => 2
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [4,2,1,3,5] => 1100 => 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [4,3,1,2,5] => 1100 => 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [4,3,2,1,5] => 1110 => 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [5,1,2,3,4] => 1000 => 1
Description
The number of runs of ones in a binary word.
Matching statistic: St001280
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => [1,1] => [1,1]
=> 0
[[.,.],.]
=> [1,2] => [2] => [2]
=> 1
[.,[.,[.,.]]]
=> [3,2,1] => [1,1,1] => [1,1,1]
=> 0
[.,[[.,.],.]]
=> [2,3,1] => [2,1] => [2,1]
=> 1
[[.,.],[.,.]]
=> [3,1,2] => [1,2] => [2,1]
=> 1
[[.,[.,.]],.]
=> [2,1,3] => [1,2] => [2,1]
=> 1
[[[.,.],.],.]
=> [1,2,3] => [3] => [3]
=> 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,1,1,1] => [1,1,1,1]
=> 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [2,1,1] => [2,1,1]
=> 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,2,1] => [2,1,1]
=> 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,2,1] => [2,1,1]
=> 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [3,1] => [3,1]
=> 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,1,2] => [2,1,1]
=> 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [2,2] => [2,2]
=> 2
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,1,2] => [2,1,1]
=> 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,3] => [3,1]
=> 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,1,2] => [2,1,1]
=> 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,2] => [2,2]
=> 2
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,3] => [3,1]
=> 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,3] => [3,1]
=> 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [4] => [4]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,1,1,1,1] => [1,1,1,1,1]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [2,1,1,1] => [2,1,1,1]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,2,1,1] => [2,1,1,1]
=> 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,1,1] => [2,1,1,1]
=> 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,1,1] => [3,1,1]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,1,2,1] => [2,1,1,1]
=> 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [2,2,1] => [2,2,1]
=> 2
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,1,2,1] => [2,1,1,1]
=> 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,3,1] => [3,1,1]
=> 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,1,2,1] => [2,1,1,1]
=> 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [2,2,1] => [2,2,1]
=> 2
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,3,1] => [3,1,1]
=> 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,3,1] => [3,1,1]
=> 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [4,1] => [4,1]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,1,1,2] => [2,1,1,1]
=> 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [2,1,2] => [2,2,1]
=> 2
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [1,2,2] => [2,2,1]
=> 2
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [1,2,2] => [2,2,1]
=> 2
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [3,2] => [3,2]
=> 2
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [1,1,1,2] => [2,1,1,1]
=> 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [2,1,2] => [2,2,1]
=> 2
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,1,3] => [3,1,1]
=> 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [2,3] => [3,2]
=> 2
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [1,1,1,2] => [2,1,1,1]
=> 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [1,2,2] => [2,2,1]
=> 2
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,1,3] => [3,1,1]
=> 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [1,1,3] => [3,1,1]
=> 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,4] => [4,1]
=> 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [1,1,1,2] => [2,1,1,1]
=> 1
[.,[.,[[.,.],[[[.,.],[.,.]],.]]]]
=> [7,5,6,8,3,4,2,1] => ? => ?
=> ? = 2
[.,[.,[[[.,.],[.,[.,.]]],[.,.]]]]
=> [8,6,5,3,4,7,2,1] => ? => ?
=> ? = 1
[.,[.,[[[.,.],[[.,.],[.,.]]],.]]]
=> [7,5,6,3,4,8,2,1] => ? => ?
=> ? = 2
[.,[.,[[[.,[[.,.],[.,.]]],.],.]]]
=> [6,4,5,3,7,8,2,1] => ? => ?
=> ? = 2
[.,[.,[[[[.,.],[[.,.],.]],.],.]]]
=> [5,6,3,4,7,8,2,1] => ? => ?
=> ? = 2
[.,[[.,.],[.,[.,[[[.,.],.],.]]]]]
=> [6,7,8,5,4,2,3,1] => ? => ?
=> ? = 2
[.,[[.,.],[.,[[.,.],[[.,.],.]]]]]
=> [7,8,5,6,4,2,3,1] => ? => ?
=> ? = 3
[.,[[.,.],[[[.,[.,.]],.],[.,.]]]]
=> [8,5,4,6,7,2,3,1] => ? => ?
=> ? = 2
[.,[[[.,.],.],[[[.,.],[.,.]],.]]]
=> [7,5,6,8,2,3,4,1] => ? => ?
=> ? = 2
[.,[[[.,[.,.]],.],[[.,[.,.]],.]]]
=> [7,6,8,3,2,4,5,1] => ? => ?
=> ? = 2
[.,[[[.,.],[[.,.],.]],[.,[.,.]]]]
=> [8,7,4,5,2,3,6,1] => ? => ?
=> ? = 2
[.,[[[.,[.,.]],[.,.]],[[.,.],.]]]
=> [7,8,5,3,2,4,6,1] => ? => ?
=> ? = 2
[.,[[[[.,.],.],[.,.]],[.,[.,.]]]]
=> [8,7,5,2,3,4,6,1] => ? => ?
=> ? = 1
[.,[[[[.,.],[.,.]],.],[.,[.,.]]]]
=> [8,7,4,2,3,5,6,1] => ? => ?
=> ? = 1
[.,[[[[.,.],.],[.,[.,.]]],[.,.]]]
=> [8,6,5,2,3,4,7,1] => ? => ?
=> ? = 1
[.,[[[[.,[.,.]],[.,.]],.],[.,.]]]
=> [8,5,3,2,4,6,7,1] => ? => ?
=> ? = 1
[.,[[.,[[[.,[.,.]],.],[.,.]]],.]]
=> [7,4,3,5,6,2,8,1] => ? => ?
=> ? = 2
[.,[[[.,[.,.]],[.,[.,[.,.]]]],.]]
=> [7,6,5,3,2,4,8,1] => ? => ?
=> ? = 1
[.,[[[.,[.,.]],[[[.,.],.],.]],.]]
=> [5,6,7,3,2,4,8,1] => ? => ?
=> ? = 2
[.,[[[[.,.],.],[.,[[.,.],.]]],.]]
=> [6,7,5,2,3,4,8,1] => ? => ?
=> ? = 2
[.,[[[[.,.],.],[[.,[.,.]],.]],.]]
=> [6,5,7,2,3,4,8,1] => ? => ?
=> ? = 2
[.,[[[.,[.,[.,[[.,.],.]]]],.],.]]
=> [5,6,4,3,2,7,8,1] => ? => ?
=> ? = 2
[.,[[[[.,[.,.]],[[.,.],.]],.],.]]
=> [5,6,3,2,4,7,8,1] => ? => ?
=> ? = 2
[.,[[[[[.,.],.],[[.,.],.]],.],.]]
=> [5,6,2,3,4,7,8,1] => ? => ?
=> ? = 2
[.,[[[[.,[.,[.,.]]],[.,.]],.],.]]
=> [6,4,3,2,5,7,8,1] => ? => ?
=> ? = 1
[[.,.],[.,[[.,[.,.]],[[.,.],.]]]]
=> [7,8,5,4,6,3,1,2] => ? => ?
=> ? = 3
[[.,.],[.,[[.,[[[.,.],.],.]],.]]]
=> [5,6,7,4,8,3,1,2] => ? => ?
=> ? = 3
[[.,.],[[[.,[[.,.],[.,.]]],.],.]]
=> [6,4,5,3,7,8,1,2] => ? => ?
=> ? = 3
[[.,[.,.]],[[[[.,.],[.,.]],.],.]]
=> [6,4,5,7,8,2,1,3] => ? => ?
=> ? = 2
[[.,[.,[[.,.],[.,.]]]],[[.,.],.]]
=> [7,8,5,3,4,2,1,6] => ? => ?
=> ? = 3
[[.,[[.,.],[.,[.,.]]]],[[.,.],.]]
=> [7,8,5,4,2,3,1,6] => ? => ?
=> ? = 3
[[[[[.,.],[.,.]],.],.],[.,[.,.]]]
=> [8,7,3,1,2,4,5,6] => ? => ?
=> ? = 1
[[.,[.,[.,[[.,[.,.]],.]]]],[.,.]]
=> [8,5,4,6,3,2,1,7] => ? => ?
=> ? = 2
[[.,[.,[[[.,.],[.,.]],.]]],[.,.]]
=> [8,5,3,4,6,2,1,7] => ? => ?
=> ? = 2
[[[[[.,[.,.]],.],.],[.,.]],[.,.]]
=> [8,6,2,1,3,4,5,7] => ? => ?
=> ? = 1
[[[.,[[.,[[.,.],.]],.]],.],[.,.]]
=> [8,3,4,2,5,1,6,7] => ? => ?
=> ? = 3
[[.,[.,[.,[[[.,.],[.,.]],.]]]],.]
=> [6,4,5,7,3,2,1,8] => ? => ?
=> ? = 2
[[.,[.,[[[.,[.,.]],.],[.,.]]]],.]
=> [7,4,3,5,6,2,1,8] => ? => ?
=> ? = 2
[[.,[.,[[[.,[.,.]],[.,.]],.]]],.]
=> [6,4,3,5,7,2,1,8] => ? => ?
=> ? = 2
[[.,[.,[[[[.,.],.],[.,.]],.]]],.]
=> [6,3,4,5,7,2,1,8] => ? => ?
=> ? = 2
[[.,[[.,.],[[[.,.],[.,.]],.]]],.]
=> [6,4,5,7,2,3,1,8] => ? => ?
=> ? = 3
[[.,[[.,[.,.]],[.,[[.,.],.]]]],.]
=> [6,7,5,3,2,4,1,8] => ? => ?
=> ? = 3
[[.,[[.,[.,[[[.,.],.],.]]],.]],.]
=> [4,5,6,3,2,7,1,8] => ? => ?
=> ? = 3
[[.,[[.,[[.,.],[.,[.,.]]]],.]],.]
=> [6,5,3,4,2,7,1,8] => ? => ?
=> ? = 3
[[.,[[[.,.],[.,[.,[.,.]]]],.]],.]
=> [6,5,4,2,3,7,1,8] => ? => ?
=> ? = 2
[[.,[[[.,[.,[.,.]]],[.,.]],.]],.]
=> [6,4,3,2,5,7,1,8] => ? => ?
=> ? = 2
[[[.,.],[.,[[.,[[.,.],.]],.]]],.]
=> [5,6,4,7,3,1,2,8] => ? => ?
=> ? = 3
[[[.,.],[[.,.],[[.,[.,.]],.]]],.]
=> [6,5,7,3,4,1,2,8] => ? => ?
=> ? = 3
[[[.,.],[[.,[[.,.],[.,.]]],.]],.]
=> [6,4,5,3,7,1,2,8] => ? => ?
=> ? = 3
[[[.,.],[[[.,[.,[.,.]]],.],.]],.]
=> [5,4,3,6,7,1,2,8] => ? => ?
=> ? = 2
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St000291
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [1,0,1,0]
=> [1,1] => 11 => 0
[[.,.],.]
=> [1,1,0,0]
=> [2] => 10 => 1
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,1,1] => 111 => 0
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,2] => 110 => 1
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [2,1] => 101 => 1
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [2,1] => 101 => 1
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [3] => 100 => 1
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1111 => 0
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 1110 => 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => 1101 => 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => 1101 => 1
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 1100 => 1
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => 1011 => 1
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 1010 => 2
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [2,1,1] => 1011 => 1
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 1001 => 1
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [2,1,1] => 1011 => 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [2,2] => 1010 => 2
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [3,1] => 1001 => 1
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [3,1] => 1001 => 1
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [4] => 1000 => 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 11111 => 0
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 11110 => 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 11101 => 1
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => 11101 => 1
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 11100 => 1
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 11011 => 1
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 11010 => 2
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => 11011 => 1
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 11001 => 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => 11011 => 1
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => 11010 => 2
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => 11001 => 1
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => 11001 => 1
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 11000 => 1
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 10111 => 1
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 10110 => 2
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 10101 => 2
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => 10101 => 2
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 10100 => 2
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => 10111 => 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => 10110 => 2
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => 10011 => 1
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => 10010 => 2
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => 10111 => 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => 10101 => 2
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => 10011 => 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,1,1] => 10011 => 1
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => 10001 => 1
[[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => 10111 => 1
[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1] => 1111111111 => ? = 0
[.,[.,[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,3] => 1111111100 => ? = 1
[.,[.,[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,4] => 1111111000 => ? = 1
[.,[.,[.,[.,[.,[[[[[.,.],.],.],.],.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,5] => 1111110000 => ? = 1
[.,[.,[.,[[[[[[[.,.],.],.],.],.],.],.]]]]
