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Your data matches 66 different statistics following compositions of up to 3 maps.
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Matching statistic: St000035
(load all 94 compositions to match this statistic)
(load all 94 compositions to match this statistic)
St000035: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,2] => 0
[2,1] => 1
[1,2,3] => 0
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 1
[3,1,2] => 1
[3,2,1] => 1
[1,2,3,4] => 0
[1,2,4,3] => 1
[1,3,2,4] => 1
[1,3,4,2] => 1
[1,4,2,3] => 1
[1,4,3,2] => 1
[2,1,3,4] => 1
[2,1,4,3] => 2
[2,3,1,4] => 1
[2,3,4,1] => 1
[2,4,1,3] => 1
[2,4,3,1] => 1
[3,1,2,4] => 1
[3,1,4,2] => 2
[3,2,1,4] => 1
[3,2,4,1] => 2
[3,4,1,2] => 1
[3,4,2,1] => 1
[4,1,2,3] => 1
[4,1,3,2] => 2
[4,2,1,3] => 1
[4,2,3,1] => 2
[4,3,1,2] => 1
[4,3,2,1] => 1
[1,2,3,4,5] => 0
[1,2,3,5,4] => 1
[1,2,4,3,5] => 1
[1,2,4,5,3] => 1
[1,2,5,3,4] => 1
[1,2,5,4,3] => 1
[1,3,2,4,5] => 1
[1,3,2,5,4] => 2
[1,3,4,2,5] => 1
[1,3,4,5,2] => 1
[1,3,5,2,4] => 1
[1,3,5,4,2] => 1
[1,4,2,3,5] => 1
[1,4,2,5,3] => 2
[1,4,3,2,5] => 1
[1,4,3,5,2] => 2
[1,4,5,2,3] => 1
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
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●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: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1]
=> 0
[1,2] => [2,1] => [1,1] => [1,1]
=> 0
[2,1] => [1,2] => [2] => [2]
=> 1
[1,2,3] => [3,2,1] => [1,1,1] => [1,1,1]
=> 0
[1,3,2] => [2,3,1] => [2,1] => [2,1]
=> 1
[2,1,3] => [3,1,2] => [1,2] => [2,1]
=> 1
[2,3,1] => [1,3,2] => [2,1] => [2,1]
=> 1
[3,1,2] => [2,1,3] => [1,2] => [2,1]
=> 1
[3,2,1] => [1,2,3] => [3] => [3]
=> 1
[1,2,3,4] => [4,3,2,1] => [1,1,1,1] => [1,1,1,1]
=> 0
[1,2,4,3] => [3,4,2,1] => [2,1,1] => [2,1,1]
=> 1
[1,3,2,4] => [4,2,3,1] => [1,2,1] => [2,1,1]
=> 1
[1,3,4,2] => [2,4,3,1] => [2,1,1] => [2,1,1]
=> 1
[1,4,2,3] => [3,2,4,1] => [1,2,1] => [2,1,1]
=> 1
[1,4,3,2] => [2,3,4,1] => [3,1] => [3,1]
=> 1
[2,1,3,4] => [4,3,1,2] => [1,1,2] => [2,1,1]
=> 1
[2,1,4,3] => [3,4,1,2] => [2,2] => [2,2]
=> 2
[2,3,1,4] => [4,1,3,2] => [1,2,1] => [2,1,1]
=> 1
[2,3,4,1] => [1,4,3,2] => [2,1,1] => [2,1,1]
=> 1
[2,4,1,3] => [3,1,4,2] => [1,2,1] => [2,1,1]
=> 1
[2,4,3,1] => [1,3,4,2] => [3,1] => [3,1]
=> 1
[3,1,2,4] => [4,2,1,3] => [1,1,2] => [2,1,1]
=> 1
[3,1,4,2] => [2,4,1,3] => [2,2] => [2,2]
=> 2
[3,2,1,4] => [4,1,2,3] => [1,3] => [3,1]
=> 1
[3,2,4,1] => [1,4,2,3] => [2,2] => [2,2]
=> 2
[3,4,1,2] => [2,1,4,3] => [1,2,1] => [2,1,1]
=> 1
[3,4,2,1] => [1,2,4,3] => [3,1] => [3,1]
=> 1
[4,1,2,3] => [3,2,1,4] => [1,1,2] => [2,1,1]
=> 1
[4,1,3,2] => [2,3,1,4] => [2,2] => [2,2]
=> 2
[4,2,1,3] => [3,1,2,4] => [1,3] => [3,1]
=> 1
[4,2,3,1] => [1,3,2,4] => [2,2] => [2,2]
=> 2
[4,3,1,2] => [2,1,3,4] => [1,3] => [3,1]
=> 1
[4,3,2,1] => [1,2,3,4] => [4] => [4]
=> 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1] => [1,1,1,1,1]
=> 0
[1,2,3,5,4] => [4,5,3,2,1] => [2,1,1,1] => [2,1,1,1]
=> 1
[1,2,4,3,5] => [5,3,4,2,1] => [1,2,1,1] => [2,1,1,1]
=> 1
[1,2,4,5,3] => [3,5,4,2,1] => [2,1,1,1] => [2,1,1,1]
=> 1
[1,2,5,3,4] => [4,3,5,2,1] => [1,2,1,1] => [2,1,1,1]
=> 1
[1,2,5,4,3] => [3,4,5,2,1] => [3,1,1] => [3,1,1]
=> 1
[1,3,2,4,5] => [5,4,2,3,1] => [1,1,2,1] => [2,1,1,1]
=> 1
[1,3,2,5,4] => [4,5,2,3,1] => [2,2,1] => [2,2,1]
=> 2
[1,3,4,2,5] => [5,2,4,3,1] => [1,2,1,1] => [2,1,1,1]
=> 1
[1,3,4,5,2] => [2,5,4,3,1] => [2,1,1,1] => [2,1,1,1]
=> 1
[1,3,5,2,4] => [4,2,5,3,1] => [1,2,1,1] => [2,1,1,1]
=> 1
[1,3,5,4,2] => [2,4,5,3,1] => [3,1,1] => [3,1,1]
=> 1
[1,4,2,3,5] => [5,3,2,4,1] => [1,1,2,1] => [2,1,1,1]
=> 1
[1,4,2,5,3] => [3,5,2,4,1] => [2,2,1] => [2,2,1]
=> 2
[1,4,3,2,5] => [5,2,3,4,1] => [1,3,1] => [3,1,1]
=> 1
[1,4,3,5,2] => [2,5,3,4,1] => [2,2,1] => [2,2,1]
=> 2
[1,4,5,2,3] => [3,2,5,4,1] => [1,2,1,1] => [2,1,1,1]
=> 1
[] => [] => [] => ?
=> ? = 0
[2,1,4,3,12,11,10,9,8,7,6,5] => [5,6,7,8,9,10,11,12,3,4,1,2] => [8,2,2] => ?
=> ? = 3
[2,1,12,5,4,11,10,9,8,7,6,3] => [3,6,7,8,9,10,11,4,5,12,1,2] => [7,3,2] => ?
=> ? = 3
[2,1,8,7,6,5,4,3,10,9,12,11] => [11,12,9,10,3,4,5,6,7,8,1,2] => [2,2,6,2] => ?
=> ? = 4
[2,1,10,9,8,7,6,5,4,3,12,11] => [11,12,3,4,5,6,7,8,9,10,1,2] => [2,8,2] => ?
=> ? = 3
[2,1,12,9,8,7,6,5,4,11,10,3] => [3,10,11,4,5,6,7,8,9,12,1,2] => [3,7,2] => ?
