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Your data matches 23 different statistics following compositions of up to 3 maps.
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Matching statistic: St000035
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Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000035: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
St000035: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => 1
[2,1] => [1,2] => [2,1] => 1
[1,2,3] => [1,2,3] => [3,2,1] => 1
[1,3,2] => [1,2,3] => [3,2,1] => 1
[2,1,3] => [1,2,3] => [3,2,1] => 1
[2,3,1] => [1,2,3] => [3,2,1] => 1
[3,1,2] => [1,3,2] => [3,1,2] => 1
[3,2,1] => [1,3,2] => [3,1,2] => 1
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 1
[1,2,4,3] => [1,2,3,4] => [4,3,2,1] => 1
[1,3,2,4] => [1,2,3,4] => [4,3,2,1] => 1
[1,3,4,2] => [1,2,3,4] => [4,3,2,1] => 1
[1,4,2,3] => [1,2,4,3] => [4,3,1,2] => 1
[1,4,3,2] => [1,2,4,3] => [4,3,1,2] => 1
[2,1,3,4] => [1,2,3,4] => [4,3,2,1] => 1
[2,1,4,3] => [1,2,3,4] => [4,3,2,1] => 1
[2,3,1,4] => [1,2,3,4] => [4,3,2,1] => 1
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => 1
[2,4,1,3] => [1,2,4,3] => [4,3,1,2] => 1
[2,4,3,1] => [1,2,4,3] => [4,3,1,2] => 1
[3,1,2,4] => [1,3,2,4] => [4,2,3,1] => 2
[3,1,4,2] => [1,3,4,2] => [4,2,1,3] => 1
[3,2,1,4] => [1,3,2,4] => [4,2,3,1] => 2
[3,2,4,1] => [1,3,4,2] => [4,2,1,3] => 1
[3,4,1,2] => [1,3,2,4] => [4,2,3,1] => 2
[3,4,2,1] => [1,3,2,4] => [4,2,3,1] => 2
[4,1,2,3] => [1,4,3,2] => [4,1,2,3] => 1
[4,1,3,2] => [1,4,2,3] => [4,1,3,2] => 2
[4,2,1,3] => [1,4,3,2] => [4,1,2,3] => 1
[4,2,3,1] => [1,4,2,3] => [4,1,3,2] => 2
[4,3,1,2] => [1,4,2,3] => [4,1,3,2] => 2
[4,3,2,1] => [1,4,2,3] => [4,1,3,2] => 2
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,2,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,2,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,2,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,2,5,3,4] => [1,2,3,5,4] => [5,4,3,1,2] => 1
[1,2,5,4,3] => [1,2,3,5,4] => [5,4,3,1,2] => 1
[1,3,2,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,3,2,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,3,4,2,5] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,3,4,5,2] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,3,5,2,4] => [1,2,3,5,4] => [5,4,3,1,2] => 1
[1,3,5,4,2] => [1,2,3,5,4] => [5,4,3,1,2] => 1
[1,4,2,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => 2
[1,4,2,5,3] => [1,2,4,5,3] => [5,4,2,1,3] => 1
[1,4,3,2,5] => [1,2,4,3,5] => [5,4,2,3,1] => 2
[1,4,3,5,2] => [1,2,4,5,3] => [5,4,2,1,3] => 1
[1,4,5,2,3] => [1,2,4,3,5] => [5,4,2,3,1] => 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: St001280
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1]
=> 0
[1,2] => [1,2] => [2] => [2]
=> 1
[2,1] => [1,2] => [2] => [2]
=> 1
[1,2,3] => [1,2,3] => [3] => [3]
=> 1
[1,3,2] => [1,2,3] => [3] => [3]
=> 1
[2,1,3] => [1,2,3] => [3] => [3]
=> 1
[2,3,1] => [1,2,3] => [3] => [3]
=> 1
[3,1,2] => [1,3,2] => [2,1] => [2,1]
=> 1
[3,2,1] => [1,3,2] => [2,1] => [2,1]
=> 1
[1,2,3,4] => [1,2,3,4] => [4] => [4]
=> 1
[1,2,4,3] => [1,2,3,4] => [4] => [4]
=> 1
[1,3,2,4] => [1,2,3,4] => [4] => [4]
=> 1
[1,3,4,2] => [1,2,3,4] => [4] => [4]
=> 1
[1,4,2,3] => [1,2,4,3] => [3,1] => [3,1]
=> 1
[1,4,3,2] => [1,2,4,3] => [3,1] => [3,1]
=> 1
[2,1,3,4] => [1,2,3,4] => [4] => [4]
=> 1
[2,1,4,3] => [1,2,3,4] => [4] => [4]
=> 1
[2,3,1,4] => [1,2,3,4] => [4] => [4]
=> 1
[2,3,4,1] => [1,2,3,4] => [4] => [4]
=> 1
[2,4,1,3] => [1,2,4,3] => [3,1] => [3,1]
=> 1
[2,4,3,1] => [1,2,4,3] => [3,1] => [3,1]
=> 1
[3,1,2,4] => [1,3,2,4] => [2,2] => [2,2]
=> 2
[3,1,4,2] => [1,3,4,2] => [3,1] => [3,1]
=> 1
[3,2,1,4] => [1,3,2,4] => [2,2] => [2,2]
=> 2
[3,2,4,1] => [1,3,4,2] => [3,1] => [3,1]
=> 1
[3,4,1,2] => [1,3,2,4] => [2,2] => [2,2]
=> 2
[3,4,2,1] => [1,3,2,4] => [2,2] => [2,2]
=> 2
[4,1,2,3] => [1,4,3,2] => [2,1,1] => [2,1,1]
=> 1
[4,1,3,2] => [1,4,2,3] => [2,2] => [2,2]
=> 2
[4,2,1,3] => [1,4,3,2] => [2,1,1] => [2,1,1]
=> 1
[4,2,3,1] => [1,4,2,3] => [2,2] => [2,2]
=> 2
[4,3,1,2] => [1,4,2,3] => [2,2] => [2,2]
=> 2
[4,3,2,1] => [1,4,2,3] => [2,2] => [2,2]
=> 2
[1,2,3,4,5] => [1,2,3,4,5] => [5] => [5]
=> 1
[1,2,3,5,4] => [1,2,3,4,5] => [5] => [5]
=> 1
[1,2,4,3,5] => [1,2,3,4,5] => [5] => [5]
=> 1
[1,2,4,5,3] => [1,2,3,4,5] => [5] => [5]
=> 1
[1,2,5,3,4] => [1,2,3,5,4] => [4,1] => [4,1]
=> 1
[1,2,5,4,3] => [1,2,3,5,4] => [4,1] => [4,1]
=> 1
[1,3,2,4,5] => [1,2,3,4,5] => [5] => [5]
=> 1
[1,3,2,5,4] => [1,2,3,4,5] => [5] => [5]
=> 1
[1,3,4,2,5] => [1,2,3,4,5] => [5] => [5]
=> 1
[1,3,4,5,2] => [1,2,3,4,5] => [5] => [5]
=> 1
[1,3,5,2,4] => [1,2,3,5,4] => [4,1] => [4,1]
=> 1
[1,3,5,4,2] => [1,2,3,5,4] => [4,1] => [4,1]
=> 1
[1,4,2,3,5] => [1,2,4,3,5] => [3,2] => [3,2]
=> 2
[1,4,2,5,3] => [1,2,4,5,3] => [4,1] => [4,1]
=> 1
[1,4,3,2,5] => [1,2,4,3,5] => [3,2] => [3,2]
=> 2
[1,4,3,5,2] => [1,2,4,5,3] => [4,1] => [4,1]
=> 1
[1,4,5,2,3] => [1,2,4,3,5] => [3,2] => [3,2]
=> 2
[] => [] => [] => ?
=> ? = 0
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St000390
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
Mp00280: Binary words —path rowmotion⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 80% ●values known / values provided: 100%●distinct values known / distinct values provided: 80%
Mp00109: Permutations —descent word⟶ Binary words
Mp00280: Binary words —path rowmotion⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 80% ●values known / values provided: 100%●distinct values known / distinct values provided: 80%
Values
[1] => [1] => => => ? = 0
[1,2] => [1,2] => 0 => 1 => 1
[2,1] => [1,2] => 0 => 1 => 1
[1,2,3] => [1,2,3] => 00 => 01 => 1
[1,3,2] => [1,2,3] => 00 => 01 => 1
[2,1,3] => [1,2,3] => 00 => 01 => 1
[2,3,1] => [1,2,3] => 00 => 01 => 1
[3,1,2] => [1,3,2] => 01 => 10 => 1
[3,2,1] => [1,3,2] => 01 => 10 => 1
[1,2,3,4] => [1,2,3,4] => 000 => 001 => 1
[1,2,4,3] => [1,2,3,4] => 000 => 001 => 1
[1,3,2,4] => [1,2,3,4] => 000 => 001 => 1
[1,3,4,2] => [1,2,3,4] => 000 => 001 => 1
[1,4,2,3] => [1,2,4,3] => 001 => 010 => 1
[1,4,3,2] => [1,2,4,3] => 001 => 010 => 1
[2,1,3,4] => [1,2,3,4] => 000 => 001 => 1
[2,1,4,3] => [1,2,3,4] => 000 => 001 => 1
[2,3,1,4] => [1,2,3,4] => 000 => 001 => 1
[2,3,4,1] => [1,2,3,4] => 000 => 001 => 1
[2,4,1,3] => [1,2,4,3] => 001 => 010 => 1
[2,4,3,1] => [1,2,4,3] => 001 => 010 => 1
[3,1,2,4] => [1,3,2,4] => 010 => 101 => 2
[3,1,4,2] => [1,3,4,2] => 001 => 010 => 1
[3,2,1,4] => [1,3,2,4] => 010 => 101 => 2
[3,2,4,1] => [1,3,4,2] => 001 => 010 => 1
[3,4,1,2] => [1,3,2,4] => 010 => 101 => 2
[3,4,2,1] => [1,3,2,4] => 010 => 101 => 2
[4,1,2,3] => [1,4,3,2] => 011 => 100 => 1
[4,1,3,2] => [1,4,2,3] => 010 => 101 => 2
[4,2,1,3] => [1,4,3,2] => 011 => 100 => 1
[4,2,3,1] => [1,4,2,3] => 010 => 101 => 2
[4,3,1,2] => [1,4,2,3] => 010 => 101 => 2
[4,3,2,1] => [1,4,2,3] => 010 => 101 => 2
[1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0001 => 1
[1,2,3,5,4] => [1,2,3,4,5] => 0000 => 0001 => 1
[1,2,4,3,5] => [1,2,3,4,5] => 0000 => 0001 => 1
[1,2,4,5,3] => [1,2,3,4,5] => 0000 => 0001 => 1
[1,2,5,3,4] => [1,2,3,5,4] => 0001 => 0010 => 1
[1,2,5,4,3] => [1,2,3,5,4] => 0001 => 0010 => 1
[1,3,2,4,5] => [1,2,3,4,5] => 0000 => 0001 => 1
[1,3,2,5,4] => [1,2,3,4,5] => 0000 => 0001 => 1
[1,3,4,2,5] => [1,2,3,4,5] => 0000 => 0001 => 1
[1,3,4,5,2] => [1,2,3,4,5] => 0000 => 0001 => 1
[1,3,5,2,4] => [1,2,3,5,4] => 0001 => 0010 => 1
[1,3,5,4,2] => [1,2,3,5,4] => 0001 => 0010 => 1
[1,4,2,3,5] => [1,2,4,3,5] => 0010 => 0101 => 2
[1,4,2,5,3] => [1,2,4,5,3] => 0001 => 0010 => 1
[1,4,3,2,5] => [1,2,4,3,5] => 0010 => 0101 => 2
[1,4,3,5,2] => [1,2,4,5,3] => 0001 => 0010 => 1
[1,4,5,2,3] => [1,2,4,3,5] => 0010 => 0101 => 2
[1,4,5,3,2] => [1,2,4,3,5] => 0010 => 0101 => 2
[] => [] => ? => ? => ? = 0
[4,8,2,3,5,7,1,6] => [1,4,3,2,8,6,7,5] => ? => ? => ? = 3
[4,8,1,3,2,7,5,6] => [1,4,3,2,8,6,7,5] => ? => ? => ? = 3
[4,8,1,3,5,7,2,6] => [1,4,3,2,8,6,7,5] => ? => ? => ? = 3
Description
The number of runs of ones in a binary word.