=> [1,0,1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,7] => 1111000000 => ? = 1
[.,[.,[[[[[[[[.,.],.],.],.],.],.],.],.]]]
=> [1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,8] => 1110000000 => ? = 1
[.,[[[[[[[[[.,.],.],.],.],.],.],.],.],.]]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,9] => 1100000000 => ? = 1
[.,[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]]
=> ?
=> ? => ? => ? = 1
[.,[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]]
=> ?
=> ? => ? => ? = 1
[.,[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]]
=> ?
=> ? => ? => ? = 1
[.,[.,[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]]]
=> ?
=> ? => ? => ? = 1
[.,[.,[.,[.,[.,[[[.,.],[.,.]],.]]]]]]
=> ?
=> ? => ? => ? = 1
[.,[.,[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]]]
=> ?
=> ? => ? => ? = 1
[.,[.,[.,[.,[[[.,.],[.,[.,.]]],.]]]]]
=> ?
=> ? => ? => ? = 1
[.,[.,[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,2,1,1,1] => 1111110111 => ? = 1
[.,[.,[.,[[[.,.],[.,[.,[.,.]]]],.]]]]
=> ?
=> ? => ? => ? = 1
[.,[.,[.,[.,[[.,[.,[.,[.,[.,.]]]]],.]]]]]
=> ?
=> ? => ? => ? = 1
[.,[.,[[[.,.],[.,[.,[.,[.,.]]]]],.]]]
=> ?
=> ? => ? => ? = 1
[.,[.,[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]]]
=> ?
=> ? => ? => ? = 1
[.,[[[.,.],[.,[.,[.,[.,[.,.]]]]]],.]]
=> ?
=> ? => ? => ? = 1
[.,[.,[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]]]
=> ?
=> ? => ? => ? = 1
[.,[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]]
=> ?
=> ? => ? => ? = 1
[[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[[[[[.,[.,.]],[.,.]],[.,.]],[.,.]],[.,.]]
=> [1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0]
=> [5,1,1,1,1,1] => 1000011111 => ? = 1
[[[[[[[[[.,.],.],.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => ? = 1
[[[[[[[.,[.,.]],.],.],.],.],.],[.,.]]
=> ?
=> ? => ? => ? = 1
[.,[[[[[[[.,[.,.]],.],.],.],.],.],.]]
=> ?
=> ? => ? => ? = 1
[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],[.,.]]
=> ?
=> ? => ? => ? = 1
[[[[[[[[.,[.,.]],.],.],.],.],.],.],[.,.]]
=> ?
=> ? => ? => ? = 1
[.,[[[[[[.,[[.,.],.]],.],.],.],.],.]]
=> ?
=> ? => ? => ? = 2
[.,[[[[[[[[.,[.,.]],.],.],.],.],.],.],.]]
=> ?
=> ? => ? => ? = 1
[.,[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.]]
=> ?
=> ? => ? => ? = 1
[[[[[[[.,.],.],.],.],.],.],[[.,.],.]]
=> ?
=> ? => ? => ? = 2
[[[[[[.,.],.],.],.],.],[[[.,.],.],.]]
=> ?
=> ? => ? => ? = 2
[[[[[.,.],.],.],.],[[[[.,.],.],.],.]]
=> ?
=> ? => ? => ? = 2
[[[[.,.],.],.],[[[[[.,.],.],.],.],.]]
=> ?
=> ? => ? => ? = 2
[[[.,.],.],[[[[[[.,.],.],.],.],.],.]]
=> ?
=> ? => ? => ? = 2
[[[[[[[[.,.],.],.],.],.],.],.],[[.,.],.]]
=> ?
=> ? => ? => ? = 2
[[[[[[[.,.],.],.],.],.],.],[[[.,.],.],.]]
=> ?
=> ? => ? => ? = 2
[[[[[[.,.],.],.],.],.],[[[[.,.],.],.],.]]
=> ?
=> ? => ? => ? = 2
[[[[[.,.],.],.],.],[[[[[.,.],.],.],.],.]]
=> ?
=> ? => ? => ? = 2
[[[[.,.],.],.],[[[[[[.,.],.],.],.],.],.]]
=> ?
=> ? => ? => ? = 2
[[[.,.],.],[[[[[[[.,.],.],.],.],.],.],.]]
=> ?
=> ? => ? => ? = 2
[[.,.],[[[[[[[[.,.],.],.],.],.],.],.],.]]
=> [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,8] => 1010000000 => ? = 2
[[.,.],[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> ?
=> ? => ? => ? = 2
[[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1,1,1,1,1,1,1] => 10111111111 => ? = 1
[[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]],.]
=> ?
=> ? => ? => ? = 2
[[.,.],[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
=> ?
=> ? => ? => ? = 2
[[.,.],[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> ?
=> ? => ? => ? = 2
Description
The number of descents of a binary word.
Matching statistic: St000157
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 100%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [1,0,1,0]
=> [1,2] => [[1,2]]
=> 0
[[.,.],.]
=> [1,1,0,0]
=> [2,1] => [[1],[2]]
=> 1
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,2,3] => [[1,2,3]]
=> 0
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,3,2] => [[1,2],[3]]
=> 1
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [2,1,3] => [[1,3],[2]]
=> 1
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [2,3,1] => [[1,3],[2]]
=> 1
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [3,1,2] => [[1,2],[3]]
=> 1
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [[1,2,3,4]]
=> 0
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [[1,2,3],[4]]
=> 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [[1,2,4],[3]]
=> 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [[1,2,4],[3]]
=> 1
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [[1,2,3],[4]]
=> 1
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [[1,3,4],[2]]
=> 1
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [[1,3],[2,4]]
=> 2
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [[1,3,4],[2]]
=> 1
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [[1,2,4],[3]]
=> 1
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [[1,3,4],[2]]
=> 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [[1,3],[2,4]]
=> 2
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [[1,2],[3,4]]
=> 1
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [[1,2],[3,4]]
=> 1
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [[1,2,3],[4]]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 1
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [[1,2,3,5],[4]]
=> 1
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [[1,2,3,4],[5]]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [[1,2,4,5],[3]]
=> 1
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [[1,2,4],[3,5]]
=> 2
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [[1,2,4,5],[3]]
=> 1
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [[1,2,3,5],[4]]
=> 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [[1,2,4,5],[3]]
=> 1
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [[1,2,4],[3,5]]
=> 2
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [[1,2,3],[4,5]]
=> 1
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [[1,2,3],[4,5]]
=> 1
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [[1,2,3,4],[5]]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [[1,3,4,5],[2]]
=> 1
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [[1,3,4],[2,5]]
=> 2
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [[1,3,5],[2,4]]
=> 2
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [[1,3,5],[2,4]]
=> 2
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [[1,3,4],[2,5]]
=> 2
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [[1,3,4,5],[2]]
=> 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [[1,3,4],[2,5]]
=> 2
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [[1,2,4,5],[3]]
=> 1
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => [[1,2,4],[3,5]]
=> 2
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [[1,3,4,5],[2]]
=> 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [[1,3,5],[2,4]]
=> 2
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => [[1,2,5],[3,4]]
=> 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => [[1,2,5],[3,4]]
=> 1
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => [[1,2,3,5],[4]]
=> 1
[[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [[1,3,4,5],[2]]
=> 1
[.,[.,[[.,.],[[.,[[.,.],.]],.]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,2,4,3,6,8,5,7] => ?
=> ? = 3
[.,[.,[[.,[.,.]],[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,2,4,5,3,6,7,8] => ?
=> ? = 1
[.,[.,[[.,[.,.]],[.,[[.,.],.]]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,2,4,5,3,6,8,7] => ?
=> ? = 2
[.,[.,[[.,[[.,.],[.,.]]],[.,.]]]]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,2,4,6,3,7,5,8] => ?
=> ? = 2
[.,[.,[[.,[[.,[.,.]],.]],[.,.]]]]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,2,4,6,7,3,5,8] => ?
=> ? = 2
[.,[.,[[[.,.],[.,[.,.]]],[.,.]]]]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,2,5,3,6,7,4,8] => ?
=> ? = 1
[.,[.,[[[.,[.,[.,.]]],.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,2,5,6,7,3,4,8] => ?
=> ? = 1
[.,[.,[[.,[[.,.],[.,[.,.]]]],.]]]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,2,4,6,3,7,8,5] => ?
=> ? = 2
[.,[.,[[.,[[.,.],[[.,.],.]]],.]]]
=> [1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,2,4,6,3,8,5,7] => ?
=> ? = 3
[.,[.,[[.,[[.,[.,.]],[.,.]]],.]]]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,2,4,6,7,3,8,5] => ?
=> ? = 2
[.,[.,[[.,[[.,[.,[.,.]]],.]],.]]]
=> [1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,2,4,6,7,8,3,5] => ?
=> ? = 2
[.,[.,[[[.,.],[.,[[.,.],.]]],.]]]
=> [1,0,1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,2,5,3,6,8,4,7] => ?
=> ? = 2
[.,[.,[[[.,.],[[.,[.,.]],.]],.]]]
=> [1,0,1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,2,5,3,7,8,4,6] => ?
=> ? = 2
[.,[.,[[[.,[.,[[.,.],.]]],.],.]]]
=> [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,2,5,6,8,3,4,7] => ?
=> ? = 2
[.,[[.,.],[.,[[[.,.],.],[.,.]]]]]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,3,2,4,7,5,6,8] => ?
=> ? = 2
[.,[[.,.],[[[.,.],.],[[.,.],.]]]]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,3,2,6,4,5,8,7] => ?
=> ? = 3
[.,[[.,.],[[.,[.,[[.,.],.]]],.]]]
=> [1,0,1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [1,3,2,5,6,8,4,7] => ?
=> ? = 3
[.,[[.,.],[[.,[[.,.],[.,.]]],.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,5,7,4,8,6] => ?
=> ? = 3
[.,[[.,.],[[[.,.],[.,[.,.]]],.]]]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [1,3,2,6,4,7,8,5] => ?
=> ? = 2
[.,[[.,.],[[[[.,.],.],[.,.]],.]]]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [1,3,2,7,4,5,8,6] => ?
=> ? = 2
[.,[[.,.],[[[[.,[.,.]],.],.],.]]]
=> [1,0,1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [1,3,2,7,8,4,5,6] => ?
=> ? = 2
[.,[[.,[.,.]],[[.,.],[.,[.,.]]]]]
=> [1,0,1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,4,2,6,5,7,8] => ?
=> ? = 2
[.,[[.,[.,.]],[[.,[[.,.],.]],.]]]
=> [1,0,1,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [1,3,4,2,6,8,5,7] => ?
=> ? = 3
[.,[[.,[.,[.,.]]],[.,[[.,.],.]]]]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,3,4,5,2,6,8,7] => ?
=> ? = 2
[.,[[.,[[.,.],.]],[[.,[.,.]],.]]]
=> [1,0,1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4,7,8,6] => ?
=> ? = 3
[.,[[[.,.],[.,.]],[[.,[.,.]],.]]]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> [1,4,2,5,3,7,8,6] => ?
=> ? = 2
[.,[[[.,[.,.]],.],[.,[[.,.],.]]]]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,1,0,0]
=> [1,4,5,2,3,6,8,7] => ?
=> ? = 2
[.,[[.,[[.,.],[.,.]]],[.,[.,.]]]]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [1,3,5,2,6,4,7,8] => ?
=> ? = 2
[.,[[.,[[.,.],[.,.]]],[[.,.],.]]]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [1,3,5,2,6,4,8,7] => ?
=> ? = 3
[.,[[[.,.],[.,[.,.]]],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [1,4,2,5,6,3,7,8] => ?
=> ? = 1
[.,[[[.,[.,[.,.]]],.],[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,1,0,0]
=> [1,4,5,6,2,3,8,7] => ?
=> ? = 2
[.,[[.,[[[.,.],.],[.,.]]],[.,.]]]
=> [1,0,1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [1,3,6,2,4,7,5,8] => ?
=> ? = 2
[.,[[[.,.],[.,[[.,.],.]]],[.,.]]]