=> ? = 3
[4,3,2,1,6,5,8,7,10,9,12,11] => [11,12,9,10,7,8,5,6,1,2,3,4] => [2,2,2,2,4] => ?
=> ? = 5
[4,3,2,1,12,11,10,9,8,7,6,5] => [5,6,7,8,9,10,11,12,1,2,3,4] => [8,4] => ?
=> ? = 2
[10,3,2,9,8,7,6,5,4,1,12,11] => [11,12,1,4,5,6,7,8,9,2,3,10] => [2,7,3] => ?
=> ? = 3
[12,3,2,11,10,9,8,7,6,5,4,1] => [1,4,5,6,7,8,9,10,11,2,3,12] => [9,3] => ?
=> ? = 2
[6,5,4,3,2,1,8,7,10,9,12,11] => [11,12,9,10,7,8,1,2,3,4,5,6] => [2,2,2,6] => ?
=> ? = 4
[6,5,4,3,2,1,8,7,12,11,10,9] => [9,10,11,12,7,8,1,2,3,4,5,6] => [4,2,6] => ?
=> ? = 3
[6,5,4,3,2,1,10,9,8,7,12,11] => [11,12,7,8,9,10,1,2,3,4,5,6] => [2,4,6] => ?
=> ? = 3
[6,5,4,3,2,1,12,9,8,11,10,7] => [7,10,11,8,9,12,1,2,3,4,5,6] => [3,3,6] => ?
=> ? = 3
[8,5,4,3,2,7,6,1,10,9,12,11] => [11,12,9,10,1,6,7,2,3,4,5,8] => [2,2,3,5] => ?
=> ? = 4
[10,5,4,3,2,7,6,9,8,1,12,11] => [11,12,1,8,9,6,7,2,3,4,5,10] => [2,3,2,5] => ?
=> ? = 4
[12,5,4,3,2,7,6,9,8,11,10,1] => [1,10,11,8,9,6,7,2,3,4,5,12] => [3,2,2,5] => ?
=> ? = 4
[12,11,4,3,10,9,8,7,6,5,2,1] => [1,2,5,6,7,8,9,10,3,4,11,12] => [8,4] => ?
=> ? = 2
[8,7,6,5,4,3,2,1,10,9,12,11] => [11,12,9,10,1,2,3,4,5,6,7,8] => [2,2,8] => ?
=> ? = 3
[8,7,6,5,4,3,2,1,12,11,10,9] => [9,10,11,12,1,2,3,4,5,6,7,8] => [4,8] => ?
=> ? = 2
[10,7,6,5,4,3,2,9,8,1,12,11] => [11,12,1,8,9,2,3,4,5,6,7,10] => [2,3,7] => ?
=> ? = 3
[12,7,6,5,4,3,2,11,10,9,8,1] => [1,8,9,10,11,2,3,4,5,6,7,12] => [5,7] => ?
=> ? = 2
[10,9,6,5,4,3,8,7,2,1,12,11] => [11,12,1,2,7,8,3,4,5,6,9,10] => [2,4,6] => ?
=> ? = 3
[12,9,6,5,4,3,8,7,2,11,10,1] => [1,10,11,2,7,8,3,4,5,6,9,12] => [3,3,6] => ?
=> ? = 3
[12,11,6,5,4,3,8,7,10,9,2,1] => [1,2,9,10,7,8,3,4,5,6,11,12] => [4,2,6] => ?
=> ? = 3
[10,9,8,7,6,5,4,3,2,1,12,11] => [11,12,1,2,3,4,5,6,7,8,9,10] => [2,10] => ?
=> ? = 2
[12,9,8,7,6,5,4,3,2,11,10,1] => [1,10,11,2,3,4,5,6,7,8,9,12] => [3,9] => ?
=> ? = 2
[12,11,8,7,6,5,4,3,10,9,2,1] => [1,2,9,10,3,4,5,6,7,8,11,12] => [4,8] => ?
=> ? = 2
[12,11,10,7,6,5,4,9,8,3,2,1] => [1,2,3,8,9,4,5,6,7,10,11,12] => [5,7] => ?
=> ? = 2
[9,2,3,1,4,5,6,7,8] => [8,7,6,5,4,1,3,2,9] => ? => ?
=> ? = 2
[8,4,6,1,7,3,2,5] => [5,2,3,7,1,6,4,8] => ? => ?
=> ? = 3
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St000390
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00109: Permutations —descent word⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
St000390: Binary words ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Values
[1] => => ? = 0
[1,2] => 0 => 0
[2,1] => 1 => 1
[1,2,3] => 00 => 0
[1,3,2] => 01 => 1
[2,1,3] => 10 => 1
[2,3,1] => 01 => 1
[3,1,2] => 10 => 1
[3,2,1] => 11 => 1
[1,2,3,4] => 000 => 0
[1,2,4,3] => 001 => 1
[1,3,2,4] => 010 => 1
[1,3,4,2] => 001 => 1
[1,4,2,3] => 010 => 1
[1,4,3,2] => 011 => 1
[2,1,3,4] => 100 => 1
[2,1,4,3] => 101 => 2
[2,3,1,4] => 010 => 1
[2,3,4,1] => 001 => 1
[2,4,1,3] => 010 => 1
[2,4,3,1] => 011 => 1
[3,1,2,4] => 100 => 1
[3,1,4,2] => 101 => 2
[3,2,1,4] => 110 => 1
[3,2,4,1] => 101 => 2
[3,4,1,2] => 010 => 1
[3,4,2,1] => 011 => 1
[4,1,2,3] => 100 => 1
[4,1,3,2] => 101 => 2
[4,2,1,3] => 110 => 1
[4,2,3,1] => 101 => 2
[4,3,1,2] => 110 => 1
[4,3,2,1] => 111 => 1
[1,2,3,4,5] => 0000 => 0
[1,2,3,5,4] => 0001 => 1
[1,2,4,3,5] => 0010 => 1
[1,2,4,5,3] => 0001 => 1
[1,2,5,3,4] => 0010 => 1
[1,2,5,4,3] => 0011 => 1
[1,3,2,4,5] => 0100 => 1
[1,3,2,5,4] => 0101 => 2
[1,3,4,2,5] => 0010 => 1
[1,3,4,5,2] => 0001 => 1
[1,3,5,2,4] => 0010 => 1
[1,3,5,4,2] => 0011 => 1
[1,4,2,3,5] => 0100 => 1
[1,4,2,5,3] => 0101 => 2
[1,4,3,2,5] => 0110 => 1
[1,4,3,5,2] => 0101 => 2
[1,4,5,2,3] => 0010 => 1
[1,4,5,3,2] => 0011 => 1
[] => ? => ? = 0
[2,1,4,3,6,5,8,7,12,11,10,9] => 10101010111 => ? = 5
[2,1,4,3,6,5,10,9,8,7,12,11] => 10101011101 => ? = 5
[2,1,4,3,6,5,12,9,8,11,10,7] => 10101011011 => ? = 5
[2,1,4,3,6,5,12,11,10,9,8,7] => 10101011111 => ? = 4
[2,1,4,3,8,7,6,5,10,9,12,11] => 10101110101 => ? = 5
[2,1,4,3,8,7,6,5,12,11,10,9] => 10101110111 => ? = 4
[2,1,4,3,10,7,6,9,8,5,12,11] => 10101101101 => ? = 5
[2,1,4,3,12,7,6,9,8,11,10,5] => 10101101011 => ? = 5
[2,1,4,3,12,7,6,11,10,9,8,5] => 10101101111 => ? = 4
[2,1,4,3,10,9,8,7,6,5,12,11] => 10101111101 => ? = 4
[2,1,4,3,12,9,8,7,6,11,10,5] => 10101111011 => ? = 4
[2,1,4,3,12,11,8,7,10,9,6,5] => 10101110111 => ? = 4
[2,1,4,3,12,11,10,9,8,7,6,5] => 10101111111 => ? = 3
[2,1,6,5,4,3,8,7,10,9,12,11] => 10111010101 => ? = 5
[2,1,6,5,4,3,8,7,12,11,10,9] => 10111010111 => ? = 4