Matching statistic: St000291
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 80% ●values known / values provided: 95%●distinct values known / distinct values provided: 80%
Mp00109: Permutations —descent word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 80% ●values known / values provided: 95%●distinct values known / distinct values provided: 80%
Values
[1] => [1] => => ? = 0 - 1
[1,2] => [1,2] => 0 => 0 = 1 - 1
[2,1] => [1,2] => 0 => 0 = 1 - 1
[1,2,3] => [1,2,3] => 00 => 0 = 1 - 1
[1,3,2] => [1,2,3] => 00 => 0 = 1 - 1
[2,1,3] => [1,2,3] => 00 => 0 = 1 - 1
[2,3,1] => [1,2,3] => 00 => 0 = 1 - 1
[3,1,2] => [1,3,2] => 01 => 0 = 1 - 1
[3,2,1] => [1,3,2] => 01 => 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => 000 => 0 = 1 - 1
[1,2,4,3] => [1,2,3,4] => 000 => 0 = 1 - 1
[1,3,2,4] => [1,2,3,4] => 000 => 0 = 1 - 1
[1,3,4,2] => [1,2,3,4] => 000 => 0 = 1 - 1
[1,4,2,3] => [1,2,4,3] => 001 => 0 = 1 - 1
[1,4,3,2] => [1,2,4,3] => 001 => 0 = 1 - 1
[2,1,3,4] => [1,2,3,4] => 000 => 0 = 1 - 1
[2,1,4,3] => [1,2,3,4] => 000 => 0 = 1 - 1
[2,3,1,4] => [1,2,3,4] => 000 => 0 = 1 - 1
[2,3,4,1] => [1,2,3,4] => 000 => 0 = 1 - 1
[2,4,1,3] => [1,2,4,3] => 001 => 0 = 1 - 1
[2,4,3,1] => [1,2,4,3] => 001 => 0 = 1 - 1
[3,1,2,4] => [1,3,2,4] => 010 => 1 = 2 - 1
[3,1,4,2] => [1,3,4,2] => 001 => 0 = 1 - 1
[3,2,1,4] => [1,3,2,4] => 010 => 1 = 2 - 1
[3,2,4,1] => [1,3,4,2] => 001 => 0 = 1 - 1
[3,4,1,2] => [1,3,2,4] => 010 => 1 = 2 - 1
[3,4,2,1] => [1,3,2,4] => 010 => 1 = 2 - 1
[4,1,2,3] => [1,4,3,2] => 011 => 0 = 1 - 1
[4,1,3,2] => [1,4,2,3] => 010 => 1 = 2 - 1
[4,2,1,3] => [1,4,3,2] => 011 => 0 = 1 - 1
[4,2,3,1] => [1,4,2,3] => 010 => 1 = 2 - 1
[4,3,1,2] => [1,4,2,3] => 010 => 1 = 2 - 1
[4,3,2,1] => [1,4,2,3] => 010 => 1 = 2 - 1
[1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4,5] => 0000 => 0 = 1 - 1
[1,2,4,3,5] => [1,2,3,4,5] => 0000 => 0 = 1 - 1
[1,2,4,5,3] => [1,2,3,4,5] => 0000 => 0 = 1 - 1
[1,2,5,3,4] => [1,2,3,5,4] => 0001 => 0 = 1 - 1
[1,2,5,4,3] => [1,2,3,5,4] => 0001 => 0 = 1 - 1
[1,3,2,4,5] => [1,2,3,4,5] => 0000 => 0 = 1 - 1
[1,3,2,5,4] => [1,2,3,4,5] => 0000 => 0 = 1 - 1
[1,3,4,2,5] => [1,2,3,4,5] => 0000 => 0 = 1 - 1
[1,3,4,5,2] => [1,2,3,4,5] => 0000 => 0 = 1 - 1
[1,3,5,2,4] => [1,2,3,5,4] => 0001 => 0 = 1 - 1
[1,3,5,4,2] => [1,2,3,5,4] => 0001 => 0 = 1 - 1
[1,4,2,3,5] => [1,2,4,3,5] => 0010 => 1 = 2 - 1
[1,4,2,5,3] => [1,2,4,5,3] => 0001 => 0 = 1 - 1
[1,4,3,2,5] => [1,2,4,3,5] => 0010 => 1 = 2 - 1
[1,4,3,5,2] => [1,2,4,5,3] => 0001 => 0 = 1 - 1
[1,4,5,2,3] => [1,2,4,3,5] => 0010 => 1 = 2 - 1
[1,4,5,3,2] => [1,2,4,3,5] => 0010 => 1 = 2 - 1
[] => [] => ? => ? = 0 - 1
[2,1,4,3,6,5,8,7,10,9,12,11] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => ? = 1 - 1
[4,8,2,3,5,7,1,6] => [1,4,3,2,8,6,7,5] => ? => ? = 3 - 1
[1,2,3,4,5,6,7,8,9,10,11,12] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => ? = 1 - 1
[4,8,1,3,2,7,5,6] => [1,4,3,2,8,6,7,5] => ? => ? = 3 - 1
[4,8,1,3,5,7,2,6] => [1,4,3,2,8,6,7,5] => ? => ? = 3 - 1
[2,3,4,5,6,7,8,9,10,11,12,1] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => ? = 1 - 1
[1,3,4,5,6,7,8,9,10,11,12,2] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => ? = 1 - 1
[2,1,4,3,6,5,8,7,10,11,12,9] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => ? = 1 - 1
[2,1,4,3,6,5,8,9,10,7,12,11] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => ? = 1 - 1
[2,1,4,3,6,5,8,9,10,11,12,7] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => ? = 1 - 1
[2,1,4,3,6,7,8,5,10,9,12,11] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => ? = 1 - 1
[2,1,4,3,6,7,8,5,10,11,12,9] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => ? = 1 - 1
[2,1,4,3,6,7,8,9,10,5,12,11] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => ? = 1 - 1
[2,1,4,3,6,7,8,9,10,11,12,5] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => ? = 1 - 1
[2,1,4,5,6,3,8,7,10,9,12,11] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => ? = 1 - 1
[2,1,4,5,6,3,8,7,10,11,12,9] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => ? = 1 - 1
[2,1,4,5,6,3,8,9,10,7,12,11] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => ? = 1 - 1
[2,1,4,5,6,3,8,9,10,11,12,7] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => ? = 1 - 1
[2,1,4,5,6,7,8,3,10,9,12,11] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => ? = 1 - 1
[2,1,4,5,6,7,8,3,10,11,12,9] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => ? = 1 - 1
[2,1,4,5,6,7,8,9,10,3,12,11] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => ? = 1 - 1
[2,1,4,5,6,7,8,9,10,11,12,3] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => ? = 1 - 1
[2,3,4,1,6,5,8,7,10,9,12,11] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => ? = 1 - 1
[2,3,4,1,6,5,8,7,10,11,12,9] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => ? = 1 - 1
[2,3,4,1,6,5,8,9,10,7,12,11] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => ? = 1 - 1
[2,3,4,1,6,5,8,9,10,11,12,7] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => ? = 1 - 1
[2,3,4,1,6,7,8,5,10,9,12,11] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => ? = 1 - 1
[2,3,4,1,6,7,8,5,10,11,12,9] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => ? = 1 - 1
[2,3,4,1,6,7,8,9,10,5,12,11] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => ? = 1 - 1
[2,3,4,1,6,7,8,9,10,11,12,5] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => ? = 1 - 1
[2,3,4,5,6,1,8,7,10,9,12,11] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => ? = 1 - 1
[2,3,4,5,6,1,8,7,10,11,12,9] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => ? = 1 - 1
[2,3,4,5,6,1,8,9,10,7,12,11] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => ? = 1 - 1
[2,3,4,5,6,1,8,9,10,11,12,7] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => ? = 1 - 1
[2,3,4,5,6,7,8,1,10,9,12,11] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => ? = 1 - 1
[2,3,4,5,6,7,8,1,10,11,12,9] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => ? = 1 - 1
[2,3,4,5,6,7,8,9,10,1,12,11] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => ? = 1 - 1
[2,1,4,3,6,5,8,7,9,11,12,10] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => ? = 1 - 1
[2,1,4,3,6,5,7,9,10,8,12,11] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => ? = 1 - 1
[2,1,4,3,6,5,7,9,8,11,12,10] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => ? = 1 - 1
[2,1,4,3,6,5,7,8,10,11,12,9] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => ? = 1 - 1
[2,1,4,3,5,7,8,6,10,9,12,11] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => ? = 1 - 1
[2,1,4,3,5,7,8,6,9,11,12,10] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => ? = 1 - 1
[2,1,4,3,5,7,6,9,10,8,12,11] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => ? = 1 - 1
[2,1,4,3,5,7,6,9,8,11,12,10] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => ? = 1 - 1
[2,1,4,3,5,7,6,8,10,11,12,9] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => ? = 1 - 1
[2,1,4,3,5,6,8,9,10,7,12,11] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => ? = 1 - 1
[2,1,4,3,5,6,8,9,7,11,12,10] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => ? = 1 - 1
Description
The number of descents of a binary word.