=> [1,0,1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> [1,4,2,5,7,3,6,8] => ?
=> ? = 2
[.,[[[[.,[.,.]],[.,.]],.],[.,.]]]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,5,6,2,7,3,4,8] => ?
=> ? = 1
[.,[[[[.,[[.,.],.]],.],.],[.,.]]]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,5,7,2,3,4,6,8] => ?
=> ? = 2
[.,[[.,[.,[.,[[.,.],[.,.]]]]],.]]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,3,4,5,7,2,8,6] => ?
=> ? = 2
[.,[[.,[.,[[.,.],[.,[.,.]]]]],.]]
=> [1,0,1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,3,4,6,2,7,8,5] => ?
=> ? = 2
[.,[[.,[.,[[[.,.],.],[.,.]]]],.]]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,3,4,7,2,5,8,6] => ?
=> ? = 2
[.,[[.,[.,[[[.,.],[.,.]],.]]],.]]
=> [1,0,1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,3,4,7,2,8,5,6] => ?
=> ? = 2
[.,[[.,[[.,.],[.,[.,[.,.]]]]],.]]
=> [1,0,1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,3,5,2,6,7,8,4] => ?
=> ? = 2
[.,[[.,[[.,[.,.]],[.,[.,.]]]],.]]
=> [1,0,1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,3,5,6,2,7,8,4] => ?
=> ? = 2
[.,[[.,[[[.,[.,.]],.],[.,.]]],.]]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [1,3,6,7,2,4,8,5] => ?
=> ? = 2
[.,[[.,[[[.,.],[[.,.],.]],.]],.]]
=> [1,0,1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,3,6,2,8,4,5,7] => ?
=> ? = 3
[.,[[.,[[[.,[.,.]],[.,.]],.]],.]]
=> [1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,3,6,7,2,8,4,5] => ?
=> ? = 2
[.,[[.,[[[.,[[.,.],.]],.],.]],.]]
=> [1,0,1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,3,6,8,2,4,5,7] => ?
=> ? = 3
[.,[[[.,.],[[.,.],[.,[.,.]]]],.]]
=> [1,0,1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [1,4,2,6,3,7,8,5] => ?
=> ? = 2
[.,[[[.,.],[[.,[.,.]],[.,.]]],.]]
=> [1,0,1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [1,4,2,6,7,3,8,5] => ?
=> ? = 2
[.,[[[.,[.,[.,.]]],[[.,.],.]],.]]
=> [1,0,1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> [1,4,5,6,2,8,3,7] => ?
=> ? = 2
[.,[[[.,[[.,[.,.]],.]],[.,.]],.]]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,4,6,7,2,3,8,5] => ?
=> ? = 2
[.,[[[[.,.],[.,[.,.]]],[.,.]],.]]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [1,5,2,6,7,3,8,4] => ?
=> ? = 1
Description
The number of descents of a standard tableau.
Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.
Matching statistic: St000010
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 100%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => [2,1] => [2]
=> 1 = 0 + 1
[[.,.],.]
=> [1,2] => [1,2] => [1,1]
=> 2 = 1 + 1
[.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => [3]
=> 1 = 0 + 1
[.,[[.,.],.]]
=> [2,3,1] => [2,3,1] => [2,1]
=> 2 = 1 + 1
[[.,.],[.,.]]
=> [3,1,2] => [3,1,2] => [2,1]
=> 2 = 1 + 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => [2,1]
=> 2 = 1 + 1
[[[.,.],.],.]
=> [1,2,3] => [1,3,2] => [2,1]
=> 2 = 1 + 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => [4]
=> 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,4,2,1] => [3,1]
=> 2 = 1 + 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [4,2,3,1] => [3,1]
=> 2 = 1 + 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,4,1] => [3,1]
=> 2 = 1 + 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,4,3,1] => [3,1]
=> 2 = 1 + 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [4,3,1,2] => [3,1]
=> 2 = 1 + 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,4,1,2] => [2,1,1]
=> 3 = 2 + 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [4,2,1,3] => [3,1]
=> 2 = 1 + 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [4,1,3,2] => [3,1]
=> 2 = 1 + 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => [3,1]
=> 2 = 1 + 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,4,1,3] => [2,1,1]
=> 3 = 2 + 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,1,4,2] => [2,2]
=> 2 = 1 + 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,4,3] => [2,2]
=> 2 = 1 + 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,4,3,2] => [3,1]
=> 2 = 1 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => [5]
=> 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,5,3,2,1] => [4,1]
=> 2 = 1 + 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [5,3,4,2,1] => [4,1]
=> 2 = 1 + 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,5,2,1] => [4,1]
=> 2 = 1 + 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,5,4,2,1] => [4,1]
=> 2 = 1 + 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [5,4,2,3,1] => [4,1]
=> 2 = 1 + 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [4,5,2,3,1] => [3,1,1]
=> 3 = 2 + 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [5,3,2,4,1] => [4,1]
=> 2 = 1 + 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [5,2,4,3,1] => [4,1]
=> 2 = 1 + 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,5,1] => [4,1]
=> 2 = 1 + 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,5,2,4,1] => [3,1,1]
=> 3 = 2 + 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [4,2,5,3,1] => [3,2]
=> 2 = 1 + 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,5,4,1] => [3,2]
=> 2 = 1 + 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,5,4,3,1] => [4,1]
=> 2 = 1 + 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [5,4,3,1,2] => [4,1]
=> 2 = 1 + 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [4,5,3,1,2] => [3,1,1]
=> 3 = 2 + 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [5,3,4,1,2] => [3,1,1]
=> 3 = 2 + 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [4,3,5,1,2] => [3,1,1]
=> 3 = 2 + 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [3,5,4,1,2] => [3,1,1]
=> 3 = 2 + 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [5,4,2,1,3] => [4,1]
=> 2 = 1 + 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [4,5,2,1,3] => [3,1,1]
=> 3 = 2 + 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [5,4,1,3,2] => [4,1]
=> 2 = 1 + 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [4,5,1,3,2] => [3,1,1]
=> 3 = 2 + 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [5,3,2,1,4] => [4,1]
=> 2 = 1 + 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [5,2,4,1,3] => [3,1,1]
=> 3 = 2 + 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [5,3,1,4,2] => [3,2]
=> 2 = 1 + 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [5,2,1,4,3] => [3,2]
=> 2 = 1 + 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [5,1,4,3,2] => [4,1]
=> 2 = 1 + 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1,5] => [4,1]
=> 2 = 1 + 1
[.,[.,[.,[[.,[.,.]],[[.,.],.]]]]]
=> [7,8,5,4,6,3,2,1] => ? => ?
=> ? = 2 + 1
[.,[.,[.,[[[.,.],.],[[.,.],.]]]]]
=> [7,8,4,5,6,3,2,1] => ? => ?
=> ? = 2 + 1
[.,[.,[.,[[[.,[.,.]],[.,.]],.]]]]
=> [7,5,4,6,8,3,2,1] => [7,5,4,8,6,3,2,1] => ?
=> ? = 1 + 1
[.,[.,[.,[[[[.,[.,.]],.],.],.]]]]
=> [5,4,6,7,8,3,2,1] => [5,4,8,7,6,3,2,1] => ?
=> ? = 1 + 1
[.,[.,[[.,.],[.,[[[.,.],.],.]]]]]
=> [6,7,8,5,3,4,2,1] => ? => ?
=> ? = 2 + 1
[.,[.,[[.,.],[[.,[.,.]],[.,.]]]]]
=> [8,6,5,7,3,4,2,1] => ? => ?
=> ? = 2 + 1
[.,[.,[[.,.],[[.,[[.,.],.]],.]]]]
=> [6,7,5,8,3,4,2,1] => [6,8,5,7,3,4,2,1] => ?
=> ? = 3 + 1
[.,[.,[[.,.],[[[.,.],[.,.]],.]]]]
=> [7,5,6,8,3,4,2,1] => ? => ?
=> ? = 2 + 1
[.,[.,[[[.,.],.],[.,[.,[.,.]]]]]]
=> [8,7,6,3,4,5,2,1] => [8,7,6,3,5,4,2,1] => ?
=> ? = 1 + 1
[.,[.,[[[.,.],.],[[.,[.,.]],.]]]]
=> [7,6,8,3,4,5,2,1] => [7,6,8,3,5,4,2,1] => ?
=> ? = 2 + 1
[.,[.,[[.,[.,[[.,.],.]]],[.,.]]]]
=> [8,5,6,4,3,7,2,1] => ? => ?
=> ? = 2 + 1
[.,[.,[[[.,.],[.,[.,.]]],[.,.]]]]
=> [8,6,5,3,4,7,2,1] => ? => ?
=> ? = 1 + 1
[.,[.,[[.,[[.,.],[.,[.,.]]]],.]]]
=> [7,6,4,5,3,8,2,1] => [7,6,4,8,3,5,2,1] => ?
=> ? = 2 + 1
[.,[.,[[.,[[.,[[.,.],.]],.]],.]]]
=> [5,6,4,7,3,8,2,1] => ? => ?
=> ? = 3 + 1
[.,[.,[[[.,.],[.,[.,[.,.]]]],.]]]
=> [7,6,5,3,4,8,2,1] => [7,6,5,3,8,4,2,1] => ?
=> ? = 1 + 1
[.,[.,[[[.,.],[.,[[.,.],.]]],.]]]
=> [6,7,5,3,4,8,2,1] => ? => ?
=> ? = 2 + 1
[.,[.,[[[.,.],[[.,.],[.,.]]],.]]]
=> [7,5,6,3,4,8,2,1] => ? => ?
=> ? = 2 + 1
[.,[.,[[[.,.],[[.,[.,.]],.]],.]]]
=> [6,5,7,3,4,8,2,1] => ? => ?
=> ? = 2 + 1
[.,[.,[[[.,.],[[[.,.],.],.]],.]]]
=> [5,6,7,3,4,8,2,1] => ? => ?
=> ? = 2 + 1
[.,[.,[[[.,[.,[[.,.],.]]],.],.]]]
=> [5,6,4,3,7,8,2,1] => ? => ?
=> ? = 2 + 1
[.,[.,[[[.,[[.,.],[.,.]]],.],.]]]
=> [6,4,5,3,7,8,2,1] => ? => ?
=> ? = 2 + 1
[.,[.,[[[[.,.],[[.,.],.]],.],.]]]
=> [5,6,3,4,7,8,2,1] => ? => ?
=> ? = 2 + 1
[.,[.,[[[[.,[[.,.],.]],.],.],.]]]
=> [4,5,3,6,7,8,2,1] => ? => ?
=> ? = 2 + 1
[.,[.,[[[[[.,.],[.,.]],.],.],.]]]
=> [5,3,4,6,7,8,2,1] => ? => ?
=> ? = 1 + 1
[.,[[.,.],[.,[.,[[[.,.],.],.]]]]]
=> [6,7,8,5,4,2,3,1] => ? => ?
=> ? = 2 + 1
[.,[[.,.],[.,[[.,.],[.,[.,.]]]]]]
=> [8,7,5,6,4,2,3,1] => [8,7,5,6,4,2,3,1] => ?
=> ? = 2 + 1
[.,[[.,.],[.,[[.,.],[[.,.],.]]]]]
=> [7,8,5,6,4,2,3,1] => ? => ?
=> ? = 3 + 1
[.,[[.,.],[.,[[[.,.],[.,.]],.]]]]
=> [7,5,6,8,4,2,3,1] => [7,5,8,6,4,2,3,1] => ?
=> ? = 2 + 1
[.,[[.,.],[[[.,.],.],[.,[.,.]]]]]
=> [8,7,4,5,6,2,3,1] => [8,7,4,6,5,2,3,1] => ?
=> ? = 2 + 1
[.,[[.,.],[[[.,.],.],[[.,.],.]]]]
=> [7,8,4,5,6,2,3,1] => [7,8,4,6,5,2,3,1] => ?
=> ? = 3 + 1
[.,[[.,.],[[[.,[.,.]],.],[.,.]]]]
=> [8,5,4,6,7,2,3,1] => ? => ?
=> ? = 2 + 1
[.,[[.,.],[[.,[.,[[.,.],.]]],.]]]
=> [6,7,5,4,8,2,3,1] => ? => ?
=> ? = 3 + 1
[.,[[.,.],[[[.,[.,.]],[.,.]],.]]]