[2,1,6,5,4,3,10,9,8,7,12,11] => 10111011101 => ? = 4
[2,1,6,5,4,3,12,9,8,11,10,7] => 10111011011 => ? = 4
[2,1,6,5,4,3,12,11,10,9,8,7] => 10111011111 => ? = 3
[2,1,8,5,4,7,6,3,10,9,12,11] => 10110110101 => ? = 5
[2,1,8,5,4,7,6,3,12,11,10,9] => 10110110111 => ? = 4
[2,1,10,5,4,7,6,9,8,3,12,11] => 10110101101 => ? = 5
[2,1,12,5,4,7,6,9,8,11,10,3] => 10110101011 => ? = 5
[2,1,12,5,4,7,6,11,10,9,8,3] => 10110101111 => ? = 4
[2,1,10,5,4,9,8,7,6,3,12,11] => 10110111101 => ? = 4
[2,1,12,5,4,9,8,7,6,11,10,3] => 10110111011 => ? = 4
[2,1,12,5,4,11,8,7,10,9,6,3] => 10110110111 => ? = 4
[2,1,12,5,4,11,10,9,8,7,6,3] => 10110111111 => ? = 3
[2,1,8,7,6,5,4,3,10,9,12,11] => 10111110101 => ? = 4
[2,1,8,7,6,5,4,3,12,11,10,9] => 10111110111 => ? = 3
[2,1,10,7,6,5,4,9,8,3,12,11] => 10111101101 => ? = 4
[2,1,12,7,6,5,4,9,8,11,10,3] => 10111101011 => ? = 4
[2,1,12,7,6,5,4,11,10,9,8,3] => 10111101111 => ? = 3
[2,1,10,9,6,5,8,7,4,3,12,11] => 10111011101 => ? = 4
[2,1,12,9,6,5,8,7,4,11,10,3] => 10111011011 => ? = 4
[2,1,12,11,6,5,8,7,10,9,4,3] => 10111010111 => ? = 4
[2,1,12,11,6,5,10,9,8,7,4,3] => 10111011111 => ? = 3
[2,1,10,9,8,7,6,5,4,3,12,11] => 10111111101 => ? = 3
[2,1,12,9,8,7,6,5,4,11,10,3] => 10111111011 => ? = 3
[2,1,12,11,8,7,6,5,10,9,4,3] => 10111110111 => ? = 3
[2,1,12,11,10,7,6,9,8,5,4,3] => 10111101111 => ? = 3
[2,1,12,11,10,9,8,7,6,5,4,3] => 10111111111 => ? = 2
[4,3,2,1,6,5,8,7,10,9,12,11] => 11101010101 => ? = 5
[4,3,2,1,6,5,8,7,12,11,10,9] => 11101010111 => ? = 4
[4,3,2,1,6,5,10,9,8,7,12,11] => 11101011101 => ? = 4
[4,3,2,1,6,5,12,9,8,11,10,7] => 11101011011 => ? = 4
[4,3,2,1,6,5,12,11,10,9,8,7] => 11101011111 => ? = 3
[4,3,2,1,8,7,6,5,10,9,12,11] => 11101110101 => ? = 4
[4,3,2,1,8,7,6,5,12,11,10,9] => 11101110111 => ? = 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)
Mp00064: Permutations —reverse⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 100%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 1 => 0
[1,2] => [2,1] => [1,1] => 11 => 0
[2,1] => [1,2] => [2] => 10 => 1
[1,2,3] => [3,2,1] => [1,1,1] => 111 => 0
[1,3,2] => [2,3,1] => [2,1] => 101 => 1
[2,1,3] => [3,1,2] => [1,2] => 110 => 1
[2,3,1] => [1,3,2] => [2,1] => 101 => 1
[3,1,2] => [2,1,3] => [1,2] => 110 => 1
[3,2,1] => [1,2,3] => [3] => 100 => 1
[1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1111 => 0
[1,2,4,3] => [3,4,2,1] => [2,1,1] => 1011 => 1
[1,3,2,4] => [4,2,3,1] => [1,2,1] => 1101 => 1
[1,3,4,2] => [2,4,3,1] => [2,1,1] => 1011 => 1
[1,4,2,3] => [3,2,4,1] => [1,2,1] => 1101 => 1
[1,4,3,2] => [2,3,4,1] => [3,1] => 1001 => 1
[2,1,3,4] => [4,3,1,2] => [1,1,2] => 1110 => 1
[2,1,4,3] => [3,4,1,2] => [2,2] => 1010 => 2
[2,3,1,4] => [4,1,3,2] => [1,2,1] => 1101 => 1
[2,3,4,1] => [1,4,3,2] => [2,1,1] => 1011 => 1
[2,4,1,3] => [3,1,4,2] => [1,2,1] => 1101 => 1
[2,4,3,1] => [1,3,4,2] => [3,1] => 1001 => 1
[3,1,2,4] => [4,2,1,3] => [1,1,2] => 1110 => 1
[3,1,4,2] => [2,4,1,3] => [2,2] => 1010 => 2
[3,2,1,4] => [4,1,2,3] => [1,3] => 1100 => 1
[3,2,4,1] => [1,4,2,3] => [2,2] => 1010 => 2
[3,4,1,2] => [2,1,4,3] => [1,2,1] => 1101 => 1
[3,4,2,1] => [1,2,4,3] => [3,1] => 1001 => 1
[4,1,2,3] => [3,2,1,4] => [1,1,2] => 1110 => 1
[4,1,3,2] => [2,3,1,4] => [2,2] => 1010 => 2
[4,2,1,3] => [3,1,2,4] => [1,3] => 1100 => 1
[4,2,3,1] => [1,3,2,4] => [2,2] => 1010 => 2
[4,3,1,2] => [2,1,3,4] => [1,3] => 1100 => 1
[4,3,2,1] => [1,2,3,4] => [4] => 1000 => 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1] => 11111 => 0
[1,2,3,5,4] => [4,5,3,2,1] => [2,1,1,1] => 10111 => 1
[1,2,4,3,5] => [5,3,4,2,1] => [1,2,1,1] => 11011 => 1
[1,2,4,5,3] => [3,5,4,2,1] => [2,1,1,1] => 10111 => 1
[1,2,5,3,4] => [4,3,5,2,1] => [1,2,1,1] => 11011 => 1
[1,2,5,4,3] => [3,4,5,2,1] => [3,1,1] => 10011 => 1
[1,3,2,4,5] => [5,4,2,3,1] => [1,1,2,1] => 11101 => 1
[1,3,2,5,4] => [4,5,2,3,1] => [2,2,1] => 10101 => 2
[1,3,4,2,5] => [5,2,4,3,1] => [1,2,1,1] => 11011 => 1
[1,3,4,5,2] => [2,5,4,3,1] => [2,1,1,1] => 10111 => 1
[1,3,5,2,4] => [4,2,5,3,1] => [1,2,1,1] => 11011 => 1
[1,3,5,4,2] => [2,4,5,3,1] => [3,1,1] => 10011 => 1
[1,4,2,3,5] => [5,3,2,4,1] => [1,1,2,1] => 11101 => 1
[1,4,2,5,3] => [3,5,2,4,1] => [2,2,1] => 10101 => 2
[1,4,3,2,5] => [5,2,3,4,1] => [1,3,1] => 11001 => 1
[1,4,3,5,2] => [2,5,3,4,1] => [2,2,1] => 10101 => 2
[1,4,5,2,3] => [3,2,5,4,1] => [1,2,1,1] => 11011 => 1