Matching statistic: St000292
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 80% ●values known / values provided: 95%●distinct values known / distinct values provided: 80%
Mp00109: Permutations —descent word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 80% ●values known / values provided: 95%●distinct values known / distinct values provided: 80%
Values
[1] => [1] => => => ? = 0 - 1
[1,2] => [1,2] => 0 => 0 => 0 = 1 - 1
[2,1] => [1,2] => 0 => 0 => 0 = 1 - 1
[1,2,3] => [1,2,3] => 00 => 00 => 0 = 1 - 1
[1,3,2] => [1,2,3] => 00 => 00 => 0 = 1 - 1
[2,1,3] => [1,2,3] => 00 => 00 => 0 = 1 - 1
[2,3,1] => [1,2,3] => 00 => 00 => 0 = 1 - 1
[3,1,2] => [1,3,2] => 01 => 10 => 0 = 1 - 1
[3,2,1] => [1,3,2] => 01 => 10 => 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => 000 => 000 => 0 = 1 - 1
[1,2,4,3] => [1,2,3,4] => 000 => 000 => 0 = 1 - 1
[1,3,2,4] => [1,2,3,4] => 000 => 000 => 0 = 1 - 1
[1,3,4,2] => [1,2,3,4] => 000 => 000 => 0 = 1 - 1
[1,4,2,3] => [1,2,4,3] => 001 => 100 => 0 = 1 - 1
[1,4,3,2] => [1,2,4,3] => 001 => 100 => 0 = 1 - 1
[2,1,3,4] => [1,2,3,4] => 000 => 000 => 0 = 1 - 1
[2,1,4,3] => [1,2,3,4] => 000 => 000 => 0 = 1 - 1
[2,3,1,4] => [1,2,3,4] => 000 => 000 => 0 = 1 - 1
[2,3,4,1] => [1,2,3,4] => 000 => 000 => 0 = 1 - 1
[2,4,1,3] => [1,2,4,3] => 001 => 100 => 0 = 1 - 1
[2,4,3,1] => [1,2,4,3] => 001 => 100 => 0 = 1 - 1
[3,1,2,4] => [1,3,2,4] => 010 => 010 => 1 = 2 - 1
[3,1,4,2] => [1,3,4,2] => 001 => 100 => 0 = 1 - 1
[3,2,1,4] => [1,3,2,4] => 010 => 010 => 1 = 2 - 1
[3,2,4,1] => [1,3,4,2] => 001 => 100 => 0 = 1 - 1
[3,4,1,2] => [1,3,2,4] => 010 => 010 => 1 = 2 - 1
[3,4,2,1] => [1,3,2,4] => 010 => 010 => 1 = 2 - 1
[4,1,2,3] => [1,4,3,2] => 011 => 110 => 0 = 1 - 1
[4,1,3,2] => [1,4,2,3] => 010 => 010 => 1 = 2 - 1
[4,2,1,3] => [1,4,3,2] => 011 => 110 => 0 = 1 - 1
[4,2,3,1] => [1,4,2,3] => 010 => 010 => 1 = 2 - 1
[4,3,1,2] => [1,4,2,3] => 010 => 010 => 1 = 2 - 1
[4,3,2,1] => [1,4,2,3] => 010 => 010 => 1 = 2 - 1
[1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0000 => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4,5] => 0000 => 0000 => 0 = 1 - 1
[1,2,4,3,5] => [1,2,3,4,5] => 0000 => 0000 => 0 = 1 - 1
[1,2,4,5,3] => [1,2,3,4,5] => 0000 => 0000 => 0 = 1 - 1
[1,2,5,3,4] => [1,2,3,5,4] => 0001 => 1000 => 0 = 1 - 1
[1,2,5,4,3] => [1,2,3,5,4] => 0001 => 1000 => 0 = 1 - 1
[1,3,2,4,5] => [1,2,3,4,5] => 0000 => 0000 => 0 = 1 - 1
[1,3,2,5,4] => [1,2,3,4,5] => 0000 => 0000 => 0 = 1 - 1
[1,3,4,2,5] => [1,2,3,4,5] => 0000 => 0000 => 0 = 1 - 1
[1,3,4,5,2] => [1,2,3,4,5] => 0000 => 0000 => 0 = 1 - 1
[1,3,5,2,4] => [1,2,3,5,4] => 0001 => 1000 => 0 = 1 - 1
[1,3,5,4,2] => [1,2,3,5,4] => 0001 => 1000 => 0 = 1 - 1
[1,4,2,3,5] => [1,2,4,3,5] => 0010 => 0100 => 1 = 2 - 1
[1,4,2,5,3] => [1,2,4,5,3] => 0001 => 1000 => 0 = 1 - 1
[1,4,3,2,5] => [1,2,4,3,5] => 0010 => 0100 => 1 = 2 - 1
[1,4,3,5,2] => [1,2,4,5,3] => 0001 => 1000 => 0 = 1 - 1
[1,4,5,2,3] => [1,2,4,3,5] => 0010 => 0100 => 1 = 2 - 1
[1,4,5,3,2] => [1,2,4,3,5] => 0010 => 0100 => 1 = 2 - 1
[] => [] => ? => ? => ? = 0 - 1
[2,1,4,3,6,5,8,7,10,9,12,11] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => 00000000000 => ? = 1 - 1
[4,8,2,3,5,7,1,6] => [1,4,3,2,8,6,7,5] => ? => ? => ? = 3 - 1
[1,2,3,4,5,6,7,8,9,10,11,12] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => 00000000000 => ? = 1 - 1
[4,8,1,3,2,7,5,6] => [1,4,3,2,8,6,7,5] => ? => ? => ? = 3 - 1
[4,8,1,3,5,7,2,6] => [1,4,3,2,8,6,7,5] => ? => ? => ? = 3 - 1
[2,3,4,5,6,7,8,9,10,11,12,1] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => 00000000000 => ? = 1 - 1
[1,3,4,5,6,7,8,9,10,11,12,2] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => 00000000000 => ? = 1 - 1
[2,1,4,3,6,5,8,7,10,11,12,9] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => 00000000000 => ? = 1 - 1
[2,1,4,3,6,5,8,9,10,7,12,11] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => 00000000000 => ? = 1 - 1
[2,1,4,3,6,5,8,9,10,11,12,7] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => 00000000000 => ? = 1 - 1
[2,1,4,3,6,7,8,5,10,9,12,11] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => 00000000000 => ? = 1 - 1
[2,1,4,3,6,7,8,5,10,11,12,9] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => 00000000000 => ? = 1 - 1
[2,1,4,3,6,7,8,9,10,5,12,11] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => 00000000000 => ? = 1 - 1
[2,1,4,3,6,7,8,9,10,11,12,5] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => 00000000000 => ? = 1 - 1
[2,1,4,5,6,3,8,7,10,9,12,11] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => 00000000000 => ? = 1 - 1
[2,1,4,5,6,3,8,7,10,11,12,9] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => 00000000000 => ? = 1 - 1
[2,1,4,5,6,3,8,9,10,7,12,11] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => 00000000000 => ? = 1 - 1
[2,1,4,5,6,3,8,9,10,11,12,7] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => 00000000000 => ? = 1 - 1
[2,1,4,5,6,7,8,3,10,9,12,11] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => 00000000000 => ? = 1 - 1
[2,1,4,5,6,7,8,3,10,11,12,9] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => 00000000000 => ? = 1 - 1
[2,1,4,5,6,7,8,9,10,3,12,11] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => 00000000000 => ? = 1 - 1
[2,1,4,5,6,7,8,9,10,11,12,3] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => 00000000000 => ? = 1 - 1
[2,3,4,1,6,5,8,7,10,9,12,11] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => 00000000000 => ? = 1 - 1
[2,3,4,1,6,5,8,7,10,11,12,9] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => 00000000000 => ? = 1 - 1
[2,3,4,1,6,5,8,9,10,7,12,11] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => 00000000000 => ? = 1 - 1
[2,3,4,1,6,5,8,9,10,11,12,7] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => 00000000000 => ? = 1 - 1
[2,3,4,1,6,7,8,5,10,9,12,11] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => 00000000000 => ? = 1 - 1
[2,3,4,1,6,7,8,5,10,11,12,9] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => 00000000000 => ? = 1 - 1
[2,3,4,1,6,7,8,9,10,5,12,11] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => 00000000000 => ? = 1 - 1
[2,3,4,1,6,7,8,9,10,11,12,5] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => 00000000000 => ? = 1 - 1
[2,3,4,5,6,1,8,7,10,9,12,11] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => 00000000000 => ? = 1 - 1
[2,3,4,5,6,1,8,7,10,11,12,9] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => 00000000000 => ? = 1 - 1
[2,3,4,5,6,1,8,9,10,7,12,11] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => 00000000000 => ? = 1 - 1
[2,3,4,5,6,1,8,9,10,11,12,7] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => 00000000000 => ? = 1 - 1
[2,3,4,5,6,7,8,1,10,9,12,11] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => 00000000000 => ? = 1 - 1
[2,3,4,5,6,7,8,1,10,11,12,9] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => 00000000000 => ? = 1 - 1
[2,3,4,5,6,7,8,9,10,1,12,11] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => 00000000000 => ? = 1 - 1
[2,1,4,3,6,5,8,7,9,11,12,10] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => 00000000000 => ? = 1 - 1
[2,1,4,3,6,5,7,9,10,8,12,11] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => 00000000000 => ? = 1 - 1
[2,1,4,3,6,5,7,9,8,11,12,10] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => 00000000000 => ? = 1 - 1
[2,1,4,3,6,5,7,8,10,11,12,9] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => 00000000000 => ? = 1 - 1
[2,1,4,3,5,7,8,6,10,9,12,11] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => 00000000000 => ? = 1 - 1
[2,1,4,3,5,7,8,6,9,11,12,10] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => 00000000000 => ? = 1 - 1
[2,1,4,3,5,7,6,9,10,8,12,11] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => 00000000000 => ? = 1 - 1
[2,1,4,3,5,7,6,9,8,11,12,10] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => 00000000000 => ? = 1 - 1
[2,1,4,3,5,7,6,8,10,11,12,9] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => 00000000000 => ? = 1 - 1
[2,1,4,3,5,6,8,9,10,7,12,11] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => 00000000000 => ? = 1 - 1
[2,1,4,3,5,6,8,9,7,11,12,10] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => 00000000000 => ? = 1 - 1
Description
The number of ascents of a binary word.