=> [7,5,4,6,8,2,3,1] => ? => ?
=> ? = 2 + 1
[.,[[.,.],[[[[.,.],.],[.,.]],.]]]
=> [7,4,5,6,8,2,3,1] => ? => ?
=> ? = 2 + 1
[.,[[.,.],[[[.,[.,[.,.]]],.],.]]]
=> [6,5,4,7,8,2,3,1] => ? => ?
=> ? = 2 + 1
[.,[[.,.],[[[.,[[.,.],.]],.],.]]]
=> [5,6,4,7,8,2,3,1] => ? => ?
=> ? = 3 + 1
[.,[[.,.],[[[[.,.],[.,.]],.],.]]]
=> [6,4,5,7,8,2,3,1] => ? => ?
=> ? = 2 + 1
[.,[[.,.],[[[[.,[.,.]],.],.],.]]]
=> [5,4,6,7,8,2,3,1] => ? => ?
=> ? = 2 + 1
[.,[[.,[.,.]],[[.,.],[[.,.],.]]]]
=> [7,8,5,6,3,2,4,1] => ? => ?
=> ? = 3 + 1
[.,[[.,[.,.]],[[.,[[.,.],.]],.]]]
=> [6,7,5,8,3,2,4,1] => [6,8,5,7,3,2,4,1] => ?
=> ? = 3 + 1
[.,[[.,[.,.]],[[[.,.],[.,.]],.]]]
=> [7,5,6,8,3,2,4,1] => ? => ?
=> ? = 2 + 1
[.,[[.,[.,.]],[[[.,[.,.]],.],.]]]
=> [6,5,7,8,3,2,4,1] => ? => ?
=> ? = 2 + 1
[.,[[[.,.],.],[.,[[[.,.],.],.]]]]
=> [6,7,8,5,2,3,4,1] => [6,8,7,5,2,4,3,1] => ?
=> ? = 2 + 1
[.,[[[.,.],.],[[[.,.],[.,.]],.]]]
=> [7,5,6,8,2,3,4,1] => ? => ?
=> ? = 2 + 1
[.,[[.,[[.,.],.]],[.,[[.,.],.]]]]
=> [7,8,6,3,4,2,5,1] => [7,8,6,3,5,2,4,1] => ?
=> ? = 3 + 1
[.,[[[.,.],[.,.]],[.,[[.,.],.]]]]
=> [7,8,6,4,2,3,5,1] => ? => ?
=> ? = 2 + 1
[.,[[[.,[.,.]],.],[[.,.],[.,.]]]]
=> [8,6,7,3,2,4,5,1] => [8,6,7,3,2,5,4,1] => ?
=> ? = 2 + 1
[.,[[[.,[.,.]],.],[[.,[.,.]],.]]]
=> [7,6,8,3,2,4,5,1] => ? => ?
=> ? = 2 + 1
[.,[[[.,[.,.]],.],[[[.,.],.],.]]]
=> [6,7,8,3,2,4,5,1] => [6,8,7,3,2,5,4,1] => ?
=> ? = 2 + 1
[.,[[[[.,.],.],.],[.,[.,[.,.]]]]]
=> [8,7,6,2,3,4,5,1] => [8,7,6,2,5,4,3,1] => ?
=> ? = 1 + 1
Description
The length of the partition.
Matching statistic: St000507
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 100%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => [2,1] => [[1],[2]]
=> 1 = 0 + 1
[[.,.],.]
=> [1,2] => [1,2] => [[1,2]]
=> 2 = 1 + 1
[.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => [[1],[2],[3]]
=> 1 = 0 + 1
[.,[[.,.],.]]
=> [2,3,1] => [2,3,1] => [[1,2],[3]]
=> 2 = 1 + 1
[[.,.],[.,.]]
=> [3,1,2] => [3,1,2] => [[1,3],[2]]
=> 2 = 1 + 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => [[1,3],[2]]
=> 2 = 1 + 1
[[[.,.],.],.]
=> [1,2,3] => [1,3,2] => [[1,2],[3]]
=> 2 = 1 + 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,4,2,1] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [4,2,3,1] => [[1,3],[2],[4]]
=> 2 = 1 + 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,4,1] => [[1,3],[2],[4]]
=> 2 = 1 + 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,4,3,1] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [4,3,1,2] => [[1,4],[2],[3]]
=> 2 = 1 + 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,4,1,2] => [[1,2],[3,4]]
=> 3 = 2 + 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [4,2,1,3] => [[1,4],[2],[3]]
=> 2 = 1 + 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [4,1,3,2] => [[1,3],[2],[4]]
=> 2 = 1 + 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => [[1,4],[2],[3]]
=> 2 = 1 + 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,4,1,3] => [[1,2],[3,4]]
=> 3 = 2 + 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,1,4,2] => [[1,3],[2,4]]
=> 2 = 1 + 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,4,3] => [[1,3],[2,4]]
=> 2 = 1 + 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,5,3,2,1] => [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [5,3,4,2,1] => [[1,3],[2],[4],[5]]
=> 2 = 1 + 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,5,2,1] => [[1,3],[2],[4],[5]]
=> 2 = 1 + 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,5,4,2,1] => [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [5,4,2,3,1] => [[1,4],[2],[3],[5]]
=> 2 = 1 + 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [4,5,2,3,1] => [[1,2],[3,4],[5]]
=> 3 = 2 + 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [5,3,2,4,1] => [[1,4],[2],[3],[5]]
=> 2 = 1 + 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [5,2,4,3,1] => [[1,3],[2],[4],[5]]
=> 2 = 1 + 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,5,1] => [[1,4],[2],[3],[5]]
=> 2 = 1 + 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,5,2,4,1] => [[1,2],[3,4],[5]]
=> 3 = 2 + 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [4,2,5,3,1] => [[1,3],[2,4],[5]]
=> 2 = 1 + 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,5,4,1] => [[1,3],[2,4],[5]]
=> 2 = 1 + 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,5,4,3,1] => [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [5,4,3,1,2] => [[1,5],[2],[3],[4]]
=> 2 = 1 + 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [4,5,3,1,2] => [[1,2],[3,5],[4]]
=> 3 = 2 + 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [5,3,4,1,2] => [[1,3],[2,5],[4]]
=> 3 = 2 + 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [4,3,5,1,2] => [[1,3],[2,5],[4]]
=> 3 = 2 + 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [3,5,4,1,2] => [[1,2],[3,5],[4]]
=> 3 = 2 + 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [5,4,2,1,3] => [[1,5],[2],[3],[4]]
=> 2 = 1 + 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [4,5,2,1,3] => [[1,2],[3,5],[4]]
=> 3 = 2 + 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [5,4,1,3,2] => [[1,4],[2],[3],[5]]
=> 2 = 1 + 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [4,5,1,3,2] => [[1,2],[3,4],[5]]
=> 3 = 2 + 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [5,3,2,1,4] => [[1,5],[2],[3],[4]]
=> 2 = 1 + 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [5,2,4,1,3] => [[1,3],[2,5],[4]]
=> 3 = 2 + 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [5,3,1,4,2] => [[1,4],[2,5],[3]]
=> 2 = 1 + 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [5,2,1,4,3] => [[1,4],[2,5],[3]]
=> 2 = 1 + 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [5,1,4,3,2] => [[1,3],[2],[4],[5]]
=> 2 = 1 + 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1,5] => [[1,5],[2],[3],[4]]
=> 2 = 1 + 1
[.,[.,[.,[[.,[.,.]],[[.,.],.]]]]]
=> [7,8,5,4,6,3,2,1] => ? => ?
=> ? = 2 + 1
[.,[.,[.,[[[.,.],.],[[.,.],.]]]]]
=> [7,8,4,5,6,3,2,1] => ? => ?
=> ? = 2 + 1
[.,[.,[.,[[[.,.],[[.,.],.]],.]]]]
=> [6,7,4,5,8,3,2,1] => [6,8,4,7,5,3,2,1] => ?
=> ? = 2 + 1
[.,[.,[.,[[[.,[.,.]],[.,.]],.]]]]
=> [7,5,4,6,8,3,2,1] => [7,5,4,8,6,3,2,1] => ?
=> ? = 1 + 1
[.,[.,[.,[[[.,[.,[.,.]]],.],.]]]]
=> [6,5,4,7,8,3,2,1] => [6,5,4,8,7,3,2,1] => ?
=> ? = 1 + 1
[.,[.,[.,[[[[.,[.,.]],.],.],.]]]]
=> [5,4,6,7,8,3,2,1] => [5,4,8,7,6,3,2,1] => ?
=> ? = 1 + 1
[.,[.,[[.,.],[.,[[[.,.],.],.]]]]]
=> [6,7,8,5,3,4,2,1] => ? => ?
=> ? = 2 + 1
[.,[.,[[.,.],[[.,[.,.]],[.,.]]]]]
=> [8,6,5,7,3,4,2,1] => ? => ?
=> ? = 2 + 1
[.,[.,[[.,.],[[.,[[.,.],.]],.]]]]
=> [6,7,5,8,3,4,2,1] => [6,8,5,7,3,4,2,1] => ?
=> ? = 3 + 1
[.,[.,[[.,.],[[[.,.],[.,.]],.]]]]
=> [7,5,6,8,3,4,2,1] => ? => ?
=> ? = 2 + 1
[.,[.,[[.,[.,[[.,.],.]]],[.,.]]]]
=> [8,5,6,4,3,7,2,1] => ? => ?
=> ? = 2 + 1
[.,[.,[[.,[[.,.],[.,.]]],[.,.]]]]
=> [8,6,4,5,3,7,2,1] => [8,6,4,7,3,5,2,1] => ?
=> ? = 2 + 1
[.,[.,[[[.,.],[.,[.,.]]],[.,.]]]]
=> [8,6,5,3,4,7,2,1] => ? => ?
=> ? = 1 + 1
[.,[.,[[[[.,.],[.,.]],.],[.,.]]]]
=> [8,5,3,4,6,7,2,1] => [8,5,3,7,6,4,2,1] => ?
=> ? = 1 + 1
[.,[.,[[.,[[.,[[.,.],.]],.]],.]]]
=> [5,6,4,7,3,8,2,1] => ? => ?
=> ? = 3 + 1
[.,[.,[[[.,.],[.,[.,[.,.]]]],.]]]
=> [7,6,5,3,4,8,2,1] => [7,6,5,3,8,4,2,1] => ?
=> ? = 1 + 1
[.,[.,[[[.,.],[.,[[.,.],.]]],.]]]
=> [6,7,5,3,4,8,2,1] => ? => ?
=> ? = 2 + 1
[.,[.,[[[.,.],[[.,.],[.,.]]],.]]]
=> [7,5,6,3,4,8,2,1] => ? => ?
=> ? = 2 + 1
[.,[.,[[[.,.],[[.,[.,.]],.]],.]]]
=> [6,5,7,3,4,8,2,1] => ? => ?
=> ? = 2 + 1
[.,[.,[[[.,.],[[[.,.],.],.]],.]]]
=> [5,6,7,3,4,8,2,1] => ? => ?
=> ? = 2 + 1
[.,[.,[[[.,[.,[[.,.],.]]],.],.]]]
=> [5,6,4,3,7,8,2,1] => ? => ?
=> ? = 2 + 1
[.,[.,[[[.,[[.,.],[.,.]]],.],.]]]
=> [6,4,5,3,7,8,2,1] => ? => ?
=> ? = 2 + 1
[.,[.,[[[[.,.],[[.,.],.]],.],.]]]
=> [5,6,3,4,7,8,2,1] => ? => ?
=> ? = 2 + 1
[.,[.,[[[[.,[[.,.],.]],.],.],.]]]
=> [4,5,3,6,7,8,2,1] => ? => ?
=> ? = 2 + 1
[.,[.,[[[[[.,.],[.,.]],.],.],.]]]
=> [5,3,4,6,7,8,2,1] => ? => ?
=> ? = 1 + 1
[.,[[.,.],[.,[.,[[[.,.],.],.]]]]]
=> [6,7,8,5,4,2,3,1] => ? => ?
=> ? = 2 + 1
[.,[[.,.],[.,[[.,.],[[.,.],.]]]]]
=> [7,8,5,6,4,2,3,1] => ? => ?
=> ? = 3 + 1
[.,[[.,.],[.,[[[.,.],[.,.]],.]]]]
=> [7,5,6,8,4,2,3,1] => [7,5,8,6,4,2,3,1] => ?