[4,3,2,1,6,5,8,7,10,9] => [9,10,7,8,5,6,1,2,3,4] => [2,2,2,4] => 1010101000 => ? = 4
[6,3,2,5,4,1,8,7,10,9] => [9,10,7,8,1,4,5,2,3,6] => [2,2,3,3] => 1010100100 => ? = 4
[8,3,2,5,4,7,6,1,10,9] => [9,10,1,6,7,4,5,2,3,8] => [2,3,2,3] => 1010010100 => ? = 4
[10,3,2,5,4,7,6,9,8,1] => [1,8,9,6,7,4,5,2,3,10] => [3,2,2,3] => 1001010100 => ? = 4
[2,1,6,5,4,3,8,7,10,9] => [9,10,7,8,3,4,5,6,1,2] => [2,2,4,2] => 1010100010 => ? = 4
[6,5,4,3,2,1,8,7,10,9] => [9,10,7,8,1,2,3,4,5,6] => [2,2,6] => 1010100000 => ? = 3
[8,5,4,3,2,7,6,1,10,9] => [9,10,1,6,7,2,3,4,5,8] => [2,3,5] => 1010010000 => ? = 3
[10,5,4,3,2,7,6,9,8,1] => [1,8,9,6,7,2,3,4,5,10] => [3,2,5] => 1001010000 => ? = 3
[2,1,8,5,4,7,6,3,10,9] => [9,10,3,6,7,4,5,8,1,2] => [2,3,3,2] => 1010010010 => ? = 4
[8,7,4,3,6,5,2,1,10,9] => [9,10,1,2,5,6,3,4,7,8] => [2,4,4] => 1010001000 => ? = 3
[10,7,4,3,6,5,2,9,8,1] => [1,8,9,2,5,6,3,4,7,10] => [3,3,4] => 1001001000 => ? = 3
[2,1,10,5,4,7,6,9,8,3] => [3,8,9,6,7,4,5,10,1,2] => [3,2,3,2] => 1001010010 => ? = 4
[10,9,4,3,6,5,8,7,2,1] => [1,2,7,8,5,6,3,4,9,10] => [4,2,4] => 1000101000 => ? = 3
[8,9,5,6,3,4,10,1,2,7] => [7,2,1,10,4,3,6,5,9,8] => [1,1,2,1,2,2,1] => 1110110101 => ? = 3
[2,1,4,3,8,7,6,5,10,9] => [9,10,5,6,7,8,3,4,1,2] => [2,4,2,2] => 1010001010 => ? = 4
[4,3,2,1,8,7,6,5,10,9] => [9,10,5,6,7,8,1,2,3,4] => [2,4,4] => 1010001000 => ? = 3
[8,3,2,7,6,5,4,1,10,9] => [9,10,1,4,5,6,7,2,3,8] => [2,5,3] => 1010000100 => ? = 3
[10,3,2,7,6,5,4,9,8,1] => [1,8,9,4,5,6,7,2,3,10] => [3,4,3] => 1001000100 => ? = 3
[2,1,8,7,6,5,4,3,10,9] => [9,10,3,4,5,6,7,8,1,2] => [2,6,2] => 1010000010 => ? = 3
[8,7,6,5,4,3,2,1,10,9] => [9,10,1,2,3,4,5,6,7,8] => [2,8] => 1010000000 => ? = 2
[9,7,6,5,4,3,2,10,1,8] => [8,1,10,2,3,4,5,6,7,9] => [1,2,7] => 1101000000 => ? = 2
[10,7,6,5,4,3,2,9,8,1] => [1,8,9,2,3,4,5,6,7,10] => [3,7] => 1001000000 => ? = 2
[2,1,10,7,6,5,4,9,8,3] => [3,8,9,4,5,6,7,10,1,2] => [3,5,2] => 1001000010 => ? = 3
[10,9,6,5,4,3,8,7,2,1] => [1,2,7,8,3,4,5,6,9,10] => [4,6] => 1000100000 => ? = 2
[2,1,4,3,10,7,6,9,8,5] => [5,8,9,6,7,10,3,4,1,2] => [3,3,2,2] => 1001001010 => ? = 4
[4,3,2,1,10,7,6,9,8,5] => [5,8,9,6,7,10,1,2,3,4] => [3,3,4] => 1001001000 => ? = 3
[10,3,2,9,6,5,8,7,4,1] => [1,4,7,8,5,6,9,2,3,10] => [4,3,3] => 1000100100 => ? = 3
[2,1,10,9,6,5,8,7,4,3] => [3,4,7,8,5,6,9,10,1,2] => [4,4,2] => 1000100010 => ? = 3
[10,9,8,5,4,7,6,3,2,1] => [1,2,3,6,7,4,5,8,9,10] => [5,5] => 1000010000 => ? = 2
[6,7,8,9,10,1,2,3,4,5] => [5,4,3,2,1,10,9,8,7,6] => [1,1,1,1,2,1,1,1,1] => 1111101111 => ? = 1
[7,3,2,8,9,10,1,4,5,6] => [6,5,4,1,10,9,8,2,3,7] => [1,1,1,2,1,1,3] => 1111011100 => ? = 2
[2,1,4,3,6,5,10,9,8,7] => [7,8,9,10,5,6,3,4,1,2] => [4,2,2,2] => 1000101010 => ? = 4
[4,3,2,1,6,5,10,9,8,7] => [7,8,9,10,5,6,1,2,3,4] => [4,2,4] => 1000101000 => ? = 3
[6,3,2,5,4,1,10,9,8,7] => [7,8,9,10,1,4,5,2,3,6] => [4,3,3] => 1000100100 => ? = 3
[10,3,2,5,4,9,8,7,6,1] => [1,6,7,8,9,4,5,2,3,10] => [5,2,3] => 1000010100 => ? = 3
[2,1,6,5,4,3,10,9,8,7] => [7,8,9,10,3,4,5,6,1,2] => [4,4,2] => 1000100010 => ? = 3
[6,5,4,3,2,1,10,9,8,7] => [7,8,9,10,1,2,3,4,5,6] => [4,6] => 1000100000 => ? = 2
[10,5,4,3,2,9,8,7,6,1] => [1,6,7,8,9,2,3,4,5,10] => [5,5] => 1000010000 => ? = 2
[2,1,10,5,4,9,8,7,6,3] => [3,6,7,8,9,4,5,10,1,2] => [5,3,2] => 1000010010 => ? = 3
[10,9,4,3,8,7,6,5,2,1] => [1,2,5,6,7,8,3,4,9,10] => [6,4] => 1000001000 => ? = 2
[4,3,2,1,10,9,8,7,6,5] => [5,6,7,8,9,10,1,2,3,4] => [6,4] => 1000001000 => ? = 2
[] => [] => [] => ? => ? = 0
[2,3,4,5,6,7,8,9,10,1] => [1,10,9,8,7,6,5,4,3,2] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[2,3,4,5,6,7,8,9,1,10] => [10,1,9,8,7,6,5,4,3,2] => [1,2,1,1,1,1,1,1,1] => 1101111111 => ? = 1
[2,1,4,3,6,5,8,7,12,11,10,9] => [9,10,11,12,7,8,5,6,3,4,1,2] => [4,2,2,2,2] => 100010101010 => ? = 5
[2,1,4,3,6,5,10,9,8,7,12,11] => [11,12,7,8,9,10,5,6,3,4,1,2] => [2,4,2,2,2] => 101000101010 => ? = 5
[2,1,4,3,6,5,12,9,8,11,10,7] => [7,10,11,8,9,12,5,6,3,4,1,2] => [3,3,2,2,2] => 100100101010 => ? = 5
[2,1,4,3,6,5,12,11,10,9,8,7] => [7,8,9,10,11,12,5,6,3,4,1,2] => [6,2,2,2] => 100000101010 => ? = 4
[2,1,4,3,8,7,6,5,10,9,12,11] => [11,12,9,10,5,6,7,8,3,4,1,2] => [2,2,4,2,2] => 101010001010 => ? = 5
[2,1,4,3,8,7,6,5,12,11,10,9] => [9,10,11,12,5,6,7,8,3,4,1,2] => [4,4,2,2] => 100010001010 => ? = 4
Description
The number of descents of a binary word.