Matching statistic: St000834
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 100%
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => 1
[2,1] => [1,2] => [1,2] => 1
[1,2,3] => [1,2,3] => [1,2,3] => 1
[1,3,2] => [1,2,3] => [1,2,3] => 1
[2,1,3] => [1,2,3] => [1,2,3] => 1
[2,3,1] => [1,2,3] => [1,2,3] => 1
[3,1,2] => [1,3,2] => [3,1,2] => 1
[3,2,1] => [1,3,2] => [3,1,2] => 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 1
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 1
[1,4,2,3] => [1,2,4,3] => [4,1,2,3] => 1
[1,4,3,2] => [1,2,4,3] => [4,1,2,3] => 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 1
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 1
[2,4,1,3] => [1,2,4,3] => [4,1,2,3] => 1
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => 1
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => 2
[3,1,4,2] => [1,3,4,2] => [3,1,2,4] => 1
[3,2,1,4] => [1,3,2,4] => [1,3,2,4] => 2
[3,2,4,1] => [1,3,4,2] => [3,1,2,4] => 1
[3,4,1,2] => [1,3,2,4] => [1,3,2,4] => 2
[3,4,2,1] => [1,3,2,4] => [1,3,2,4] => 2
[4,1,2,3] => [1,4,3,2] => [4,3,1,2] => 1
[4,1,3,2] => [1,4,2,3] => [1,4,2,3] => 2
[4,2,1,3] => [1,4,3,2] => [4,3,1,2] => 1
[4,2,3,1] => [1,4,2,3] => [1,4,2,3] => 2
[4,3,1,2] => [1,4,2,3] => [1,4,2,3] => 2
[4,3,2,1] => [1,4,2,3] => [1,4,2,3] => 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,5,3,4] => [1,2,3,5,4] => [5,1,2,3,4] => 1
[1,2,5,4,3] => [1,2,3,5,4] => [5,1,2,3,4] => 1
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,3,5,2,4] => [1,2,3,5,4] => [5,1,2,3,4] => 1
[1,3,5,4,2] => [1,2,3,5,4] => [5,1,2,3,4] => 1
[1,4,2,3,5] => [1,2,4,3,5] => [1,4,2,3,5] => 2
[1,4,2,5,3] => [1,2,4,5,3] => [4,1,2,3,5] => 1
[1,4,3,2,5] => [1,2,4,3,5] => [1,4,2,3,5] => 2
[1,4,3,5,2] => [1,2,4,5,3] => [4,1,2,3,5] => 1
[1,4,5,2,3] => [1,2,4,3,5] => [1,4,2,3,5] => 2
[1,2,5,7,3,4,6] => [1,2,3,5,4,7,6] => [5,1,7,2,3,4,6] => ? = 2
[1,2,5,7,3,6,4] => [1,2,3,5,4,7,6] => [5,1,7,2,3,4,6] => ? = 2
[1,2,5,7,4,3,6] => [1,2,3,5,4,7,6] => [5,1,7,2,3,4,6] => ? = 2
[1,2,5,7,4,6,3] => [1,2,3,5,4,7,6] => [5,1,7,2,3,4,6] => ? = 2
[1,3,5,7,2,4,6] => [1,2,3,5,4,7,6] => [5,1,7,2,3,4,6] => ? = 2
[1,3,5,7,2,6,4] => [1,2,3,5,4,7,6] => [5,1,7,2,3,4,6] => ? = 2
[1,3,5,7,4,2,6] => [1,2,3,5,4,7,6] => [5,1,7,2,3,4,6] => ? = 2
[1,3,5,7,4,6,2] => [1,2,3,5,4,7,6] => [5,1,7,2,3,4,6] => ? = 2
[1,4,7,2,3,5,6] => [1,2,4,3,7,6,5] => [7,4,1,6,2,3,5] => ? = 2
[1,4,7,2,5,3,6] => [1,2,4,3,7,6,5] => [7,4,1,6,2,3,5] => ? = 2
[1,4,7,3,2,5,6] => [1,2,4,3,7,6,5] => [7,4,1,6,2,3,5] => ? = 2
[1,4,7,3,5,2,6] => [1,2,4,3,7,6,5] => [7,4,1,6,2,3,5] => ? = 2
[2,1,5,7,3,4,6] => [1,2,3,5,4,7,6] => [5,1,7,2,3,4,6] => ? = 2
[2,1,5,7,3,6,4] => [1,2,3,5,4,7,6] => [5,1,7,2,3,4,6] => ? = 2
[2,1,5,7,4,3,6] => [1,2,3,5,4,7,6] => [5,1,7,2,3,4,6] => ? = 2
[2,1,5,7,4,6,3] => [1,2,3,5,4,7,6] => [5,1,7,2,3,4,6] => ? = 2
[2,3,5,7,1,4,6] => [1,2,3,5,4,7,6] => [5,1,7,2,3,4,6] => ? = 2
[2,3,5,7,1,6,4] => [1,2,3,5,4,7,6] => [5,1,7,2,3,4,6] => ? = 2
[2,3,5,7,4,1,6] => [1,2,3,5,4,7,6] => [5,1,7,2,3,4,6] => ? = 2
[2,3,5,7,4,6,1] => [1,2,3,5,4,7,6] => [5,1,7,2,3,4,6] => ? = 2
[2,4,7,1,3,5,6] => [1,2,4,3,7,6,5] => [7,4,1,6,2,3,5] => ? = 2
[2,4,7,1,5,3,6] => [1,2,4,3,7,6,5] => [7,4,1,6,2,3,5] => ? = 2
[2,4,7,3,1,5,6] => [1,2,4,3,7,6,5] => [7,4,1,6,2,3,5] => ? = 2
[2,4,7,3,5,1,6] => [1,2,4,3,7,6,5] => [7,4,1,6,2,3,5] => ? = 2
[3,1,7,2,4,5,6] => [1,3,7,6,5,4,2] => [7,6,5,3,1,2,4] => ? = 1
[3,2,7,1,4,5,6] => [1,3,7,6,5,4,2] => [7,6,5,3,1,2,4] => ? = 1
[3,5,1,7,2,4,6] => [1,3,2,5,4,7,6] => [3,1,5,2,7,4,6] => ? = 3
[3,5,1,7,2,6,4] => [1,3,2,5,4,7,6] => [3,1,5,2,7,4,6] => ? = 3
[3,5,1,7,4,2,6] => [1,3,2,5,4,7,6] => [3,1,5,2,7,4,6] => ? = 3
[3,5,1,7,4,6,2] => [1,3,2,5,4,7,6] => [3,1,5,2,7,4,6] => ? = 3
[3,5,2,7,1,4,6] => [1,3,2,5,4,7,6] => [3,1,5,2,7,4,6] => ? = 3
[3,5,2,7,1,6,4] => [1,3,2,5,4,7,6] => [3,1,5,2,7,4,6] => ? = 3
[3,5,2,7,4,1,6] => [1,3,2,5,4,7,6] => [3,1,5,2,7,4,6] => ? = 3
[3,5,2,7,4,6,1] => [1,3,2,5,4,7,6] => [3,1,5,2,7,4,6] => ? = 3
[3,7,1,2,4,5,6] => [1,3,2,7,6,5,4] => [7,6,3,1,5,2,4] => ? = 2
[3,7,1,4,2,5,6] => [1,3,2,7,6,5,4] => [7,6,3,1,5,2,4] => ? = 2
[3,7,2,1,4,5,6] => [1,3,2,7,6,5,4] => [7,6,3,1,5,2,4] => ? = 2
[3,7,2,4,1,5,6] => [1,3,2,7,6,5,4] => [7,6,3,1,5,2,4] => ? = 2
[4,1,2,6,3,7,5] => [1,4,6,7,5,3,2] => [6,4,3,1,2,5,7] => ? = 1
[4,1,2,7,3,5,6] => [1,4,7,6,5,3,2] => [7,6,4,3,1,2,5] => ? = 1
[4,2,1,6,3,7,5] => [1,4,6,7,5,3,2] => [6,4,3,1,2,5,7] => ? = 1
[4,2,1,7,3,5,6] => [1,4,7,6,5,3,2] => [7,6,4,3,1,2,5] => ? = 1
[4,7,1,3,2,5,6] => [1,4,3,2,7,6,5] => [4,3,1,7,6,2,5] => ? = 2
[4,7,1,3,5,2,6] => [1,4,3,2,7,6,5] => [4,3,1,7,6,2,5] => ? = 2
[4,7,2,3,1,5,6] => [1,4,3,2,7,6,5] => [4,3,1,7,6,2,5] => ? = 2
[4,7,2,3,5,1,6] => [1,4,3,2,7,6,5] => [4,3,1,7,6,2,5] => ? = 2
[5,1,2,3,6,7,4] => [1,5,6,7,4,3,2] => [5,4,3,1,2,6,7] => ? = 1
[5,1,2,3,7,4,6] => [1,5,7,6,4,3,2] => [7,5,4,3,1,2,6] => ? = 1
[5,1,2,7,4,3,6] => [1,5,4,7,6,3,2] => [5,4,3,1,7,2,6] => ? = 2
[5,1,2,7,6,4,3] => [1,5,6,4,7,3,2] => [5,4,1,3,2,6,7] => ? = 2
Description
The number of right outer peaks of a permutation.