=> ? = 2 + 1
[.,[[.,.],[[.,.],[[[.,.],.],.]]]]
=> [6,7,8,4,5,2,3,1] => [6,8,7,4,5,2,3,1] => ?
=> ? = 3 + 1
[.,[[.,.],[[[.,.],.],[[.,.],.]]]]
=> [7,8,4,5,6,2,3,1] => [7,8,4,6,5,2,3,1] => ?
=> ? = 3 + 1
[.,[[.,.],[[[.,[.,.]],.],[.,.]]]]
=> [8,5,4,6,7,2,3,1] => ? => ?
=> ? = 2 + 1
[.,[[.,.],[[.,[.,[[.,.],.]]],.]]]
=> [6,7,5,4,8,2,3,1] => ? => ?
=> ? = 3 + 1
[.,[[.,.],[[[.,.],[[.,.],.]],.]]]
=> [6,7,4,5,8,2,3,1] => [6,8,4,7,5,2,3,1] => ?
=> ? = 3 + 1
[.,[[.,.],[[[.,[.,.]],[.,.]],.]]]
=> [7,5,4,6,8,2,3,1] => ? => ?
=> ? = 2 + 1
[.,[[.,.],[[[[.,.],.],[.,.]],.]]]
=> [7,4,5,6,8,2,3,1] => ? => ?
=> ? = 2 + 1
[.,[[.,.],[[[.,[.,[.,.]]],.],.]]]
=> [6,5,4,7,8,2,3,1] => ? => ?
=> ? = 2 + 1
[.,[[.,.],[[[.,[[.,.],.]],.],.]]]
=> [5,6,4,7,8,2,3,1] => ? => ?
=> ? = 3 + 1
[.,[[.,.],[[[[.,.],[.,.]],.],.]]]
=> [6,4,5,7,8,2,3,1] => ? => ?
=> ? = 2 + 1
[.,[[.,.],[[[[.,[.,.]],.],.],.]]]
=> [5,4,6,7,8,2,3,1] => ? => ?
=> ? = 2 + 1
[.,[[.,[.,.]],[[.,.],[[.,.],.]]]]
=> [7,8,5,6,3,2,4,1] => ? => ?
=> ? = 3 + 1
[.,[[.,[.,.]],[[.,[[.,.],.]],.]]]
=> [6,7,5,8,3,2,4,1] => [6,8,5,7,3,2,4,1] => ?
=> ? = 3 + 1
[.,[[.,[.,.]],[[[.,.],[.,.]],.]]]
=> [7,5,6,8,3,2,4,1] => ? => ?
=> ? = 2 + 1
[.,[[.,[.,.]],[[[.,[.,.]],.],.]]]
=> [6,5,7,8,3,2,4,1] => ? => ?
=> ? = 2 + 1
[.,[[[.,.],.],[.,[[[.,.],.],.]]]]
=> [6,7,8,5,2,3,4,1] => [6,8,7,5,2,4,3,1] => ?
=> ? = 2 + 1
[.,[[[.,.],.],[[[.,.],[.,.]],.]]]
=> [7,5,6,8,2,3,4,1] => ? => ?
=> ? = 2 + 1
[.,[[.,[[.,.],.]],[.,[[.,.],.]]]]
=> [7,8,6,3,4,2,5,1] => [7,8,6,3,5,2,4,1] => ?
=> ? = 3 + 1
[.,[[[.,.],[.,.]],[.,[.,[.,.]]]]]
=> [8,7,6,4,2,3,5,1] => [8,7,6,4,2,5,3,1] => ?
=> ? = 1 + 1
[.,[[[.,.],[.,.]],[.,[[.,.],.]]]]
=> [7,8,6,4,2,3,5,1] => ? => ?
=> ? = 2 + 1
[.,[[[.,.],[.,.]],[[.,.],[.,.]]]]
=> [8,6,7,4,2,3,5,1] => [8,6,7,4,2,5,3,1] => ?
=> ? = 2 + 1
[.,[[[.,[.,.]],.],[[.,.],[.,.]]]]
=> [8,6,7,3,2,4,5,1] => [8,6,7,3,2,5,4,1] => ?
=> ? = 2 + 1
Description
The number of ascents of a standard tableau.
Entry $i$ of a standard Young tableau is an '''ascent''' if $i+1$ appears to the right or above $i$ in the tableau (with respect to the English notation for tableaux).
Matching statistic: St000659
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 71%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 71%
Values
[.,[.,.]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 0
[[.,.],.]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 1
[.,[.,[.,.]]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 0
[.,[[.,.],.]]
=> [2,3,1] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 1
[[.,.],[.,.]]
=> [3,1,2] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[[.,[.,.]],.]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[[[.,.],.],.]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [[[[.,[.,.]],.],.],.]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [[[[.,.],[.,.]],.],.]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [[[[.,.],[.,.]],.],.]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [[[[.,.],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [[[.,[.,.]],[.,.]],.]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [[[[.,.],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [[[.,[.,.]],[.,.]],.]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [[[.,.],[.,[.,.]]],.]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [[[.,.],[.,[.,.]]],.]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]
=> [6,7,8,5,4,3,2,1] => [[[[[[.,[.,[.,.]]],.],.],.],.],.]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
=> [5,6,7,8,4,3,2,1] => [[[[[.,[.,[.,[.,.]]]],.],.],.],.]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[.,[.,[.,[[.,.],[[.,[.,.]],.]]]]]
=> [7,6,8,4,5,3,2,1] => ?
=> ?
=> ? = 2
[.,[.,[.,[[.,.],[[[.,.],.],.]]]]]
=> [6,7,8,4,5,3,2,1] => ?
=> ?
=> ? = 2
[.,[.,[.,[[.,[.,.]],[[.,.],.]]]]]
=> [7,8,5,4,6,3,2,1] => ?
=> ?
=> ? = 2
[.,[.,[.,[[[.,.],.],[[.,.],.]]]]]
=> [7,8,4,5,6,3,2,1] => ?
=> ?
=> ? = 2
[.,[.,[.,[[.,[.,[.,.]]],[.,.]]]]]
=> [8,6,5,4,7,3,2,1] => ?
=> ?
=> ? = 1
[.,[.,[.,[[[.,[.,.]],.],[.,.]]]]]
=> [8,5,4,6,7,3,2,1] => ?
=> ?
=> ? = 1
[.,[.,[.,[[.,[[[.,.],.],.]],.]]]]
=> [5,6,7,4,8,3,2,1] => [[[[[.,[.,[.,.]]],[.,.]],.],.],.]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2
[.,[.,[.,[[[[[.,.],.],.],.],.]]]]
=> [4,5,6,7,8,3,2,1] => [[[[.,[.,[.,[.,[.,.]]]]],.],.],.]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[.,[.,[[.,.],[.,[[[.,.],.],.]]]]]
=> [6,7,8,5,3,4,2,1] => ?
=> ?
=> ? = 2
[.,[.,[[.,.],[[.,[.,.]],[.,.]]]]]
=> [8,6,5,7,3,4,2,1] => ?
=> ?
=> ? = 2
[.,[.,[[.,.],[[[.,.],[.,.]],.]]]]
=> [7,5,6,8,3,4,2,1] => ?
=> ?
=> ? = 2
[.,[.,[[.,.],[[[[.,.],.],.],.]]]]
=> [5,6,7,8,3,4,2,1] => [[[[.,[.,[.,[.,.]]]],[.,.]],.],.]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 2
[.,[.,[[.,[.,.]],[[.,.],[.,.]]]]]
=> [8,6,7,4,3,5,2,1] => ?
=> ?
=> ? = 2
[.,[.,[[.,[.,.]],[[[.,.],.],.]]]]
=> [6,7,8,4,3,5,2,1] => [[[[[.,[.,[.,.]]],.],[.,.]],.],.]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 2
[.,[.,[[[.,.],.],[[[.,.],.],.]]]]
=> [6,7,8,3,4,5,2,1] => [[[[.,[.,[.,.]]],[.,[.,.]]],.],.]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 2
[.,[.,[[.,[[.,.],.]],[.,[.,.]]]]]
=> [8,7,4,5,3,6,2,1] => ?
=> ?
=> ? = 2
[.,[.,[[.,[.,[[.,.],.]]],[.,.]]]]
=> [8,5,6,4,3,7,2,1] => ?
=> ?
=> ? = 2
[.,[.,[[.,[[.,.],[.,.]]],[.,.]]]]
=> [8,6,4,5,3,7,2,1] => ?
=> ?
=> ? = 2
[.,[.,[[[.,.],[.,[.,.]]],[.,.]]]]
=> [8,6,5,3,4,7,2,1] => ?
=> ?
=> ? = 1
[.,[.,[[[.,[.,[.,.]]],.],[.,.]]]]
=> [8,5,4,3,6,7,2,1] => ?
=> ?
=> ? = 1
[.,[.,[[.,[.,[[[.,.],.],.]]],.]]]
=> [5,6,7,4,3,8,2,1] => [[[[[.,[.,[.,.]]],.],[.,.]],.],.]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 2
[.,[.,[[.,[[.,[[.,.],.]],.]],.]]]
=> [5,6,4,7,3,8,2,1] => ?
=> ?
=> ? = 3
[.,[.,[[.,[[[[.,.],.],.],.]],.]]]
=> [4,5,6,7,3,8,2,1] => [[[[.,[.,[.,[.,.]]]],[.,.]],.],.]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 2
[.,[.,[[[.,.],[.,[[.,.],.]]],.]]]
=> [6,7,5,3,4,8,2,1] => ?
=> ?
=> ? = 2
[.,[.,[[[.,.],[[.,.],[.,.]]],.]]]
=> [7,5,6,3,4,8,2,1] => ?
=> ?
=> ? = 2
[.,[.,[[[.,.],[[.,[.,.]],.]],.]]]
=> [6,5,7,3,4,8,2,1] => ?
=> ?
=> ? = 2
[.,[.,[[[.,.],[[[.,.],.],.]],.]]]
=> [5,6,7,3,4,8,2,1] => ?
=> ?
=> ? = 2
[.,[.,[[[.,[.,[[.,.],.]]],.],.]]]
=> [5,6,4,3,7,8,2,1] => ?
=> ?
=> ? = 2
[.,[.,[[[.,[[.,.],[.,.]]],.],.]]]
=> [6,4,5,3,7,8,2,1] => ?
=> ?
=> ? = 2
[.,[.,[[[.,[[[.,.],.],.]],.],.]]]
=> [4,5,6,3,7,8,2,1] => ?
=> ?
=> ? = 2
[.,[.,[[[[.,.],[[.,.],.]],.],.]]]
=> [5,6,3,4,7,8,2,1] => ?
=> ?
=> ? = 2
[.,[.,[[[[.,[[.,.],.]],.],.],.]]]
=> [4,5,3,6,7,8,2,1] => ?
=> ?
=> ? = 2
[.,[.,[[[[[[.,.],.],.],.],.],.]]]
=> [3,4,5,6,7,8,2,1] => [[[.,[.,[.,[.,[.,[.,.]]]]]],.],.]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[.,[[.,.],[.,[.,[[[.,.],.],.]]]]]
=> [6,7,8,5,4,2,3,1] => ?
=> ?
=> ? = 2
[.,[[.,.],[.,[[.,.],[[.,.],.]]]]]
=> [7,8,5,6,4,2,3,1] => ?
=> ?
=> ? = 3
[.,[[.,.],[.,[[[[.,.],.],.],.]]]]
=> [5,6,7,8,4,2,3,1] => [[[[.,[.,[.,[.,.]]]],.],[.,.]],.]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 2
[.,[[.,.],[[.,.],[[[.,.],.],.]]]]
=> [6,7,8,4,5,2,3,1] => [[[[.,[.,[.,.]]],[.,.]],[.,.]],.]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3
[.,[[.,.],[[.,[.,.]],[.,[.,.]]]]]
=> [8,7,5,4,6,2,3,1] => ?
=> ?
=> ? = 2
[.,[[.,.],[[.,[.,[.,.]]],[.,.]]]]
=> [8,6,5,4,7,2,3,1] => ?
=> ?
=> ? = 2
[.,[[.,.],[[[.,[.,.]],.],[.,.]]]]
=> [8,5,4,6,7,2,3,1] => ?
=> ?
=> ? = 2
[.,[[.,.],[[.,[.,[.,[.,.]]]],.]]]