Matching statistic: St000658
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St000658: Dyck paths ⟶ ℤResult quality: 71% ●values known / values provided: 73%●distinct values known / distinct values provided: 71%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St000658: Dyck paths ⟶ ℤResult quality: 71% ●values known / values provided: 73%●distinct values known / distinct values provided: 71%
Values
[1] => [1] => [1,0]
=> [1,0]
=> ? = 0
[1,2] => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 0
[2,1] => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,4,5,3,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[7,6,5,4,8,3,2,1] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 2
[8,7,5,6,3,4,2,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 3
[8,7,5,6,4,2,3,1] => [1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> ? = 3
[8,7,6,4,5,2,3,1] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 3
[8,6,5,7,3,2,4,1] => [1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> ? = 3
[7,6,5,4,3,2,8,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 2
[6,5,4,3,7,2,8,1] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 3
[8,7,6,5,4,3,1,2] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 1
[8,7,5,6,4,3,1,2] => [1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 2
[8,7,6,4,5,3,1,2] => [1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 2
[8,7,6,5,3,4,1,2] => [1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 2
[8,7,5,6,3,4,1,2] => [1,1,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3
[8,7,6,5,4,2,1,3] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 1
[8,7,6,5,4,1,2,3] => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 1
[8,7,6,4,5,1,2,3] => [1,1,1,2,3] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 2
[7,6,5,8,3,2,1,4] => [1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 2
[8,7,6,5,1,2,3,4] => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 1
[8,7,6,4,3,2,1,5] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 1
[8,7,6,4,2,1,3,5] => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 1
[8,7,6,1,2,3,4,5] => [1,1,1,5] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[8,6,5,4,3,2,1,7] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 1
[8,6,5,4,3,1,2,7] => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 1
[8,6,5,4,2,1,3,7] => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 1
[8,6,4,3,2,1,5,7] => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 1
[8,6,4,2,1,3,5,7] => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 1
[8,3,2,1,4,5,6,7] => [1,1,1,5] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[7,6,5,4,3,2,1,8] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 1
[7,6,4,5,3,2,1,8] => [1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 2
[6,5,4,7,3,2,1,8] => [1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 2
[7,6,4,3,5,2,1,8] => [1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 2
[7,5,4,3,6,2,1,8] => [1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 2
[6,5,4,3,2,1,7,8] => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 1
[5,4,3,2,1,6,7,8] => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 1
[4,3,2,1,5,6,7,8] => [1,1,1,5] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[3,2,1,8,7,6,5,4] => [1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2
[3,2,1,6,5,8,7,4] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 3
[4,3,2,1,8,7,6,5] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 2
[5,4,3,2,1,8,7,6] => [1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> ? = 2
[6,5,4,3,2,1,8,7] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 2
[4,3,2,1,6,5,8,7] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 3
[5,4,3,2,7,6,1,8] => [1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 2
[4,3,2,7,5,6,8,1] => [1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3
[4,3,2,8,5,7,6,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 3
[6,3,2,5,4,1,8,7] => [1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> ? = 3
[8,3,2,5,4,7,6,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 3
[8,3,2,7,6,5,4,1] => [1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2
[7,4,3,6,5,2,1,8] => [1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 2
[7,6,3,5,4,2,1,8] => [1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 2
Description
The number of rises of length 2 of a Dyck path.
This is also the number of $(1,1)$ steps of the associated Łukasiewicz path, see [1].
A related statistic is the number of double rises in a Dyck path, [[St000024]].
Matching statistic: St000659
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 71% ●values known / values provided: 73%●distinct values known / distinct values provided: 71%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 71% ●values known / values provided: 73%●distinct values known / distinct values provided: 71%
Values
[1] => [1] => [1,0]
=> [1,0]
=> ? = 0
[1,2] => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 0
[2,1] => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,4,5,3,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[7,6,5,4,8,3,2,1] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 2
[8,7,5,6,3,4,2,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 3
[8,7,5,6,4,2,3,1] => [1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> ? = 3
[8,7,6,4,5,2,3,1] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 3
[8,6,5,7,3,2,4,1] => [1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> ? = 3
[7,6,5,4,3,2,8,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 2
[6,5,4,3,7,2,8,1] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 3
[8,7,6,5,4,3,1,2] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 1
[8,7,5,6,4,3,1,2] => [1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 2
[8,7,6,4,5,3,1,2] => [1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 2
[8,7,6,5,3,4,1,2] => [1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 2
[8,7,5,6,3,4,1,2] => [1,1,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3
[8,7,6,5,4,2,1,3] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 1
[8,7,6,5,4,1,2,3] => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 1
[8,7,6,4,5,1,2,3] => [1,1,1,2,3] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 2
[7,6,5,8,3,2,1,4] => [1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 2
[8,7,6,5,1,2,3,4] => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 1
[8,7,6,4,3,2,1,5] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 1
[8,7,6,4,2,1,3,5] => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 1
[8,7,6,1,2,3,4,5] => [1,1,1,5] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[8,6,5,4,3,2,1,7] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 1
[8,6,5,4,3,1,2,7] => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 1
[8,6,5,4,2,1,3,7] => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 1
[8,6,4,3,2,1,5,7] => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 1
[8,6,4,2,1,3,5,7] => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 1
[8,3,2,1,4,5,6,7] => [1,1,1,5] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[7,6,5,4,3,2,1,8] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 1
[7,6,4,5,3,2,1,8] => [1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 2
[6,5,4,7,3,2,1,8] => [1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 2
[7,6,4,3,5,2,1,8] => [1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 2
[7,5,4,3,6,2,1,8] => [1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 2
[6,5,4,3,2,1,7,8] => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 1
[5,4,3,2,1,6,7,8] => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 1
[4,3,2,1,5,6,7,8] => [1,1,1,5] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[3,2,1,8,7,6,5,4] => [1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2
[3,2,1,6,5,8,7,4] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 3
[4,3,2,1,8,7,6,5] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 2
[5,4,3,2,1,8,7,6] => [1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> ? = 2
[6,5,4,3,2,1,8,7] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 2
[4,3,2,1,6,5,8,7] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 3
[5,4,3,2,7,6,1,8] => [1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 2
[4,3,2,7,5,6,8,1] => [1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3
[4,3,2,8,5,7,6,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 3
[6,3,2,5,4,1,8,7] => [1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> ? = 3
[8,3,2,5,4,7,6,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 3
[8,3,2,7,6,5,4,1] => [1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2
[7,4,3,6,5,2,1,8] => [1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 2
[7,6,3,5,4,2,1,8] => [1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 2
Description
The number of rises of length at least 2 of a Dyck path.