A right outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $n$ if $w_n > w_{n-1}$.
In other words, it is a peak in the word $[w_1,..., w_n,0]$.
Matching statistic: St000659
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 80%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 80%
Values
[1] => [1] => [1] => [1,0]
=> ? = 0
[1,2] => [1,2] => [2] => [1,1,0,0]
=> 1
[2,1] => [1,2] => [2] => [1,1,0,0]
=> 1
[1,2,3] => [1,2,3] => [3] => [1,1,1,0,0,0]
=> 1
[1,3,2] => [1,2,3] => [3] => [1,1,1,0,0,0]
=> 1
[2,1,3] => [1,2,3] => [3] => [1,1,1,0,0,0]
=> 1
[2,3,1] => [1,2,3] => [3] => [1,1,1,0,0,0]
=> 1
[3,1,2] => [1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[3,2,1] => [1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,2,3,4] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,2,4,3] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,3,2,4] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,3,4,2] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,4,2,3] => [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,4,3,2] => [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[2,1,3,4] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[2,1,4,3] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[2,3,1,4] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[2,3,4,1] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[2,4,1,3] => [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[2,4,3,1] => [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[3,1,2,4] => [1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,1,4,2] => [1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[3,2,1,4] => [1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,2,4,1] => [1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[3,4,1,2] => [1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,4,2,1] => [1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[4,1,2,3] => [1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[4,1,3,2] => [1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[4,2,1,3] => [1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[4,2,3,1] => [1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[4,3,1,2] => [1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[4,3,2,1] => [1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,2,3,4,5] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,2,3,5,4] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,2,4,3,5] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,2,4,5,3] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,2,5,3,4] => [1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,2,5,4,3] => [1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,3,2,4,5] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,3,2,5,4] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,3,4,2,5] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,3,4,5,2] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,3,5,2,4] => [1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,3,5,4,2] => [1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,4,2,3,5] => [1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,4,2,5,3] => [1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,4,3,2,5] => [1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,4,3,5,2] => [1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,4,5,2,3] => [1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,4,5,3,2] => [1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[4,5,6,7,8,3,2,1] => [1,4,7,2,5,8,3,6] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3
[5,6,3,4,7,2,8,1] => [1,5,7,8,2,6,3,4] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 3
[5,4,3,6,7,2,8,1] => [1,5,7,8,2,4,6,3] => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 2
[7,6,5,3,2,4,8,1] => [1,7,8,2,6,4,3,5] => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3
[7,6,5,2,3,4,8,1] => [1,7,8,2,6,4,3,5] => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3
[3,4,5,6,2,7,8,1] => [1,3,5,2,4,6,7,8] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[3,2,4,5,6,7,8,1] => [1,3,4,5,6,7,8,2] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
[2,3,4,5,6,7,8,1] => [1,2,3,4,5,6,7,8] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[4,5,6,7,8,3,1,2] => [1,4,7,2,5,8,3,6] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3
[4,5,6,7,8,2,1,3] => [1,4,7,2,5,8,3,6] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3
[4,5,6,7,8,1,2,3] => [1,4,7,2,5,8,3,6] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3
[6,3,4,5,2,7,1,8] => [1,6,7,2,3,4,5,8] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[3,4,5,6,2,7,1,8] => [1,3,5,2,4,6,7,8] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[6,3,2,4,5,7,1,8] => [1,6,7,2,3,4,5,8] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[6,2,3,4,5,7,1,8] => [1,6,7,2,3,4,5,8] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[4,3,2,5,6,7,1,8] => [1,4,5,6,7,2,3,8] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 2
[4,2,3,5,6,7,1,8] => [1,4,5,6,7,2,3,8] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 2
[2,3,4,5,6,7,1,8] => [1,2,3,4,5,6,7,8] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[3,4,5,6,2,1,7,8] => [1,3,5,2,4,6,7,8] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[2,3,4,5,6,1,7,8] => [1,2,3,4,5,6,7,8] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[3,4,5,6,1,2,7,8] => [1,3,5,2,4,6,7,8] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[3,4,5,2,1,6,7,8] => [1,3,5,2,4,6,7,8] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[4,3,2,5,1,6,7,8] => [1,4,5,2,3,6,7,8] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[4,2,3,5,1,6,7,8] => [1,4,5,2,3,6,7,8] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[2,3,4,5,1,6,7,8] => [1,2,3,4,5,6,7,8] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[3,4,5,1,2,6,7,8] => [1,3,5,2,4,6,7,8] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[2,3,4,1,5,6,7,8] => [1,2,3,4,5,6,7,8] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[2,3,1,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[2,1,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[5,4,7,6,8,3,2,1] => [1,5,8,2,4,6,3,7] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3
[4,5,7,6,8,3,2,1] => [1,4,6,3,7,2,5,8] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3
[3,6,8,7,5,4,2,1] => [1,3,8,2,6,4,7,5] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3
[4,5,3,7,8,6,2,1] => [1,4,7,2,5,8,3,6] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3
[2,6,7,8,5,4,3,1] => [1,2,6,4,8,3,7,5] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3
[3,2,7,8,6,5,4,1] => [1,3,7,4,8,2,5,6] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3
[2,3,6,5,8,7,4,1] => [1,2,3,6,7,4,5,8] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 2
[2,3,4,8,7,6,5,1] => [1,2,3,4,8,5,7,6] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 2
[2,4,5,3,7,8,6,1] => [1,2,4,3,5,7,6,8] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3
[2,3,5,6,4,8,7,1] => [1,2,3,5,4,6,8,7] => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 2
[2,4,5,3,6,8,7,1] => [1,2,4,3,5,6,8,7] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 2
[2,3,4,5,6,8,7,1] => [1,2,3,4,5,6,8,7] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
[2,5,4,7,6,3,8,1] => [1,2,5,6,3,4,7,8] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 2
[2,3,4,7,6,5,8,1] => [1,2,3,4,7,8,5,6] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 2
[2,3,4,5,7,6,8,1] => [1,2,3,4,5,7,8,6] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
[2,3,5,4,6,7,8,1] => [1,2,3,5,6,7,8,4] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
[1,6,7,8,5,4,3,2] => [1,2,6,4,8,3,7,5] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3
[1,7,6,5,8,4,3,2] => [1,2,7,3,6,4,5,8] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3
[1,3,4,8,7,6,5,2] => [1,2,3,4,8,5,7,6] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 2