=> [7,6,5,4,8,2,3,1] => ?
=> ?
=> ? = 2
[.,[[.,.],[[.,[.,[[.,.],.]]],.]]]
=> [6,7,5,4,8,2,3,1] => ?
=> ?
=> ? = 3
[.,[[.,.],[[[.,[.,.]],[.,.]],.]]]
=> [7,5,4,6,8,2,3,1] => ?
=> ?
=> ? = 2
[.,[[.,.],[[[[.,.],.],[.,.]],.]]]
=> [7,4,5,6,8,2,3,1] => ?
=> ?
=> ? = 2
[.,[[.,.],[[[.,[.,[.,.]]],.],.]]]
=> [6,5,4,7,8,2,3,1] => ?
=> ?
=> ? = 2
[.,[[.,.],[[[.,[[.,.],.]],.],.]]]
=> [5,6,4,7,8,2,3,1] => ?
=> ?
=> ? = 3
[.,[[.,.],[[[[.,.],[.,.]],.],.]]]
=> [6,4,5,7,8,2,3,1] => ?
=> ?
=> ? = 2
[.,[[.,.],[[[[.,[.,.]],.],.],.]]]
=> [5,4,6,7,8,2,3,1] => ?
=> ?
=> ? = 2
Description
The number of rises of length at least 2 of a Dyck path.
Matching statistic: St000919
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000919: Binary trees ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 71%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000919: Binary trees ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 71%
Values
[.,[.,.]]
=> [1,0,1,0]
=> [1,2] => [.,[.,.]]
=> 0
[[.,.],.]
=> [1,1,0,0]
=> [2,1] => [[.,.],.]
=> 1
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> 0
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,3,2] => [.,[[.,.],.]]
=> 1
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [2,1,3] => [[.,.],[.,.]]
=> 1
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [2,3,1] => [[.,.],[.,.]]
=> 1
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [3,1,2] => [[.,[.,.]],.]
=> 1
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [.,[[.,.],[.,.]]]
=> 1
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [.,[[.,[.,.]],.]]
=> 1
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> 1
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> 2
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [[.,.],[.,[.,.]]]
=> 1
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [[.,[.,.]],[.,.]]
=> 1
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [[.,.],[.,[.,.]]]
=> 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [[.,.],[[.,.],.]]
=> 2
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [[.,[.,.]],[.,.]]
=> 1
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> 1
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> 1
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> 1
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> 1
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> 2
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> 1
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> 1
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> 2
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> 1
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> 1
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [.,[[.,[.,[.,.]]],.]]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> 1
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> 2
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> 2
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
=> 2
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [[.,.],[[.,[.,.]],.]]
=> 2
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
=> 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
=> 2
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
=> 1
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => [[.,[.,.]],[[.,.],.]]
=> 2
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
=> 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
=> 2
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => [[.,[.,.]],[.,[.,.]]]
=> 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> 1
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => [[.,[.,[.,.]]],[.,.]]
=> 1
[[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> 1
[.,[.,[[.,[.,.]],[.,[[.,.],.]]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,2,4,5,3,6,8,7] => ?
=> ? = 2
[.,[.,[[.,[[.,[.,.]],.]],[.,.]]]]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,2,4,6,7,3,5,8] => ?
=> ? = 2
[.,[.,[[[.,.],[.,[.,.]]],[.,.]]]]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,2,5,3,6,7,4,8] => ?
=> ? = 1
[.,[.,[[[.,[.,[.,.]]],.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,2,5,6,7,3,4,8] => ?
=> ? = 1
[.,[.,[[.,[[.,.],[.,[.,.]]]],.]]]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,2,4,6,3,7,8,5] => ?
=> ? = 2
[.,[.,[[.,[[.,[.,.]],[.,.]]],.]]]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,2,4,6,7,3,8,5] => ?
=> ? = 2
[.,[.,[[[.,.],[.,[[.,.],.]]],.]]]
=> [1,0,1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,2,5,3,6,8,4,7] => ?
=> ? = 2
[.,[[.,.],[.,[[[.,.],.],[.,.]]]]]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,3,2,4,7,5,6,8] => ?
=> ? = 2
[.,[[.,.],[[[.,.],.],[[.,.],.]]]]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,3,2,6,4,5,8,7] => ?
=> ? = 3
[.,[[.,.],[[.,[.,[[.,.],.]]],.]]]
=> [1,0,1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [1,3,2,5,6,8,4,7] => ?
=> ? = 3
[.,[[.,.],[[.,[[.,.],[.,.]]],.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,5,7,4,8,6] => ?
=> ? = 3
[.,[[.,.],[[[.,.],[.,[.,.]]],.]]]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [1,3,2,6,4,7,8,5] => ?
=> ? = 2
[.,[[.,.],[[[[.,.],.],[.,.]],.]]]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [1,3,2,7,4,5,8,6] => ?
=> ? = 2
[.,[[.,.],[[[[.,[.,.]],.],.],.]]]
=> [1,0,1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [1,3,2,7,8,4,5,6] => ?
=> ? = 2
[.,[[.,[.,.]],[[.,.],[.,[.,.]]]]]
=> [1,0,1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,4,2,6,5,7,8] => ?
=> ? = 2
[.,[[.,[.,.]],[[.,[[.,.],.]],.]]]
=> [1,0,1,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [1,3,4,2,6,8,5,7] => ?
=> ? = 3
[.,[[.,[.,[.,.]]],[.,[[.,.],.]]]]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,3,4,5,2,6,8,7] => ?
=> ? = 2
[.,[[.,[[.,.],.]],[[.,[.,.]],.]]]
=> [1,0,1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4,7,8,6] => ?
=> ? = 3
[.,[[[.,.],[.,.]],[[.,[.,.]],.]]]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> [1,4,2,5,3,7,8,6] => ?
=> ? = 2
[.,[[[.,[.,.]],.],[.,[[.,.],.]]]]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,1,0,0]
=> [1,4,5,2,3,6,8,7] => ?
=> ? = 2
[.,[[.,[[.,.],[.,.]]],[.,[.,.]]]]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [1,3,5,2,6,4,7,8] => ?
=> ? = 2
[.,[[[.,.],[.,[.,.]]],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [1,4,2,5,6,3,7,8] => ?
=> ? = 1
[.,[[[.,[.,[.,.]]],.],[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,1,0,0]
=> [1,4,5,6,2,3,8,7] => ?
=> ? = 2
[.,[[[[.,[.,.]],[.,.]],.],[.,.]]]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,5,6,2,7,3,4,8] => ?
=> ? = 1
[.,[[[[.,[[.,.],.]],.],.],[.,.]]]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,5,7,2,3,4,6,8] => ?
=> ? = 2
[.,[[.,[.,[.,[[.,.],[.,.]]]]],.]]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,3,4,5,7,2,8,6] => ?
=> ? = 2
[.,[[.,[.,[[.,.],[.,[.,.]]]]],.]]
=> [1,0,1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,3,4,6,2,7,8,5] => ?
=> ? = 2
[.,[[.,[.,[[[.,.],[.,.]],.]]],.]]
=> [1,0,1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,3,4,7,2,8,5,6] => ?
=> ? = 2
[.,[[.,[[.,.],[.,[.,[.,.]]]]],.]]
=> [1,0,1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,3,5,2,6,7,8,4] => ?
=> ? = 2
[.,[[.,[[[.,.],[[.,.],.]],.]],.]]
=> [1,0,1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,3,6,2,8,4,5,7] => ?
=> ? = 3
[.,[[.,[[[.,[.,.]],[.,.]],.]],.]]
=> [1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,3,6,7,2,8,4,5] => ?
=> ? = 2
[.,[[.,[[[.,[[.,.],.]],.],.]],.]]
=> [1,0,1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,3,6,8,2,4,5,7] => ?
=> ? = 3
[.,[[[.,.],[[.,.],[.,[.,.]]]],.]]
=> [1,0,1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [1,4,2,6,3,7,8,5] => ?
=> ? = 2
[.,[[[.,.],[[.,[.,.]],[.,.]]],.]]
=> [1,0,1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [1,4,2,6,7,3,8,5] => ?
=> ? = 2
[.,[[[.,[.,[.,.]]],[[.,.],.]],.]]
=> [1,0,1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> [1,4,5,6,2,8,3,7] => ?
=> ? = 2
[.,[[[.,[[.,[.,.]],.]],[.,.]],.]]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,4,6,7,2,3,8,5] => ?
=> ? = 2
[.,[[[[.,.],[.,[.,.]]],[.,.]],.]]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [1,5,2,6,7,3,8,4] => ?
=> ? = 1
[.,[[[[.,[.,.]],[.,.]],[.,.]],.]]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [1,5,6,2,7,3,8,4] => ?
=> ? = 1
[.,[[[.,[.,[[.,.],[.,.]]]],.],.]]
=> [1,0,1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [1,4,5,7,2,8,3,6] => ?
=> ? = 2
[.,[[[.,[.,[[[.,.],.],.]]],.],.]]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,4,5,8,2,3,6,7] => ?
=> ? = 2
[[.,.],[.,[.,[[.,[.,.]],[.,.]]]]]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,4,6,7,5,8] => ?
=> ? = 2
[[.,.],[.,[.,[[.,[[.,.],.]],.]]]]
=> [1,1,0,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,1,3,4,6,8,5,7] => ?
=> ? = 3
[[.,.],[.,[[.,.],[[[.,.],.],.]]]]
=> [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,3,5,4,8,6,7] => ?
=> ? = 3
[[.,.],[.,[[.,[.,.]],[.,[.,.]]]]]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,1,3,5,6,4,7,8] => ?
=> ? = 2
[[.,.],[.,[[.,[[.,.],[.,.]]],.]]]
=> [1,1,0,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [2,1,3,5,7,4,8,6] => ?
=> ? = 3
[[.,.],[.,[[.,[[.,[.,.]],.]],.]]]
=> [1,1,0,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,1,3,5,7,8,4,6] => ?
=> ? = 3
[[.,.],[.,[[[.,.],[.,[.,.]]],.]]]
=> [1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,1,3,6,4,7,8,5] => ?
=> ? = 2
[[.,.],[.,[[[[.,.],.],[.,.]],.]]]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,1,3,7,4,5,8,6] => ?
=> ? = 2
[[.,.],[.,[[[.,[[.,.],.]],.],.]]]
=> [1,1,0,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,1,3,6,8,4,5,7] => ?
=> ? = 3
[[.,.],[[[.,.],.],[[.,.],[.,.]]]]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,5,3,4,7,6,8] => ?
=> ? = 3
Description
The number of maximal left branches of a binary tree.
A maximal left branch of a binary tree is an inclusion wise maximal path which consists of left edges only. This statistic records the number of distinct maximal left branches in the tree.
Matching statistic: St000374
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => [[.,.],.]
=> [1,2] => 0
[[.,.],.]
=> [1,2] => [.,[.,.]]
=> [2,1] => 1
[.,[.,[.,.]]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,2,3] => 0
[.,[[.,.],.]]
=> [2,3,1] => [[.,[.,.]],.]
=> [2,1,3] => 1
[[.,.],[.,.]]
=> [3,1,2] => [[.,.],[.,.]]
=> [1,3,2] => 1
[[.,[.,.]],.]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => 1
[[[.,.],.],.]
=> [1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 2
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 2
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [[[[.,.],[.,.]],.],.]
=> [1,3,2,4,5] => 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [[[[.,.],[.,.]],.],.]
=> [1,3,2,4,5] => 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [[[.,[.,.]],[.,.]],.]
=> [2,1,4,3,5] => 2
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [[[.,[.,.]],[.,.]],.]
=> [2,1,4,3,5] => 2
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => 2
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => 2
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => 2
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => 2
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => 2
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 2
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => 2
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => 1
[.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [6,7,4,5,3,2,1] => [[[[[.,[.,.]],[.,.]],.],.],.]
=> [2,1,4,3,5,6,7] => ? = 2
[.,[.,[.,[[.,[[.,.],.]],.]]]]
=> [5,6,4,7,3,2,1] => [[[[[.,[.,.]],[.,.]],.],.],.]
=> [2,1,4,3,5,6,7] => ? = 2
[.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [6,7,5,3,4,2,1] => [[[[[.,[.,.]],.],[.,.]],.],.]