Matching statistic: St000340
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000340: Dyck paths ⟶ ℤResult quality: 57% ●values known / values provided: 58%●distinct values known / distinct values provided: 57%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000340: Dyck paths ⟶ ℤResult quality: 57% ●values known / values provided: 58%●distinct values known / distinct values provided: 57%
Values
[1] => [1] => [1,0]
=> [1,0]
=> 0
[1,2] => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 0
[2,1] => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[7,6,8,5,4,3,2,1] => [1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 2
[7,8,5,6,4,3,2,1] => [2,2,1,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> ? = 2
[7,8,6,4,5,3,2,1] => [2,1,2,1,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 2
[8,6,7,4,5,3,2,1] => [1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 3
[7,6,5,4,8,3,2,1] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2
[6,5,7,4,8,3,2,1] => [1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 3
[7,8,6,5,3,4,2,1] => [2,1,1,2,1,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 2
[8,6,7,5,3,4,2,1] => [1,2,1,2,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 3
[8,7,5,6,3,4,2,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 3
[6,7,8,3,4,5,2,1] => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,0]
=> ? = 2
[7,8,6,5,4,2,3,1] => [2,1,1,1,2,1] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 2
[8,6,7,5,4,2,3,1] => [1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 3
[8,7,5,6,4,2,3,1] => [1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 3
[8,7,6,4,5,2,3,1] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 3
[8,6,5,7,3,2,4,1] => [1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 3
[6,7,8,5,2,3,4,1] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 2
[8,5,6,7,2,3,4,1] => [1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 3
[7,8,3,2,4,5,6,1] => [2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 2
[7,6,5,4,3,2,8,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2
[6,5,7,4,3,2,8,1] => [1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 3
[6,5,4,3,7,2,8,1] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 3
[5,4,6,3,7,2,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 4
[3,4,5,2,6,7,8,1] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 2
[5,2,3,4,6,7,8,1] => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 2
[4,3,2,5,6,7,8,1] => [1,1,5,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 2
[3,4,2,5,6,7,8,1] => [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 2
[4,2,3,5,6,7,8,1] => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 2
[3,2,4,5,6,7,8,1] => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 2
[2,3,4,5,6,7,8,1] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[6,5,8,7,4,3,2,1] => [1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 2
[3,5,8,7,6,4,2,1] => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 1
[4,3,6,5,8,7,2,1] => [1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 3
[2,6,8,7,5,4,3,1] => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 1
[2,3,5,6,8,7,4,1] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 1
[3,2,6,5,4,8,7,1] => [1,2,1,2,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 3
[4,3,6,5,7,2,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 4
[3,2,6,5,7,4,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 4
[4,3,5,2,7,6,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 4
[3,2,5,4,7,6,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 4
[1,8,7,6,5,4,3,2] => [2,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 1
[1,7,8,6,5,4,3,2] => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 1
[1,6,8,7,5,4,3,2] => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 1
[1,7,6,8,5,4,3,2] => [2,2,1,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> ? = 2
[1,6,7,8,5,4,3,2] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 1
[1,5,8,7,6,4,3,2] => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 1
[1,7,6,5,8,4,3,2] => [2,1,2,1,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 2
[1,3,5,7,8,6,4,2] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 1
[1,4,3,8,7,6,5,2] => [2,2,1,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> ? = 2
[1,6,5,4,3,8,7,2] => [2,1,1,2,1,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 2
Description
The number of non-final maximal constant sub-paths of length greater than one.
This is the total number of occurrences of the patterns $110$ and $001$.
Matching statistic: St000834
(load all 46 compositions to match this statistic)
(load all 46 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 71%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 71%
Values
[1] => [1] => [1,0]
=> [1] => 0
[1,2] => [2] => [1,1,0,0]
=> [2,1] => 0
[2,1] => [1,1] => [1,0,1,0]
=> [1,2] => 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> [3,2,1] => 0
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 0
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 1
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 0
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 2
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 2
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 2
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
[1,2,3,6,4,7,5] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 2
[1,2,6,3,4,7,5] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,2,1,6,5,4,7] => ? = 2
[1,3,7,2,4,5,6] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,2,1,7,6,5,4] => ? = 1
[1,4,6,7,2,3,5] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,3,2,1,7,6,5] => ? = 1
[1,4,7,2,3,5,6] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,2,1,7,6,5,4] => ? = 1
[1,5,2,3,4,7,6] => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,6,5,4,3,7] => ? = 2
[1,5,2,7,3,4,6] => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,7,6,5] => ? = 2
[1,5,4,7,6,3,2] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,5,6,7] => ? = 2
[1,5,6,2,7,3,4] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,2,1,5,4,7,6] => ? = 2
[1,5,6,4,7,3,2] => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [3,2,1,5,4,6,7] => ? = 2
[1,5,6,7,2,3,4] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,3,2,1,7,6,5] => ? = 1
[1,5,7,2,3,4,6] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,2,1,7,6,5,4] => ? = 1
[1,6,2,3,4,7,5] => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,6,5,4,3,7] => ? = 2
[1,6,2,3,7,4,5] => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [2,1,5,4,3,7,6] => ? = 2
[1,6,2,7,3,4,5] => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,7,6,5] => ? = 2
[1,6,5,4,3,7,2] => [2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,3,4,6,5,7] => ? = 2
[1,6,5,4,7,3,2] => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,3,5,4,6,7] => ? = 2
[1,6,5,7,4,3,2] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,5,6,7] => ? = 2
[1,6,7,2,3,4,5] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,2,1,7,6,5,4] => ? = 1
[1,7,3,5,6,4,2] => [2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,5,4,3,6,7] => ? = 2
[1,7,3,6,5,4,2] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,5,6,7] => ? = 2
[1,7,4,3,6,5,2] => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,3,5,4,6,7] => ? = 2
[1,7,4,5,3,6,2] => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5,7] => ? = 3
[1,7,4,5,6,3,2] => [2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,5,4,3,6,7] => ? = 2
[1,7,4,6,5,3,2] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,5,6,7] => ? = 2
[1,7,5,4,3,6,2] => [2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,3,4,6,5,7] => ? = 2
[1,7,5,4,6,3,2] => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,3,5,4,6,7] => ? = 2
[1,7,6,4,5,3,2] => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,3,5,4,6,7] => ? = 2
[2,1,3,4,7,5,6] => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,5,4,3,2,7,6] => ? = 2
[2,3,4,1,5,6,7] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,2,1,7,6,5,4] => ? = 1
[2,3,4,1,5,7,6] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,2,1,6,5,4,7] => ? = 2
[2,3,4,5,1,6,7] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,3,2,1,7,6,5] => ? = 1
[2,3,4,5,1,7,6] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 2
[2,3,4,6,1,7,5] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 2
[2,3,4,7,1,5,6] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,3,2,1,7,6,5] => ? = 1
[2,5,7,1,3,4,6] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,2,1,7,6,5,4] => ? = 1
[2,6,1,7,3,4,5] => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,7,6,5] => ? = 2
[2,6,5,4,3,1,7] => [2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,5,7,6] => ? = 1
[2,6,7,1,3,4,5] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,2,1,7,6,5,4] => ? = 1
[3,1,2,4,5,7,6] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,6,5,4,3,2,7] => ? = 2
[3,2,4,5,6,7,1] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,6,5,4,3,2,7] => ? = 2
[3,4,2,5,6,7,1] => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,6,5,4,3,7] => ? = 2
[3,4,5,2,6,7,1] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,2,1,6,5,4,7] => ? = 2
[3,4,5,6,2,7,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 2
[3,5,6,1,2,4,7] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,2,1,7,6,5,4] => ? = 1
[3,5,6,4,2,1,7] => [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [3,2,1,4,5,7,6] => ? = 1
[3,6,5,4,2,1,7] => [2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,5,7,6] => ? = 1
[4,1,2,3,5,7,6] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,6,5,4,3,2,7] => ? = 2
[4,2,3,5,6,7,1] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,6,5,4,3,2,7] => ? = 2
[4,5,1,6,2,3,7] => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,7,6,5] => ? = 2
Description
The number of right outer peaks of a permutation.