Description
The number of rises of length at least 2 of a Dyck path.
Matching statistic: St000994
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000994: Permutations ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000994: Permutations ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => [2,1] => 1
[2,1] => [1,2] => [2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => [3,2,1] => [3,1,2] => 1
[1,3,2] => [1,2,3] => [3,2,1] => [3,1,2] => 1
[2,1,3] => [1,2,3] => [3,2,1] => [3,1,2] => 1
[2,3,1] => [1,2,3] => [3,2,1] => [3,1,2] => 1
[3,1,2] => [1,3,2] => [3,1,2] => [2,3,1] => 1
[3,2,1] => [1,3,2] => [3,1,2] => [2,3,1] => 1
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [4,1,2,3] => 1
[1,2,4,3] => [1,2,3,4] => [4,3,2,1] => [4,1,2,3] => 1
[1,3,2,4] => [1,2,3,4] => [4,3,2,1] => [4,1,2,3] => 1
[1,3,4,2] => [1,2,3,4] => [4,3,2,1] => [4,1,2,3] => 1
[1,4,2,3] => [1,2,4,3] => [4,3,1,2] => [2,4,1,3] => 1
[1,4,3,2] => [1,2,4,3] => [4,3,1,2] => [2,4,1,3] => 1
[2,1,3,4] => [1,2,3,4] => [4,3,2,1] => [4,1,2,3] => 1
[2,1,4,3] => [1,2,3,4] => [4,3,2,1] => [4,1,2,3] => 1
[2,3,1,4] => [1,2,3,4] => [4,3,2,1] => [4,1,2,3] => 1
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => [4,1,2,3] => 1
[2,4,1,3] => [1,2,4,3] => [4,3,1,2] => [2,4,1,3] => 1
[2,4,3,1] => [1,2,4,3] => [4,3,1,2] => [2,4,1,3] => 1
[3,1,2,4] => [1,3,2,4] => [4,2,3,1] => [4,3,1,2] => 2
[3,1,4,2] => [1,3,4,2] => [4,2,1,3] => [3,1,4,2] => 1
[3,2,1,4] => [1,3,2,4] => [4,2,3,1] => [4,3,1,2] => 2
[3,2,4,1] => [1,3,4,2] => [4,2,1,3] => [3,1,4,2] => 1
[3,4,1,2] => [1,3,2,4] => [4,2,3,1] => [4,3,1,2] => 2
[3,4,2,1] => [1,3,2,4] => [4,2,3,1] => [4,3,1,2] => 2
[4,1,2,3] => [1,4,3,2] => [4,1,2,3] => [2,3,4,1] => 1
[4,1,3,2] => [1,4,2,3] => [4,1,3,2] => [3,4,2,1] => 2
[4,2,1,3] => [1,4,3,2] => [4,1,2,3] => [2,3,4,1] => 1
[4,2,3,1] => [1,4,2,3] => [4,1,3,2] => [3,4,2,1] => 2
[4,3,1,2] => [1,4,2,3] => [4,1,3,2] => [3,4,2,1] => 2
[4,3,2,1] => [1,4,2,3] => [4,1,3,2] => [3,4,2,1] => 2
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [5,1,2,3,4] => 1
[1,2,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [5,1,2,3,4] => 1
[1,2,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => [5,1,2,3,4] => 1
[1,2,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => [5,1,2,3,4] => 1
[1,2,5,3,4] => [1,2,3,5,4] => [5,4,3,1,2] => [2,5,1,3,4] => 1
[1,2,5,4,3] => [1,2,3,5,4] => [5,4,3,1,2] => [2,5,1,3,4] => 1
[1,3,2,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [5,1,2,3,4] => 1
[1,3,2,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [5,1,2,3,4] => 1
[1,3,4,2,5] => [1,2,3,4,5] => [5,4,3,2,1] => [5,1,2,3,4] => 1
[1,3,4,5,2] => [1,2,3,4,5] => [5,4,3,2,1] => [5,1,2,3,4] => 1
[1,3,5,2,4] => [1,2,3,5,4] => [5,4,3,1,2] => [2,5,1,3,4] => 1
[1,3,5,4,2] => [1,2,3,5,4] => [5,4,3,1,2] => [2,5,1,3,4] => 1
[1,4,2,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => [5,3,1,2,4] => 2
[1,4,2,5,3] => [1,2,4,5,3] => [5,4,2,1,3] => [3,1,5,2,4] => 1
[1,4,3,2,5] => [1,2,4,3,5] => [5,4,2,3,1] => [5,3,1,2,4] => 2
[1,4,3,5,2] => [1,2,4,5,3] => [5,4,2,1,3] => [3,1,5,2,4] => 1
[1,4,5,2,3] => [1,2,4,3,5] => [5,4,2,3,1] => [5,3,1,2,4] => 2
[1,2,3,7,4,5,6] => [1,2,3,4,7,6,5] => [7,6,5,4,1,2,3] => [2,3,7,1,4,5,6] => ? = 1
[1,2,3,7,5,4,6] => [1,2,3,4,7,6,5] => [7,6,5,4,1,2,3] => [2,3,7,1,4,5,6] => ? = 1
[1,2,4,7,3,5,6] => [1,2,3,4,7,6,5] => [7,6,5,4,1,2,3] => [2,3,7,1,4,5,6] => ? = 1
[1,2,4,7,5,3,6] => [1,2,3,4,7,6,5] => [7,6,5,4,1,2,3] => [2,3,7,1,4,5,6] => ? = 1
[1,2,5,7,3,4,6] => [1,2,3,5,4,7,6] => [7,6,5,3,4,1,2] => [2,7,4,1,3,5,6] => ? = 2
[1,2,5,7,3,6,4] => [1,2,3,5,4,7,6] => [7,6,5,3,4,1,2] => [2,7,4,1,3,5,6] => ? = 2
[1,2,5,7,4,3,6] => [1,2,3,5,4,7,6] => [7,6,5,3,4,1,2] => [2,7,4,1,3,5,6] => ? = 2
[1,2,5,7,4,6,3] => [1,2,3,5,4,7,6] => [7,6,5,3,4,1,2] => [2,7,4,1,3,5,6] => ? = 2
[1,3,2,7,4,5,6] => [1,2,3,4,7,6,5] => [7,6,5,4,1,2,3] => [2,3,7,1,4,5,6] => ? = 1
[1,3,2,7,5,4,6] => [1,2,3,4,7,6,5] => [7,6,5,4,1,2,3] => [2,3,7,1,4,5,6] => ? = 1
[1,3,4,7,2,5,6] => [1,2,3,4,7,6,5] => [7,6,5,4,1,2,3] => [2,3,7,1,4,5,6] => ? = 1
[1,3,4,7,5,2,6] => [1,2,3,4,7,6,5] => [7,6,5,4,1,2,3] => [2,3,7,1,4,5,6] => ? = 1
[1,3,5,7,2,4,6] => [1,2,3,5,4,7,6] => [7,6,5,3,4,1,2] => [2,7,4,1,3,5,6] => ? = 2
[1,3,5,7,2,6,4] => [1,2,3,5,4,7,6] => [7,6,5,3,4,1,2] => [2,7,4,1,3,5,6] => ? = 2
[1,3,5,7,4,2,6] => [1,2,3,5,4,7,6] => [7,6,5,3,4,1,2] => [2,7,4,1,3,5,6] => ? = 2
[1,3,5,7,4,6,2] => [1,2,3,5,4,7,6] => [7,6,5,3,4,1,2] => [2,7,4,1,3,5,6] => ? = 2
[1,4,7,2,3,5,6] => [1,2,4,3,7,6,5] => [7,6,4,5,1,2,3] => [2,3,7,5,1,4,6] => ? = 2
[1,4,7,2,5,3,6] => [1,2,4,3,7,6,5] => [7,6,4,5,1,2,3] => [2,3,7,5,1,4,6] => ? = 2
[1,4,7,3,2,5,6] => [1,2,4,3,7,6,5] => [7,6,4,5,1,2,3] => [2,3,7,5,1,4,6] => ? = 2
[1,4,7,3,5,2,6] => [1,2,4,3,7,6,5] => [7,6,4,5,1,2,3] => [2,3,7,5,1,4,6] => ? = 2
[2,1,3,7,4,5,6] => [1,2,3,4,7,6,5] => [7,6,5,4,1,2,3] => [2,3,7,1,4,5,6] => ? = 1
[2,1,3,7,5,4,6] => [1,2,3,4,7,6,5] => [7,6,5,4,1,2,3] => [2,3,7,1,4,5,6] => ? = 1
[2,1,4,7,3,5,6] => [1,2,3,4,7,6,5] => [7,6,5,4,1,2,3] => [2,3,7,1,4,5,6] => ? = 1
[2,1,4,7,5,3,6] => [1,2,3,4,7,6,5] => [7,6,5,4,1,2,3] => [2,3,7,1,4,5,6] => ? = 1
[2,1,5,7,3,4,6] => [1,2,3,5,4,7,6] => [7,6,5,3,4,1,2] => [2,7,4,1,3,5,6] => ? = 2
[2,1,5,7,3,6,4] => [1,2,3,5,4,7,6] => [7,6,5,3,4,1,2] => [2,7,4,1,3,5,6] => ? = 2
[2,1,5,7,4,3,6] => [1,2,3,5,4,7,6] => [7,6,5,3,4,1,2] => [2,7,4,1,3,5,6] => ? = 2
[2,1,5,7,4,6,3] => [1,2,3,5,4,7,6] => [7,6,5,3,4,1,2] => [2,7,4,1,3,5,6] => ? = 2
[2,3,1,7,4,5,6] => [1,2,3,4,7,6,5] => [7,6,5,4,1,2,3] => [2,3,7,1,4,5,6] => ? = 1
[2,3,1,7,5,4,6] => [1,2,3,4,7,6,5] => [7,6,5,4,1,2,3] => [2,3,7,1,4,5,6] => ? = 1
[2,3,4,7,1,5,6] => [1,2,3,4,7,6,5] => [7,6,5,4,1,2,3] => [2,3,7,1,4,5,6] => ? = 1
[2,3,4,7,5,1,6] => [1,2,3,4,7,6,5] => [7,6,5,4,1,2,3] => [2,3,7,1,4,5,6] => ? = 1
[2,3,5,7,1,4,6] => [1,2,3,5,4,7,6] => [7,6,5,3,4,1,2] => [2,7,4,1,3,5,6] => ? = 2
[2,3,5,7,1,6,4] => [1,2,3,5,4,7,6] => [7,6,5,3,4,1,2] => [2,7,4,1,3,5,6] => ? = 2
[2,3,5,7,4,1,6] => [1,2,3,5,4,7,6] => [7,6,5,3,4,1,2] => [2,7,4,1,3,5,6] => ? = 2
[2,3,5,7,4,6,1] => [1,2,3,5,4,7,6] => [7,6,5,3,4,1,2] => [2,7,4,1,3,5,6] => ? = 2
[2,4,7,1,3,5,6] => [1,2,4,3,7,6,5] => [7,6,4,5,1,2,3] => [2,3,7,5,1,4,6] => ? = 2
[2,4,7,1,5,3,6] => [1,2,4,3,7,6,5] => [7,6,4,5,1,2,3] => [2,3,7,5,1,4,6] => ? = 2
[2,4,7,3,1,5,6] => [1,2,4,3,7,6,5] => [7,6,4,5,1,2,3] => [2,3,7,5,1,4,6] => ? = 2
[2,4,7,3,5,1,6] => [1,2,4,3,7,6,5] => [7,6,4,5,1,2,3] => [2,3,7,5,1,4,6] => ? = 2
[3,5,1,7,2,4,6] => [1,3,2,5,4,7,6] => [7,5,6,3,4,1,2] => [2,7,4,1,6,3,5] => ? = 3
[3,5,1,7,2,6,4] => [1,3,2,5,4,7,6] => [7,5,6,3,4,1,2] => [2,7,4,1,6,3,5] => ? = 3
[3,5,1,7,4,2,6] => [1,3,2,5,4,7,6] => [7,5,6,3,4,1,2] => [2,7,4,1,6,3,5] => ? = 3
[3,5,1,7,4,6,2] => [1,3,2,5,4,7,6] => [7,5,6,3,4,1,2] => [2,7,4,1,6,3,5] => ? = 3
[3,5,2,7,1,4,6] => [1,3,2,5,4,7,6] => [7,5,6,3,4,1,2] => [2,7,4,1,6,3,5] => ? = 3
[3,5,2,7,1,6,4] => [1,3,2,5,4,7,6] => [7,5,6,3,4,1,2] => [2,7,4,1,6,3,5] => ? = 3
[3,5,2,7,4,1,6] => [1,3,2,5,4,7,6] => [7,5,6,3,4,1,2] => [2,7,4,1,6,3,5] => ? = 3
[3,5,2,7,4,6,1] => [1,3,2,5,4,7,6] => [7,5,6,3,4,1,2] => [2,7,4,1,6,3,5] => ? = 3
[3,7,1,2,4,5,6] => [1,3,2,7,6,5,4] => [7,5,6,1,2,3,4] => [2,3,4,7,6,1,5] => ? = 2
[3,7,1,4,2,5,6] => [1,3,2,7,6,5,4] => [7,5,6,1,2,3,4] => [2,3,4,7,6,1,5] => ? = 2
Description
The number of cycle peaks and the number of cycle valleys of a permutation.
A '''cycle peak''' of a permutation $\pi$ is an index $i$ such that $\pi^{-1}(i) < i > \pi(i)$. Analogously, a '''cycle valley''' is an index $i$ such that $\pi^{-1}(i) > i < \pi(i)$.
Clearly, every cycle of $\pi$ contains as many peaks as valleys.
Matching statistic: St000636
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000636: Graphs ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 80%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000636: Graphs ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 80%
Values
[1] => [1] => [.,.]