=> [2,1,3,5,4,6,7] => ? = 2
[.,[.,[[.,.],[[[.,.],.],.]]]]
=> [5,6,7,3,4,2,1] => [[[[.,[.,[.,.]]],[.,.]],.],.]
=> [3,2,1,5,4,6,7] => ? = 2
[.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [6,7,4,3,5,2,1] => [[[[[.,[.,.]],.],[.,.]],.],.]
=> [2,1,3,5,4,6,7] => ? = 2
[.,[.,[[[.,.],.],[[.,.],.]]]]
=> [6,7,3,4,5,2,1] => [[[[.,[.,.]],[.,[.,.]]],.],.]
=> [2,1,5,4,3,6,7] => ? = 2
[.,[.,[[[[.,.],.],.],[.,.]]]]
=> [7,3,4,5,6,2,1] => [[[[.,.],[.,[.,[.,.]]]],.],.]
=> [1,5,4,3,2,6,7] => ? = 1
[.,[.,[[.,[.,[[.,.],.]]],.]]]
=> [5,6,4,3,7,2,1] => [[[[[.,[.,.]],.],[.,.]],.],.]
=> [2,1,3,5,4,6,7] => ? = 2
[.,[.,[[.,[[[.,.],.],.]],.]]]
=> [4,5,6,3,7,2,1] => [[[[.,[.,[.,.]]],[.,.]],.],.]
=> [3,2,1,5,4,6,7] => ? = 2
[.,[.,[[[.,.],[[.,.],.]],.]]]
=> [5,6,3,4,7,2,1] => [[[[.,[.,.]],[.,[.,.]]],.],.]
=> [2,1,5,4,3,6,7] => ? = 2
[.,[.,[[[[.,.],.],[.,.]],.]]]
=> [6,3,4,5,7,2,1] => [[[[.,.],[.,[.,[.,.]]]],.],.]
=> [1,5,4,3,2,6,7] => ? = 1
[.,[.,[[[.,[[.,.],.]],.],.]]]
=> [4,5,3,6,7,2,1] => [[[[.,[.,.]],[.,[.,.]]],.],.]
=> [2,1,5,4,3,6,7] => ? = 2
[.,[.,[[[[.,.],[.,.]],.],.]]]
=> [5,3,4,6,7,2,1] => [[[[.,.],[.,[.,[.,.]]]],.],.]
=> [1,5,4,3,2,6,7] => ? = 1
[.,[.,[[[[.,[.,.]],.],.],.]]]
=> [4,3,5,6,7,2,1] => [[[[.,.],[.,[.,[.,.]]]],.],.]
=> [1,5,4,3,2,6,7] => ? = 1
[.,[[.,.],[.,[.,[[.,.],.]]]]]
=> [6,7,5,4,2,3,1] => [[[[[.,[.,.]],.],.],[.,.]],.]
=> [2,1,3,4,6,5,7] => ? = 2
[.,[[.,.],[.,[[[.,.],.],.]]]]
=> [5,6,7,4,2,3,1] => [[[[.,[.,[.,.]]],.],[.,.]],.]
=> [3,2,1,4,6,5,7] => ? = 2
[.,[[.,.],[[.,.],[[.,.],.]]]]
=> [6,7,4,5,2,3,1] => [[[[.,[.,.]],[.,.]],[.,.]],.]
=> [2,1,4,3,6,5,7] => ? = 3
[.,[[.,.],[[.,[[.,.],.]],.]]]
=> [5,6,4,7,2,3,1] => [[[[.,[.,.]],[.,.]],[.,.]],.]
=> [2,1,4,3,6,5,7] => ? = 3
[.,[[.,.],[[[[.,.],.],.],.]]]
=> [4,5,6,7,2,3,1] => [[[.,[.,[.,[.,.]]]],[.,.]],.]
=> [4,3,2,1,6,5,7] => ? = 2
[.,[[.,[.,.]],[.,[[.,.],.]]]]
=> [6,7,5,3,2,4,1] => [[[[[.,[.,.]],.],.],[.,.]],.]
=> [2,1,3,4,6,5,7] => ? = 2
[.,[[.,[.,.]],[[[.,.],.],.]]]
=> [5,6,7,3,2,4,1] => [[[[.,[.,[.,.]]],.],[.,.]],.]
=> [3,2,1,4,6,5,7] => ? = 2
[.,[[[.,.],.],[.,[[.,.],.]]]]
=> [6,7,5,2,3,4,1] => [[[[.,[.,.]],.],[.,[.,.]]],.]
=> [2,1,3,6,5,4,7] => ? = 2
[.,[[[.,.],.],[[[.,.],.],.]]]
=> [5,6,7,2,3,4,1] => [[[.,[.,[.,.]]],[.,[.,.]]],.]
=> [3,2,1,6,5,4,7] => ? = 2
[.,[[.,[.,[.,.]]],[[.,.],.]]]
=> [6,7,4,3,2,5,1] => [[[[[.,[.,.]],.],.],[.,.]],.]
=> [2,1,3,4,6,5,7] => ? = 2
[.,[[.,[[.,.],.]],[[.,.],.]]]
=> [6,7,3,4,2,5,1] => [[[[.,[.,.]],[.,.]],[.,.]],.]
=> [2,1,4,3,6,5,7] => ? = 3
[.,[[[.,.],[.,.]],[[.,.],.]]]
=> [6,7,4,2,3,5,1] => [[[[.,[.,.]],.],[.,[.,.]]],.]
=> [2,1,3,6,5,4,7] => ? = 2
[.,[[[.,[.,.]],.],[[.,.],.]]]
=> [6,7,3,2,4,5,1] => [[[[.,[.,.]],.],[.,[.,.]]],.]
=> [2,1,3,6,5,4,7] => ? = 2
[.,[[[[.,.],.],.],[[.,.],.]]]
=> [6,7,2,3,4,5,1] => [[[.,[.,.]],[.,[.,[.,.]]]],.]
=> [2,1,6,5,4,3,7] => ? = 2
[.,[[[[[.,.],.],.],.],[.,.]]]
=> [7,2,3,4,5,6,1] => [[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [1,6,5,4,3,2,7] => ? = 1
[.,[[.,[.,[.,[[.,.],.]]]],.]]
=> [5,6,4,3,2,7,1] => [[[[[.,[.,.]],.],.],[.,.]],.]
=> [2,1,3,4,6,5,7] => ? = 2
[.,[[.,[.,[[[.,.],.],.]]],.]]
=> [4,5,6,3,2,7,1] => [[[[.,[.,[.,.]]],.],[.,.]],.]
=> [3,2,1,4,6,5,7] => ? = 2
[.,[[.,[[.,.],[[.,.],.]]],.]]
=> [5,6,3,4,2,7,1] => [[[[.,[.,.]],[.,.]],[.,.]],.]
=> [2,1,4,3,6,5,7] => ? = 3
[.,[[.,[[.,[[.,.],.]],.]],.]]
=> [4,5,3,6,2,7,1] => [[[[.,[.,.]],[.,.]],[.,.]],.]
=> [2,1,4,3,6,5,7] => ? = 3
[.,[[.,[[[[.,.],.],.],.]],.]]
=> [3,4,5,6,2,7,1] => [[[.,[.,[.,[.,.]]]],[.,.]],.]
=> [4,3,2,1,6,5,7] => ? = 2
[.,[[[.,.],[.,[[.,.],.]]],.]]
=> [5,6,4,2,3,7,1] => [[[[.,[.,.]],.],[.,[.,.]]],.]
=> [2,1,3,6,5,4,7] => ? = 2
[.,[[[.,.],[[[.,.],.],.]],.]]
=> [4,5,6,2,3,7,1] => [[[.,[.,[.,.]]],[.,[.,.]]],.]
=> [3,2,1,6,5,4,7] => ? = 2
[.,[[[.,[.,.]],[[.,.],.]],.]]
=> [5,6,3,2,4,7,1] => [[[[.,[.,.]],.],[.,[.,.]]],.]
=> [2,1,3,6,5,4,7] => ? = 2
[.,[[[[.,.],.],[[.,.],.]],.]]
=> [5,6,2,3,4,7,1] => [[[.,[.,.]],[.,[.,[.,.]]]],.]
=> [2,1,6,5,4,3,7] => ? = 2
[.,[[[[[.,.],.],.],[.,.]],.]]
=> [6,2,3,4,5,7,1] => [[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [1,6,5,4,3,2,7] => ? = 1
[.,[[[.,[.,[[.,.],.]]],.],.]]
=> [4,5,3,2,6,7,1] => [[[[.,[.,.]],.],[.,[.,.]]],.]
=> [2,1,3,6,5,4,7] => ? = 2
[.,[[[.,[[[.,.],.],.]],.],.]]
=> [3,4,5,2,6,7,1] => [[[.,[.,[.,.]]],[.,[.,.]]],.]
=> [3,2,1,6,5,4,7] => ? = 2
[.,[[[[.,.],[[.,.],.]],.],.]]
=> [4,5,2,3,6,7,1] => [[[.,[.,.]],[.,[.,[.,.]]]],.]
=> [2,1,6,5,4,3,7] => ? = 2
[.,[[[[[.,.],.],[.,.]],.],.]]
=> [5,2,3,4,6,7,1] => [[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [1,6,5,4,3,2,7] => ? = 1
[.,[[[[.,[[.,.],.]],.],.],.]]
=> [3,4,2,5,6,7,1] => [[[.,[.,.]],[.,[.,[.,.]]]],.]
=> [2,1,6,5,4,3,7] => ? = 2
[.,[[[[[.,.],[.,.]],.],.],.]]
=> [4,2,3,5,6,7,1] => [[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [1,6,5,4,3,2,7] => ? = 1
[.,[[[[[.,[.,.]],.],.],.],.]]
=> [3,2,4,5,6,7,1] => [[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [1,6,5,4,3,2,7] => ? = 1
[[.,.],[.,[.,[.,[[.,.],.]]]]]
=> [6,7,5,4,3,1,2] => [[[[[.,[.,.]],.],.],.],[.,.]]
=> [2,1,3,4,5,7,6] => ? = 2
[[.,.],[.,[.,[[[.,.],.],.]]]]
=> [5,6,7,4,3,1,2] => [[[[.,[.,[.,.]]],.],.],[.,.]]
=> [3,2,1,4,5,7,6] => ? = 2
[[.,.],[.,[[.,.],[[.,.],.]]]]
=> [6,7,4,5,3,1,2] => [[[[.,[.,.]],[.,.]],.],[.,.]]
=> [2,1,4,3,5,7,6] => ? = 3
[[.,.],[.,[[.,[[.,.],.]],.]]]
=> [5,6,4,7,3,1,2] => [[[[.,[.,.]],[.,.]],.],[.,.]]
=> [2,1,4,3,5,7,6] => ? = 3
Description
The number of exclusive right-to-left minima of a permutation.
This is the number of right-to-left minima that are not left-to-right maxima.
This is also the number of non weak exceedences of a permutation that are also not mid-points of a decreasing subsequence of length 3.
Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j < j$ and there do not exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$.
See also [[St000213]] and [[St000119]].
Matching statistic: St000996
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => [[.,.],.]
=> [1,2] => 0
[[.,.],.]
=> [1,2] => [.,[.,.]]
=> [2,1] => 1
[.,[.,[.,.]]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,2,3] => 0
[.,[[.,.],.]]
=> [2,3,1] => [[.,[.,.]],.]
=> [2,1,3] => 1
[[.,.],[.,.]]
=> [3,1,2] => [[.,.],[.,.]]
=> [1,3,2] => 1
[[.,[.,.]],.]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => 1
[[[.,.],.],.]
=> [1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 2
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 2
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [[[[.,.],[.,.]],.],.]
=> [1,3,2,4,5] => 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [[[[.,.],[.,.]],.],.]
=> [1,3,2,4,5] => 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [[[.,[.,.]],[.,.]],.]
=> [2,1,4,3,5] => 2
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [[[.,[.,.]],[.,.]],.]
=> [2,1,4,3,5] => 2
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => 2
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => 2
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => 2
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => 2
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => 2
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 2
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => 2
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => 1
[.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [6,7,4,5,3,2,1] => [[[[[.,[.,.]],[.,.]],.],.],.]
=> [2,1,4,3,5,6,7] => ? = 2
[.,[.,[.,[[.,[[.,.],.]],.]]]]