A right outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $n$ if $w_n > w_{n-1}$.
In other words, it is a peak in the word $[w_1,..., w_n,0]$.
Matching statistic: St000884
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
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
St000884: Permutations ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 86%
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
St000884: Permutations ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 86%
Values
[1] => [.,.]
=> [1,0]
=> [1] => 0
[1,2] => [.,[.,.]]
=> [1,0,1,0]
=> [1,2] => 0
[2,1] => [[.,.],.]
=> [1,1,0,0]
=> [2,1] => 1
[1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1
[2,3,1] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[3,1,2] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1
[3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [3,1,2] => 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 2
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 2
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 2
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 2
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => 2
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 1
[1,7,4,3,6,5,2] => [.,[[[[.,.],.],[[.,.],.]],.]]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,5,2,3,7,4,6] => ? = 2
[1,7,4,5,3,6,2] => [.,[[[[.,.],[.,.]],[.,.]],.]]
=> [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,5,2,6,3,7,4] => ? = 3
[1,7,4,5,6,3,2] => [.,[[[[.,.],[.,[.,.]]],.],.]]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,5,2,6,7,3,4] => ? = 2
[2,1,3,4,7,5,6] => [[.,.],[.,[.,[[.,.],[.,.]]]]]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,3,4,6,5,7] => ? = 2
[2,1,3,7,4,5,6] => [[.,.],[.,[[.,.],[.,[.,.]]]]]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,3,5,4,6,7] => ? = 2
[2,1,6,3,4,5,7] => [[.,.],[[.,.],[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,5,6,7] => ? = 2
[2,1,7,3,4,5,6] => [[.,.],[[.,.],[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,5,6,7] => ? = 2
[2,6,1,7,3,4,5] => [[.,[.,.]],[[.,.],[.,[.,.]]]]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,4,6,7] => ? = 2
[3,1,2,4,5,7,6] => [[.,.],[.,[.,[.,[[.,.],.]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,5,7,6] => ? = 2
[3,2,4,5,6,7,1] => [[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,1,4,5,6,7,2] => ? = 2
[3,4,2,5,6,7,1] => [[[.,[.,.]],[.,[.,[.,.]]]],.]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [3,4,1,5,6,7,2] => ? = 2
[3,4,5,2,6,7,1] => [[[.,[.,[.,.]]],[.,[.,.]]],.]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [3,4,5,1,6,7,2] => ? = 2
[3,4,5,6,2,7,1] => [[[.,[.,[.,[.,.]]]],[.,.]],.]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [3,4,5,6,1,7,2] => ? = 2
[4,1,2,3,5,7,6] => [[.,.],[.,[.,[.,[[.,.],.]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,5,7,6] => ? = 2
[4,1,6,2,3,5,7] => [[.,.],[[.,.],[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,5,6,7] => ? = 2
[4,2,3,5,6,7,1] => [[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,1,4,5,6,7,2] => ? = 2
[4,2,5,1,3,6,7] => [[[.,.],[.,.]],[.,[.,[.,.]]]]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5,6,7] => ? = 2
[4,3,2,5,6,7,1] => [[[[.,.],.],[.,[.,[.,.]]]],.]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [4,1,2,5,6,7,3] => ? = 2
[4,3,5,2,6,7,1] => [[[[.,.],[.,.]],[.,[.,.]]],.]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> [4,1,5,2,6,7,3] => ? = 3
[4,5,1,6,2,3,7] => [[.,[.,.]],[[.,.],[.,[.,.]]]]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,4,6,7] => ? = 2
[4,5,2,3,6,7,1] => [[[.,[.,.]],[.,[.,[.,.]]]],.]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [3,4,1,5,6,7,2] => ? = 2
[4,6,1,2,3,7,5] => [[.,[.,.]],[.,[.,[[.,.],.]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [2,3,1,4,5,7,6] => ? = 2
[5,1,2,3,4,7,6] => [[.,.],[.,[.,[.,[[.,.],.]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,5,7,6] => ? = 2
[5,1,2,3,6,4,7] => [[.,.],[.,[.,[[.,.],[.,.]]]]]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,3,4,6,5,7] => ? = 2
[5,1,2,3,6,7,4] => [[.,.],[.,[.,[[.,[.,.]],.]]]]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [2,1,3,4,6,7,5] => ? = 2
[5,1,2,3,7,4,6] => [[.,.],[.,[.,[[.,.],[.,.]]]]]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,3,4,6,5,7] => ? = 2
[5,1,2,6,3,4,7] => [[.,.],[.,[[.,.],[.,[.,.]]]]]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,3,5,4,6,7] => ? = 2
[5,1,6,2,3,4,7] => [[.,.],[[.,.],[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,5,6,7] => ? = 2
[5,1,6,2,3,7,4] => [[.,.],[[.,.],[.,[[.,.],.]]]]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,4,3,5,7,6] => ? = 3
[5,2,3,4,6,7,1] => [[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,1,4,5,6,7,2] => ? = 2
[5,2,6,7,1,3,4] => [[[.,.],[.,[.,.]]],[.,[.,.]]]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2,6,7] => ? = 2
[5,3,2,4,6,7,1] => [[[[.,.],.],[.,[.,[.,.]]]],.]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [4,1,2,5,6,7,3] => ? = 2
[5,3,2,6,1,4,7] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [4,1,2,5,3,6,7] => ? = 2
[5,3,6,7,1,2,4] => [[[.,.],[.,[.,.]]],[.,[.,.]]]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2,6,7] => ? = 2
[5,4,6,7,1,2,3] => [[[.,.],[.,[.,.]]],[.,[.,.]]]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2,6,7] => ? = 2
[5,6,1,2,3,7,4] => [[.,[.,.]],[.,[.,[[.,.],.]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [2,3,1,4,5,7,6] => ? = 2
[5,6,3,4,1,2,7] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [3,4,1,5,2,6,7] => ? = 2
[6,1,2,3,4,7,5] => [[.,.],[.,[.,[.,[[.,.],.]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,5,7,6] => ? = 2
[6,1,2,3,7,4,5] => [[.,.],[.,[.,[[.,.],[.,.]]]]]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,3,4,6,5,7] => ? = 2
[6,1,2,7,3,4,5] => [[.,.],[.,[[.,.],[.,[.,.]]]]]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,3,5,4,6,7] => ? = 2
[6,2,3,4,5,7,1] => [[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,1,4,5,6,7,2] => ? = 2
[6,2,4,5,3,1,7] => [[[.,.],[[.,[.,.]],.]],[.,.]]