=> ([],1)
=> 1 = 0 + 1
[1,2] => [1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> 2 = 1 + 1
[2,1] => [1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,3,2] => [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,1,3] => [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,3,1] => [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[3,1,2] => [1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[3,2,1] => [1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,2,4,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,3,2,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,3,4,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,4,2,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,4,3,2] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[2,1,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[2,1,4,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[2,3,1,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[2,3,4,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[2,4,1,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[2,4,3,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[3,1,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,1,4,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[3,2,1,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,2,4,1] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[3,4,1,2] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,4,2,1] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,1,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[4,1,3,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,2,1,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[4,2,3,1] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,3,1,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,3,2,1] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,2,3,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,2,4,3,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,2,4,5,3] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,2,5,3,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,2,5,4,3] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,3,2,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,3,2,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,3,4,2,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,3,4,5,2] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,3,5,2,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,3,5,4,2] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,4,2,3,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 2 + 1
[1,4,2,5,3] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,4,3,2,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 2 + 1
[1,4,3,5,2] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,4,5,2,3] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 2 + 1
[4,5,6,7,8,3,2,1] => [1,4,7,2,5,8,3,6] => [.,[[.,[.,.]],[[.,[.,.]],[.,.]]]]
=> ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 3 + 1
[5,6,7,8,3,4,2,1] => [1,5,3,7,2,6,4,8] => [.,[[[.,.],[.,.]],[[.,.],[.,.]]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 4 + 1
[7,5,4,6,3,8,2,1] => [1,7,2,5,3,4,6,8] => [.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 3 + 1
[7,6,8,5,4,2,3,1] => [1,7,3,8,2,6,4,5] => [.,[[[.,.],[.,.]],[[.,.],[.,.]]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 4 + 1
[7,6,8,4,5,2,3,1] => [1,7,3,8,2,6,4,5] => [.,[[[.,.],[.,.]],[[.,.],[.,.]]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 4 + 1
[8,4,5,3,6,2,7,1] => [1,8,2,4,3,5,6,7] => [.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 3 + 1
[8,4,5,3,2,6,7,1] => [1,8,2,4,3,5,6,7] => [.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 3 + 1
[8,4,5,2,3,6,7,1] => [1,8,2,4,3,5,6,7] => [.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 3 + 1
[8,4,3,2,5,6,7,1] => [1,8,2,4,3,5,6,7] => [.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 3 + 1
[8,4,2,3,5,6,7,1] => [1,8,2,4,3,5,6,7] => [.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 3 + 1
[6,5,4,7,3,2,8,1] => [1,6,2,5,3,4,7,8] => [.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 3 + 1
[6,4,5,7,3,2,8,1] => [1,6,2,4,7,8,3,5] => [.,[[.,.],[[.,[.,[.,.]]],[.,.]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 3 + 1
[5,6,3,4,7,2,8,1] => [1,5,7,8,2,6,3,4] => [.,[[.,[.,[.,.]]],[[.,.],[.,.]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 3 + 1
[5,4,3,6,7,2,8,1] => [1,5,7,8,2,4,6,3] => [.,[[.,[.,[.,.]]],[[.,[.,.]],.]]]
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ? = 2 + 1
[7,6,5,3,2,4,8,1] => [1,7,8,2,6,4,3,5] => [.,[[.,[.,.]],[[[.,.],.],[.,.]]]]
=> ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 3 + 1
[7,6,5,2,3,4,8,1] => [1,7,8,2,6,4,3,5] => [.,[[.,[.,.]],[[[.,.],.],[.,.]]]]
=> ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 3 + 1
[3,4,5,6,2,7,8,1] => [1,3,5,2,4,6,7,8] => [.,[[.,[.,.]],[.,[.,[.,[.,.]]]]]]
=> ([(0,6),(1,5),(2,7),(3,4),(3,5),(4,7),(6,7)],8)
=> ? = 2 + 1
[3,2,4,5,6,7,8,1] => [1,3,4,5,6,7,8,2] => [.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ? = 1 + 1
[2,3,4,5,6,7,8,1] => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ? = 1 + 1
[4,5,6,7,8,3,1,2] => [1,4,7,2,5,8,3,6] => [.,[[.,[.,.]],[[.,[.,.]],[.,.]]]]
=> ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 3 + 1
[5,6,7,8,3,4,1,2] => [1,5,3,7,2,6,4,8] => [.,[[[.,.],[.,.]],[[.,.],[.,.]]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 4 + 1
[7,5,4,6,3,8,1,2] => [1,7,2,5,3,4,6,8] => [.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 3 + 1
[4,5,6,7,8,2,1,3] => [1,4,7,2,5,8,3,6] => [.,[[.,[.,.]],[[.,[.,.]],[.,.]]]]
=> ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 3 + 1
[4,5,6,7,8,1,2,3] => [1,4,7,2,5,8,3,6] => [.,[[.,[.,.]],[[.,[.,.]],[.,.]]]]
=> ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 3 + 1
[5,6,7,8,3,2,1,4] => [1,5,3,7,2,6,4,8] => [.,[[[.,.],[.,.]],[[.,.],[.,.]]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 4 + 1
[5,6,7,8,3,1,2,4] => [1,5,3,7,2,6,4,8] => [.,[[[.,.],[.,.]],[[.,.],[.,.]]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 4 + 1
[6,7,8,4,2,3,1,5] => [1,6,3,8,5,2,7,4] => [.,[[[.,.],[[.,.],.]],[[.,.],.]]]
=> ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 3 + 1
[6,7,8,2,1,3,4,5] => [1,6,3,8,5,2,7,4] => [.,[[[.,.],[[.,.],.]],[[.,.],.]]]
=> ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 3 + 1
[6,7,8,1,2,3,4,5] => [1,6,3,8,5,2,7,4] => [.,[[[.,.],[[.,.],.]],[[.,.],.]]]
=> ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 3 + 1
[8,6,4,3,2,5,1,7] => [1,8,7,2,6,5,3,4] => [.,[[[.,.],.],[[[.,.],.],[.,.]]]]
=> ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 3 + 1
[8,6,4,2,3,5,1,7] => [1,8,7,2,6,5,3,4] => [.,[[[.,.],.],[[[.,.],.],[.,.]]]]
=> ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 3 + 1
[8,2,1,3,4,5,6,7] => [1,8,7,6,5,4,3,2] => [.,[[[[[[[.,.],.],.],.],.],.],.]]
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ? = 1 + 1
[8,1,2,3,4,5,6,7] => [1,8,7,6,5,4,3,2] => [.,[[[[[[[.,.],.],.],.],.],.],.]]
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ? = 1 + 1
[5,6,7,4,3,2,1,8] => [1,5,3,7,2,6,4,8] => [.,[[[.,.],[.,.]],[[.,.],[.,.]]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 4 + 1
[7,5,4,6,3,2,1,8] => [1,7,2,5,3,4,6,8] => [.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 3 + 1
[6,5,4,7,3,2,1,8] => [1,6,2,5,3,4,7,8] => [.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 3 + 1
[7,5,4,6,2,3,1,8] => [1,7,2,5,3,4,6,8] => [.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 3 + 1
[5,6,7,2,3,4,1,8] => [1,5,3,7,2,6,4,8] => [.,[[[.,.],[.,.]],[[.,.],[.,.]]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 4 + 1
[7,5,4,3,2,6,1,8] => [1,7,2,5,3,4,6,8] => [.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 3 + 1
[7,5,3,4,2,6,1,8] => [1,7,2,5,3,4,6,8] => [.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 3 + 1
[7,5,4,2,3,6,1,8] => [1,7,2,5,3,4,6,8] => [.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 3 + 1
[6,3,4,5,2,7,1,8] => [1,6,7,2,3,4,5,8] => [.,[[.,[.,.]],[.,[.,[.,[.,.]]]]]]
=> ([(0,6),(1,5),(2,7),(3,4),(3,5),(4,7),(6,7)],8)
=> ? = 2 + 1
[3,4,5,6,2,7,1,8] => [1,3,5,2,4,6,7,8] => [.,[[.,[.,.]],[.,[.,[.,[.,.]]]]]]
=> ([(0,6),(1,5),(2,7),(3,4),(3,5),(4,7),(6,7)],8)
=> ? = 2 + 1
[6,3,2,4,5,7,1,8] => [1,6,7,2,3,4,5,8] => [.,[[.,[.,.]],[.,[.,[.,[.,.]]]]]]
=> ([(0,6),(1,5),(2,7),(3,4),(3,5),(4,7),(6,7)],8)
=> ? = 2 + 1
[6,2,3,4,5,7,1,8] => [1,6,7,2,3,4,5,8] => [.,[[.,[.,.]],[.,[.,[.,[.,.]]]]]]
=> ([(0,6),(1,5),(2,7),(3,4),(3,5),(4,7),(6,7)],8)
=> ? = 2 + 1
[4,3,2,5,6,7,1,8] => [1,4,5,6,7,2,3,8] => [.,[[.,[.,[.,[.,.]]]],[.,[.,.]]]]
=> ([(0,6),(1,5),(2,7),(3,4),(3,5),(4,7),(6,7)],8)
=> ? = 2 + 1
[4,2,3,5,6,7,1,8] => [1,4,5,6,7,2,3,8] => [.,[[.,[.,[.,[.,.]]]],[.,[.,.]]]]
=> ([(0,6),(1,5),(2,7),(3,4),(3,5),(4,7),(6,7)],8)
=> ? = 2 + 1
[2,3,4,5,6,7,1,8] => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ? = 1 + 1
[7,6,3,2,4,1,5,8] => [1,7,5,4,2,6,3,8] => [.,[[[[.,.],.],.],[[.,.],[.,.]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 3 + 1
[6,5,4,3,2,1,7,8] => [1,6,2,5,3,4,7,8] => [.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 3 + 1
Description
The hull number of a graph.
The convex hull of a set of vertices $S$ of a graph is the smallest set $h(S)$ such that for any pair $u,v\in h(S)$ all vertices on a shortest path from $u$ to $v$ are also in $h(S)$.
The hull number is the size of the smallest set $S$ such that $h(S)$ is the set of all vertices.
Matching statistic: St001654
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St001654: Graphs ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 80%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St001654: Graphs ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 80%
Values
[1] => [1] => [.,.]