=> [5,6,4,7,3,2,1] => [[[[[.,[.,.]],[.,.]],.],.],.]
=> [2,1,4,3,5,6,7] => ? = 2
[.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [6,7,5,3,4,2,1] => [[[[[.,[.,.]],.],[.,.]],.],.]
=> [2,1,3,5,4,6,7] => ? = 2
[.,[.,[[.,.],[[[.,.],.],.]]]]
=> [5,6,7,3,4,2,1] => [[[[.,[.,[.,.]]],[.,.]],.],.]
=> [3,2,1,5,4,6,7] => ? = 2
[.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [6,7,4,3,5,2,1] => [[[[[.,[.,.]],.],[.,.]],.],.]
=> [2,1,3,5,4,6,7] => ? = 2
[.,[.,[[[.,.],.],[[.,.],.]]]]
=> [6,7,3,4,5,2,1] => [[[[.,[.,.]],[.,[.,.]]],.],.]
=> [2,1,5,4,3,6,7] => ? = 2
[.,[.,[[[[.,.],.],.],[.,.]]]]
=> [7,3,4,5,6,2,1] => [[[[.,.],[.,[.,[.,.]]]],.],.]
=> [1,5,4,3,2,6,7] => ? = 1
[.,[.,[[.,[.,[[.,.],.]]],.]]]
=> [5,6,4,3,7,2,1] => [[[[[.,[.,.]],.],[.,.]],.],.]
=> [2,1,3,5,4,6,7] => ? = 2
[.,[.,[[.,[[[.,.],.],.]],.]]]
=> [4,5,6,3,7,2,1] => [[[[.,[.,[.,.]]],[.,.]],.],.]
=> [3,2,1,5,4,6,7] => ? = 2
[.,[.,[[[.,.],[[.,.],.]],.]]]
=> [5,6,3,4,7,2,1] => [[[[.,[.,.]],[.,[.,.]]],.],.]
=> [2,1,5,4,3,6,7] => ? = 2
[.,[.,[[[[.,.],.],[.,.]],.]]]
=> [6,3,4,5,7,2,1] => [[[[.,.],[.,[.,[.,.]]]],.],.]
=> [1,5,4,3,2,6,7] => ? = 1
[.,[.,[[[.,[[.,.],.]],.],.]]]
=> [4,5,3,6,7,2,1] => [[[[.,[.,.]],[.,[.,.]]],.],.]
=> [2,1,5,4,3,6,7] => ? = 2
[.,[.,[[[[.,.],[.,.]],.],.]]]
=> [5,3,4,6,7,2,1] => [[[[.,.],[.,[.,[.,.]]]],.],.]
=> [1,5,4,3,2,6,7] => ? = 1
[.,[.,[[[[.,[.,.]],.],.],.]]]
=> [4,3,5,6,7,2,1] => [[[[.,.],[.,[.,[.,.]]]],.],.]
=> [1,5,4,3,2,6,7] => ? = 1
[.,[[.,.],[.,[.,[[.,.],.]]]]]
=> [6,7,5,4,2,3,1] => [[[[[.,[.,.]],.],.],[.,.]],.]
=> [2,1,3,4,6,5,7] => ? = 2
[.,[[.,.],[.,[[[.,.],.],.]]]]
=> [5,6,7,4,2,3,1] => [[[[.,[.,[.,.]]],.],[.,.]],.]
=> [3,2,1,4,6,5,7] => ? = 2
[.,[[.,.],[[.,.],[[.,.],.]]]]
=> [6,7,4,5,2,3,1] => [[[[.,[.,.]],[.,.]],[.,.]],.]
=> [2,1,4,3,6,5,7] => ? = 3
[.,[[.,.],[[.,[[.,.],.]],.]]]
=> [5,6,4,7,2,3,1] => [[[[.,[.,.]],[.,.]],[.,.]],.]
=> [2,1,4,3,6,5,7] => ? = 3
[.,[[.,.],[[[[.,.],.],.],.]]]
=> [4,5,6,7,2,3,1] => [[[.,[.,[.,[.,.]]]],[.,.]],.]
=> [4,3,2,1,6,5,7] => ? = 2
[.,[[.,[.,.]],[.,[[.,.],.]]]]
=> [6,7,5,3,2,4,1] => [[[[[.,[.,.]],.],.],[.,.]],.]
=> [2,1,3,4,6,5,7] => ? = 2
[.,[[.,[.,.]],[[[.,.],.],.]]]
=> [5,6,7,3,2,4,1] => [[[[.,[.,[.,.]]],.],[.,.]],.]
=> [3,2,1,4,6,5,7] => ? = 2
[.,[[[.,.],.],[.,[[.,.],.]]]]
=> [6,7,5,2,3,4,1] => [[[[.,[.,.]],.],[.,[.,.]]],.]
=> [2,1,3,6,5,4,7] => ? = 2
[.,[[[.,.],.],[[[.,.],.],.]]]
=> [5,6,7,2,3,4,1] => [[[.,[.,[.,.]]],[.,[.,.]]],.]
=> [3,2,1,6,5,4,7] => ? = 2
[.,[[.,[.,[.,.]]],[[.,.],.]]]
=> [6,7,4,3,2,5,1] => [[[[[.,[.,.]],.],.],[.,.]],.]
=> [2,1,3,4,6,5,7] => ? = 2
[.,[[.,[[.,.],.]],[[.,.],.]]]
=> [6,7,3,4,2,5,1] => [[[[.,[.,.]],[.,.]],[.,.]],.]
=> [2,1,4,3,6,5,7] => ? = 3
[.,[[[.,.],[.,.]],[[.,.],.]]]
=> [6,7,4,2,3,5,1] => [[[[.,[.,.]],.],[.,[.,.]]],.]
=> [2,1,3,6,5,4,7] => ? = 2
[.,[[[.,[.,.]],.],[[.,.],.]]]
=> [6,7,3,2,4,5,1] => [[[[.,[.,.]],.],[.,[.,.]]],.]
=> [2,1,3,6,5,4,7] => ? = 2
[.,[[[[.,.],.],.],[[.,.],.]]]
=> [6,7,2,3,4,5,1] => [[[.,[.,.]],[.,[.,[.,.]]]],.]
=> [2,1,6,5,4,3,7] => ? = 2
[.,[[[[[.,.],.],.],.],[.,.]]]
=> [7,2,3,4,5,6,1] => [[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [1,6,5,4,3,2,7] => ? = 1
[.,[[.,[.,[.,[[.,.],.]]]],.]]
=> [5,6,4,3,2,7,1] => [[[[[.,[.,.]],.],.],[.,.]],.]
=> [2,1,3,4,6,5,7] => ? = 2
[.,[[.,[.,[[[.,.],.],.]]],.]]
=> [4,5,6,3,2,7,1] => [[[[.,[.,[.,.]]],.],[.,.]],.]
=> [3,2,1,4,6,5,7] => ? = 2
[.,[[.,[[.,.],[[.,.],.]]],.]]
=> [5,6,3,4,2,7,1] => [[[[.,[.,.]],[.,.]],[.,.]],.]
=> [2,1,4,3,6,5,7] => ? = 3
[.,[[.,[[.,[[.,.],.]],.]],.]]
=> [4,5,3,6,2,7,1] => [[[[.,[.,.]],[.,.]],[.,.]],.]
=> [2,1,4,3,6,5,7] => ? = 3
[.,[[.,[[[[.,.],.],.],.]],.]]
=> [3,4,5,6,2,7,1] => [[[.,[.,[.,[.,.]]]],[.,.]],.]
=> [4,3,2,1,6,5,7] => ? = 2
[.,[[[.,.],[.,[[.,.],.]]],.]]
=> [5,6,4,2,3,7,1] => [[[[.,[.,.]],.],[.,[.,.]]],.]
=> [2,1,3,6,5,4,7] => ? = 2
[.,[[[.,.],[[[.,.],.],.]],.]]
=> [4,5,6,2,3,7,1] => [[[.,[.,[.,.]]],[.,[.,.]]],.]
=> [3,2,1,6,5,4,7] => ? = 2
[.,[[[.,[.,.]],[[.,.],.]],.]]
=> [5,6,3,2,4,7,1] => [[[[.,[.,.]],.],[.,[.,.]]],.]
=> [2,1,3,6,5,4,7] => ? = 2
[.,[[[[.,.],.],[[.,.],.]],.]]
=> [5,6,2,3,4,7,1] => [[[.,[.,.]],[.,[.,[.,.]]]],.]
=> [2,1,6,5,4,3,7] => ? = 2
[.,[[[[[.,.],.],.],[.,.]],.]]
=> [6,2,3,4,5,7,1] => [[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [1,6,5,4,3,2,7] => ? = 1
[.,[[[.,[.,[[.,.],.]]],.],.]]
=> [4,5,3,2,6,7,1] => [[[[.,[.,.]],.],[.,[.,.]]],.]
=> [2,1,3,6,5,4,7] => ? = 2
[.,[[[.,[[[.,.],.],.]],.],.]]
=> [3,4,5,2,6,7,1] => [[[.,[.,[.,.]]],[.,[.,.]]],.]
=> [3,2,1,6,5,4,7] => ? = 2
[.,[[[[.,.],[[.,.],.]],.],.]]
=> [4,5,2,3,6,7,1] => [[[.,[.,.]],[.,[.,[.,.]]]],.]
=> [2,1,6,5,4,3,7] => ? = 2
[.,[[[[[.,.],.],[.,.]],.],.]]
=> [5,2,3,4,6,7,1] => [[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [1,6,5,4,3,2,7] => ? = 1
[.,[[[[.,[[.,.],.]],.],.],.]]
=> [3,4,2,5,6,7,1] => [[[.,[.,.]],[.,[.,[.,.]]]],.]
=> [2,1,6,5,4,3,7] => ? = 2
[.,[[[[[.,.],[.,.]],.],.],.]]
=> [4,2,3,5,6,7,1] => [[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [1,6,5,4,3,2,7] => ? = 1
[.,[[[[[.,[.,.]],.],.],.],.]]
=> [3,2,4,5,6,7,1] => [[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [1,6,5,4,3,2,7] => ? = 1
[[.,.],[.,[.,[.,[[.,.],.]]]]]
=> [6,7,5,4,3,1,2] => [[[[[.,[.,.]],.],.],.],[.,.]]
=> [2,1,3,4,5,7,6] => ? = 2
[[.,.],[.,[.,[[[.,.],.],.]]]]
=> [5,6,7,4,3,1,2] => [[[[.,[.,[.,.]]],.],.],[.,.]]
=> [3,2,1,4,5,7,6] => ? = 2
[[.,.],[.,[[.,.],[[.,.],.]]]]
=> [6,7,4,5,3,1,2] => [[[[.,[.,.]],[.,.]],.],[.,.]]
=> [2,1,4,3,5,7,6] => ? = 3
[[.,.],[.,[[.,[[.,.],.]],.]]]
=> [5,6,4,7,3,1,2] => [[[[.,[.,.]],[.,.]],.],[.,.]]
=> [2,1,4,3,5,7,6] => ? = 3
Description
The number of exclusive left-to-right maxima of a permutation.
This is the number of left-to-right maxima that are not right-to-left minima.
The following 43 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000251The number of nonsingleton blocks of a set partition. St000670The reversal length of a permutation. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000884The number of isolated descents of a permutation. St001354The number of series nodes in the modular decomposition of a graph. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000834The number of right outer peaks of a permutation. St000703The number of deficiencies of a permutation. St001737The number of descents of type 2 in a permutation. St000035The number of left outer peaks of a permutation. St000386The number of factors DDU in a Dyck path. St000742The number of big ascents of a permutation after prepending zero. St000354The number of recoils of a permutation. St001729The number of visible descents of a permutation. St001928The number of non-overlapping descents in a permutation. St000470The number of runs in a permutation. St000245The number of ascents of a permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001665The number of pure excedances of a permutation. St000702The number of weak deficiencies of a permutation. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St000021The number of descents of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000155The number of exceedances (also excedences) of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St001840The number of descents of a set partition. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000325The width of the tree associated to a permutation. St000647The number of big descents of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000646The number of big ascents of a permutation. St000711The number of big exceedences of a permutation. St000779The tier of a permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001896The number of right descents of a signed permutations. St001597The Frobenius rank of a skew partition. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001935The number of ascents in a parking function. St001946The number of descents in a parking function. St000920The logarithmic height of a Dyck path.
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