=> [1,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> [3,1,5,6,2,4,7] => ? = 2
[6,2,5,4,3,1,7] => [[[.,.],[[[.,.],.],.]],[.,.]]
=> [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [3,1,6,2,4,5,7] => ? = 2
[6,3,2,5,4,1,7] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> [1,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> [4,1,2,6,3,5,7] => ? = 2
[6,3,4,2,5,1,7] => [[[[.,.],[.,.]],[.,.]],[.,.]]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [4,1,5,2,6,3,7] => ? = 3
[6,3,4,5,2,1,7] => [[[[.,.],[.,[.,.]]],.],[.,.]]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [4,1,5,6,2,3,7] => ? = 2
[6,3,4,7,1,2,5] => [[[.,.],[.,[.,.]]],[.,[.,.]]]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2,6,7] => ? = 2
[6,4,5,7,1,2,3] => [[[.,.],[.,[.,.]]],[.,[.,.]]]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2,6,7] => ? = 2
[6,4,7,2,5,1,3] => [[[[.,.],[.,.]],[.,.]],[.,.]]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [4,1,5,2,6,3,7] => ? = 3
[6,4,7,3,5,1,2] => [[[[.,.],[.,.]],[.,.]],[.,.]]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [4,1,5,2,6,3,7] => ? = 3
Description
The number of isolated descents of a permutation.
A descent $i$ is isolated if neither $i+1$ nor $i-1$ are descents. If a permutation has only isolated descents, then it is called primitive in [1].
Matching statistic: St000374
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> [1] => 0
[1,2] => [2] => [1,1,0,0]
=> [1,2] => 0
[2,1] => [1,1] => [1,0,1,0]
=> [2,1] => 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> [1,2,3] => 0
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> [1,3,2] => 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> [2,1,3] => 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> [1,3,2] => 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> [2,1,3] => 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [2,3,1] => 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 1
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 1
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 2
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 1
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 2
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 2
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 1
[2,1,3,4,7,5,6] => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,1,3,4,6,5,7] => ? = 2
[2,1,3,7,4,5,6] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,1,3,5,4,6,7] => ? = 2
[2,1,6,3,4,5,7] => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,4,3,5,6,7] => ? = 2
[2,1,7,3,4,5,6] => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,4,3,5,6,7] => ? = 2
[2,1,7,6,5,4,3] => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,4,5,6,7,3] => ? = 2
[3,1,2,4,5,7,6] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,1,3,4,5,7,6] => ? = 2
[3,2,4,5,6,7,1] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,1,3,4,5,7,6] => ? = 2
[4,1,2,3,5,7,6] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,1,3,4,5,7,6] => ? = 2
[4,1,6,2,3,5,7] => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,4,3,5,6,7] => ? = 2
[4,2,3,5,6,7,1] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,1,3,4,5,7,6] => ? = 2
[4,2,5,1,3,6,7] => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,4,3,5,6,7] => ? = 2
[4,3,2,5,6,7,1] => [1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,3,1,4,5,7,6] => ? = 2
[4,3,5,2,6,7,1] => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,4,3,5,7,6] => ? = 3
[4,3,6,5,2,1,7] => [1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,4,5,6,3,7] => ? = 2
[5,1,2,3,4,7,6] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,1,3,4,5,7,6] => ? = 2
[5,1,2,3,6,4,7] => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,1,3,4,6,5,7] => ? = 2
[5,1,2,3,6,7,4] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,1,3,4,5,7,6] => ? = 2
[5,1,2,3,7,4,6] => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,1,3,4,6,5,7] => ? = 2
[5,1,2,6,3,4,7] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,1,3,5,4,6,7] => ? = 2
[5,1,6,2,3,4,7] => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,4,3,5,6,7] => ? = 2
[5,1,6,2,3,7,4] => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,4,3,5,7,6] => ? = 3
[5,2,3,4,6,7,1] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,1,3,4,5,7,6] => ? = 2
[5,2,6,7,1,3,4] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,1,3,5,4,6,7] => ? = 2
[5,3,2,4,6,7,1] => [1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,3,1,4,5,7,6] => ? = 2
[5,3,2,6,1,4,7] => [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,3,1,5,4,6,7] => ? = 2
[5,3,6,7,1,2,4] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,1,3,5,4,6,7] => ? = 2
[5,4,3,2,6,1,7] => [1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,3,4,1,6,5,7] => ? = 2
[5,4,3,6,2,1,7] => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,3,1,5,6,4,7] => ? = 2
[5,4,6,3,2,1,7] => [1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,4,5,6,3,7] => ? = 2
[5,4,6,7,1,2,3] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,1,3,5,4,6,7] => ? = 2
[6,1,2,3,4,7,5] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,1,3,4,5,7,6] => ? = 2
[6,1,2,3,7,4,5] => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,1,3,4,6,5,7] => ? = 2
[6,1,2,7,3,4,5] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,1,3,5,4,6,7] => ? = 2
[6,2,3,4,5,7,1] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,1,3,4,5,7,6] => ? = 2
[6,2,4,5,3,1,7] => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,1,3,5,6,4,7] => ? = 2
[6,2,5,4,3,1,7] => [1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,4,5,6,3,7] => ? = 2
[6,3,2,5,4,1,7] => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,3,1,5,6,4,7] => ? = 2
[6,3,4,2,5,1,7] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5,7] => ? = 3
[6,3,4,5,2,1,7] => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,1,3,5,6,4,7] => ? = 2
[6,3,4,7,1,2,5] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,1,3,5,4,6,7] => ? = 2
[6,3,5,4,2,1,7] => [1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,4,5,6,3,7] => ? = 2
[6,4,3,2,5,1,7] => [1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,3,4,1,6,5,7] => ? = 2
[6,4,3,2,7,1,5] => [1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,3,4,1,6,5,7] => ? = 2
[6,4,3,5,2,1,7] => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,3,1,5,6,4,7] => ? = 2
[6,4,5,7,1,2,3] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,1,3,5,4,6,7] => ? = 2
[6,4,7,2,5,1,3] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5,7] => ? = 3
[6,4,7,3,5,1,2] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5,7] => ? = 3
[6,5,3,4,2,1,7] => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,3,1,5,6,4,7] => ? = 2
[6,5,7,3,4,1,2] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5,7] => ? = 3
[7,2,3,1,4,5,6] => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,4,3,5,6,7] => ? = 2
Description
The number of exclusive right-to-left minima of a permutation.
This is the number of right-to-left minima that are not left-to-right maxima.
This is also the number of non weak exceedences of a permutation that are also not mid-points of a decreasing subsequence of length 3.
Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j < j$ and there do not exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$.
See also [[St000213]] and [[St000119]].
The following 56 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000703The number of deficiencies of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000251The number of nonsingleton blocks of a set partition. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001512The minimum rank of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St000024The number of double up and double down steps of a Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001354The number of series nodes in the modular decomposition of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001665The number of pure excedances of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001928The number of non-overlapping descents in a permutation. St000470The number of runs in a permutation. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity 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. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension 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. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000238The number of indices that are not small weak excedances. St000316The number of non-left-to-right-maxima of a permutation. St001874Lusztig's a-function for the symmetric group. St000325The width of the tree associated to a permutation. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000353The number of inner valleys of a permutation. St000711The number of big exceedences 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. St000092The number of outer peaks of a permutation. St001720The minimal length of a chain of small intervals in a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001597The Frobenius rank of a skew partition. St001624The breadth of a lattice.
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