=> ([],1)
=> 1 = 0 + 1
[1,2] => [1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> 2 = 1 + 1
[2,1] => [1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,3,2] => [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,1,3] => [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,3,1] => [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[3,1,2] => [1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[3,2,1] => [1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,2,4,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,3,2,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,3,4,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,4,2,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,4,3,2] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[2,1,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[2,1,4,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[2,3,1,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[2,3,4,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[2,4,1,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[2,4,3,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[3,1,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,1,4,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[3,2,1,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,2,4,1] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[3,4,1,2] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,4,2,1] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,1,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[4,1,3,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,2,1,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[4,2,3,1] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,3,1,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,3,2,1] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,2,3,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,2,4,3,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,2,4,5,3] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,2,5,3,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,2,5,4,3] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,3,2,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,3,2,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,3,4,2,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,3,4,5,2] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,3,5,2,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,3,5,4,2] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,4,2,3,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 2 + 1
[1,4,2,5,3] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,4,3,2,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 2 + 1
[1,4,3,5,2] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,4,5,2,3] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 2 + 1
[4,5,6,7,8,3,2,1] => [1,4,7,2,5,8,3,6] => [.,[[.,[.,.]],[[.,[.,.]],[.,.]]]]
=> ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 3 + 1
[5,6,7,8,3,4,2,1] => [1,5,3,7,2,6,4,8] => [.,[[[.,.],[.,.]],[[.,.],[.,.]]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 4 + 1
[7,5,4,6,3,8,2,1] => [1,7,2,5,3,4,6,8] => [.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 3 + 1
[7,6,8,5,4,2,3,1] => [1,7,3,8,2,6,4,5] => [.,[[[.,.],[.,.]],[[.,.],[.,.]]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 4 + 1
[7,6,8,4,5,2,3,1] => [1,7,3,8,2,6,4,5] => [.,[[[.,.],[.,.]],[[.,.],[.,.]]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 4 + 1
[8,4,5,3,6,2,7,1] => [1,8,2,4,3,5,6,7] => [.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 3 + 1
[8,4,5,3,2,6,7,1] => [1,8,2,4,3,5,6,7] => [.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 3 + 1
[8,4,5,2,3,6,7,1] => [1,8,2,4,3,5,6,7] => [.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 3 + 1
[8,4,3,2,5,6,7,1] => [1,8,2,4,3,5,6,7] => [.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 3 + 1
[8,4,2,3,5,6,7,1] => [1,8,2,4,3,5,6,7] => [.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 3 + 1
[6,5,4,7,3,2,8,1] => [1,6,2,5,3,4,7,8] => [.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 3 + 1
[6,4,5,7,3,2,8,1] => [1,6,2,4,7,8,3,5] => [.,[[.,.],[[.,[.,[.,.]]],[.,.]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 3 + 1
[5,6,3,4,7,2,8,1] => [1,5,7,8,2,6,3,4] => [.,[[.,[.,[.,.]]],[[.,.],[.,.]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 3 + 1
[5,4,3,6,7,2,8,1] => [1,5,7,8,2,4,6,3] => [.,[[.,[.,[.,.]]],[[.,[.,.]],.]]]
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ? = 2 + 1
[7,6,5,3,2,4,8,1] => [1,7,8,2,6,4,3,5] => [.,[[.,[.,.]],[[[.,.],.],[.,.]]]]
=> ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 3 + 1
[7,6,5,2,3,4,8,1] => [1,7,8,2,6,4,3,5] => [.,[[.,[.,.]],[[[.,.],.],[.,.]]]]
=> ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 3 + 1
[3,4,5,6,2,7,8,1] => [1,3,5,2,4,6,7,8] => [.,[[.,[.,.]],[.,[.,[.,[.,.]]]]]]
=> ([(0,6),(1,5),(2,7),(3,4),(3,5),(4,7),(6,7)],8)
=> ? = 2 + 1
[3,2,4,5,6,7,8,1] => [1,3,4,5,6,7,8,2] => [.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ? = 1 + 1
[2,3,4,5,6,7,8,1] => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ? = 1 + 1
[4,5,6,7,8,3,1,2] => [1,4,7,2,5,8,3,6] => [.,[[.,[.,.]],[[.,[.,.]],[.,.]]]]
=> ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 3 + 1
[5,6,7,8,3,4,1,2] => [1,5,3,7,2,6,4,8] => [.,[[[.,.],[.,.]],[[.,.],[.,.]]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 4 + 1
[7,5,4,6,3,8,1,2] => [1,7,2,5,3,4,6,8] => [.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 3 + 1
[4,5,6,7,8,2,1,3] => [1,4,7,2,5,8,3,6] => [.,[[.,[.,.]],[[.,[.,.]],[.,.]]]]
=> ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 3 + 1
[4,5,6,7,8,1,2,3] => [1,4,7,2,5,8,3,6] => [.,[[.,[.,.]],[[.,[.,.]],[.,.]]]]
=> ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 3 + 1
[5,6,7,8,3,2,1,4] => [1,5,3,7,2,6,4,8] => [.,[[[.,.],[.,.]],[[.,.],[.,.]]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 4 + 1
[5,6,7,8,3,1,2,4] => [1,5,3,7,2,6,4,8] => [.,[[[.,.],[.,.]],[[.,.],[.,.]]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 4 + 1
[6,7,8,4,2,3,1,5] => [1,6,3,8,5,2,7,4] => [.,[[[.,.],[[.,.],.]],[[.,.],.]]]
=> ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 3 + 1
[6,7,8,2,1,3,4,5] => [1,6,3,8,5,2,7,4] => [.,[[[.,.],[[.,.],.]],[[.,.],.]]]
=> ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 3 + 1
[6,7,8,1,2,3,4,5] => [1,6,3,8,5,2,7,4] => [.,[[[.,.],[[.,.],.]],[[.,.],.]]]
=> ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 3 + 1
[8,6,4,3,2,5,1,7] => [1,8,7,2,6,5,3,4] => [.,[[[.,.],.],[[[.,.],.],[.,.]]]]
=> ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 3 + 1
[8,6,4,2,3,5,1,7] => [1,8,7,2,6,5,3,4] => [.,[[[.,.],.],[[[.,.],.],[.,.]]]]
=> ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 3 + 1
[8,2,1,3,4,5,6,7] => [1,8,7,6,5,4,3,2] => [.,[[[[[[[.,.],.],.],.],.],.],.]]
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ? = 1 + 1
[8,1,2,3,4,5,6,7] => [1,8,7,6,5,4,3,2] => [.,[[[[[[[.,.],.],.],.],.],.],.]]
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ? = 1 + 1
[5,6,7,4,3,2,1,8] => [1,5,3,7,2,6,4,8] => [.,[[[.,.],[.,.]],[[.,.],[.,.]]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 4 + 1
[7,5,4,6,3,2,1,8] => [1,7,2,5,3,4,6,8] => [.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 3 + 1
[6,5,4,7,3,2,1,8] => [1,6,2,5,3,4,7,8] => [.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 3 + 1
[7,5,4,6,2,3,1,8] => [1,7,2,5,3,4,6,8] => [.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 3 + 1
[5,6,7,2,3,4,1,8] => [1,5,3,7,2,6,4,8] => [.,[[[.,.],[.,.]],[[.,.],[.,.]]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 4 + 1
[7,5,4,3,2,6,1,8] => [1,7,2,5,3,4,6,8] => [.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 3 + 1
[7,5,3,4,2,6,1,8] => [1,7,2,5,3,4,6,8] => [.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 3 + 1
[7,5,4,2,3,6,1,8] => [1,7,2,5,3,4,6,8] => [.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 3 + 1
[6,3,4,5,2,7,1,8] => [1,6,7,2,3,4,5,8] => [.,[[.,[.,.]],[.,[.,[.,[.,.]]]]]]
=> ([(0,6),(1,5),(2,7),(3,4),(3,5),(4,7),(6,7)],8)
=> ? = 2 + 1
[3,4,5,6,2,7,1,8] => [1,3,5,2,4,6,7,8] => [.,[[.,[.,.]],[.,[.,[.,[.,.]]]]]]
=> ([(0,6),(1,5),(2,7),(3,4),(3,5),(4,7),(6,7)],8)
=> ? = 2 + 1
[6,3,2,4,5,7,1,8] => [1,6,7,2,3,4,5,8] => [.,[[.,[.,.]],[.,[.,[.,[.,.]]]]]]
=> ([(0,6),(1,5),(2,7),(3,4),(3,5),(4,7),(6,7)],8)
=> ? = 2 + 1
[6,2,3,4,5,7,1,8] => [1,6,7,2,3,4,5,8] => [.,[[.,[.,.]],[.,[.,[.,[.,.]]]]]]
=> ([(0,6),(1,5),(2,7),(3,4),(3,5),(4,7),(6,7)],8)
=> ? = 2 + 1
[4,3,2,5,6,7,1,8] => [1,4,5,6,7,2,3,8] => [.,[[.,[.,[.,[.,.]]]],[.,[.,.]]]]
=> ([(0,6),(1,5),(2,7),(3,4),(3,5),(4,7),(6,7)],8)
=> ? = 2 + 1
[4,2,3,5,6,7,1,8] => [1,4,5,6,7,2,3,8] => [.,[[.,[.,[.,[.,.]]]],[.,[.,.]]]]
=> ([(0,6),(1,5),(2,7),(3,4),(3,5),(4,7),(6,7)],8)
=> ? = 2 + 1
[2,3,4,5,6,7,1,8] => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ? = 1 + 1
[7,6,3,2,4,1,5,8] => [1,7,5,4,2,6,3,8] => [.,[[[[.,.],.],.],[[.,.],[.,.]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 3 + 1
[6,5,4,3,2,1,7,8] => [1,6,2,5,3,4,7,8] => [.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 3 + 1
Description
The monophonic hull number of a graph.
The monophonic hull of a set of vertices $M$ of a graph $G$ is the set of vertices that lie on at least one induced path between vertices in $M$. The monophonic hull number is the size of the smallest set $M$ such that the monophonic hull of $M$ is all of $G$.
For example, the monophonic hull number of a graph $G$ with $n$ vertices is $n$ if and only if $G$ is a disjoint union of complete graphs.
The following 13 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001655The general position number of a graph. St001656The monophonic position number of a graph. St001883The mutual visibility number of a graph. St000099The number of valleys of a permutation, including the boundary. St000353The number of inner valleys of a permutation. St000243The number of cyclic valleys and cyclic peaks 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. St001960The number of descents of a permutation minus one if its first entry is not one. St000914The sum of the values of the Möbius function of a poset. St001946The number of descents in a parking function. St000679The pruning number of an ordered tree. St001890The maximum magnitude of the Möbius function of a poset. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
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