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Your data matches 41 different statistics following compositions of up to 3 maps.
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Matching statistic: St000035
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000035: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000035: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> [1] => 0
[1,2] => [1,2] => [1,0,1,0]
=> [1,2] => 0
[2,1] => [2,1] => [1,1,0,0]
=> [2,1] => 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,3,2] => 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [2,1,3] => 1
[2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> [3,2,1] => 1
[3,1,2] => [3,2,1] => [1,1,1,0,0,0]
=> [3,2,1] => 1
[3,2,1] => [2,3,1] => [1,1,0,1,0,0]
=> [2,3,1] => 1
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
[1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[1,4,3,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
[2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 1
[2,4,1,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 1
[2,4,3,1] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 1
[3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
[3,1,4,2] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 1
[3,2,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1
[3,2,4,1] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 1
[3,4,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 2
[3,4,2,1] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 1
[4,1,2,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 1
[4,1,3,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 1
[4,2,1,3] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 1
[4,2,3,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1
[4,3,1,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 1
[4,3,2,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 2
[1,2,3,4,5] => [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] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 1
[1,2,4,5,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 1
[1,2,5,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 1
[1,2,5,4,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 1
[1,3,2,4,5] => [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] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 2
[1,3,4,2,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 1
[1,3,4,5,2] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 1
[1,3,5,2,4] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 1
[1,3,5,4,2] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 1
[1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 1
[1,4,2,5,3] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 1
[1,4,3,2,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 1
[1,4,3,5,2] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 1
[1,4,5,2,3] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 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: St000291
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 71% ●values known / values provided: 78%●distinct values known / distinct values provided: 71%
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 71% ●values known / values provided: 78%●distinct values known / distinct values provided: 71%
Values
[1] => {{1}}
=> [1] => 1 => 0
[1,2] => {{1},{2}}
=> [1,1] => 11 => 0
[2,1] => {{1,2}}
=> [2] => 10 => 1
[1,2,3] => {{1},{2},{3}}
=> [1,1,1] => 111 => 0
[1,3,2] => {{1},{2,3}}
=> [1,2] => 110 => 1
[2,1,3] => {{1,2},{3}}
=> [2,1] => 101 => 1
[2,3,1] => {{1,2,3}}
=> [3] => 100 => 1
[3,1,2] => {{1,2,3}}
=> [3] => 100 => 1
[3,2,1] => {{1,3},{2}}
=> [2,1] => 101 => 1
[1,2,3,4] => {{1},{2},{3},{4}}
=> [1,1,1,1] => 1111 => 0
[1,2,4,3] => {{1},{2},{3,4}}
=> [1,1,2] => 1110 => 1
[1,3,2,4] => {{1},{2,3},{4}}
=> [1,2,1] => 1101 => 1
[1,3,4,2] => {{1},{2,3,4}}
=> [1,3] => 1100 => 1
[1,4,2,3] => {{1},{2,3,4}}
=> [1,3] => 1100 => 1
[1,4,3,2] => {{1},{2,4},{3}}
=> [1,2,1] => 1101 => 1
[2,1,3,4] => {{1,2},{3},{4}}
=> [2,1,1] => 1011 => 1
[2,1,4,3] => {{1,2},{3,4}}
=> [2,2] => 1010 => 2
[2,3,1,4] => {{1,2,3},{4}}
=> [3,1] => 1001 => 1
[2,3,4,1] => {{1,2,3,4}}
=> [4] => 1000 => 1
[2,4,1,3] => {{1,2,3,4}}
=> [4] => 1000 => 1
[2,4,3,1] => {{1,2,4},{3}}
=> [3,1] => 1001 => 1
[3,1,2,4] => {{1,2,3},{4}}
=> [3,1] => 1001 => 1
[3,1,4,2] => {{1,2,3,4}}
=> [4] => 1000 => 1
[3,2,1,4] => {{1,3},{2},{4}}
=> [2,1,1] => 1011 => 1
[3,2,4,1] => {{1,3,4},{2}}
=> [3,1] => 1001 => 1
[3,4,1,2] => {{1,3},{2,4}}
=> [2,2] => 1010 => 2
[3,4,2,1] => {{1,2,3,4}}
=> [4] => 1000 => 1
[4,1,2,3] => {{1,2,3,4}}
=> [4] => 1000 => 1
[4,1,3,2] => {{1,2,4},{3}}
=> [3,1] => 1001 => 1
[4,2,1,3] => {{1,3,4},{2}}
=> [3,1] => 1001 => 1
[4,2,3,1] => {{1,4},{2},{3}}
=> [2,1,1] => 1011 => 1
[4,3,1,2] => {{1,2,3,4}}
=> [4] => 1000 => 1
[4,3,2,1] => {{1,4},{2,3}}
=> [2,2] => 1010 => 2
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,1,1,1,1] => 11111 => 0
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> [1,1,1,2] => 11110 => 1
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> [1,1,2,1] => 11101 => 1
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> [1,1,3] => 11100 => 1
[1,2,5,3,4] => {{1},{2},{3,4,5}}
=> [1,1,3] => 11100 => 1
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> [1,1,2,1] => 11101 => 1
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> [1,2,1,1] => 11011 => 1
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> [1,2,2] => 11010 => 2
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> [1,3,1] => 11001 => 1
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> [1,4] => 11000 => 1
[1,3,5,2,4] => {{1},{2,3,4,5}}
=> [1,4] => 11000 => 1
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> [1,3,1] => 11001 => 1
[1,4,2,3,5] => {{1},{2,3,4},{5}}
=> [1,3,1] => 11001 => 1
[1,4,2,5,3] => {{1},{2,3,4,5}}
=> [1,4] => 11000 => 1
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> [1,2,1,1] => 11011 => 1
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> [1,3,1] => 11001 => 1
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> [1,2,2] => 11010 => 2
[8,7,5,6,4,3,2,1] => {{1,8},{2,7},{3,4,5,6}}
=> ? => ? => ? = 3
[8,5,6,7,4,3,2,1] => {{1,8},{2,4,5,7},{3,6}}
=> ? => ? => ? = 3
[8,7,4,5,6,3,2,1] => {{1,8},{2,7},{3,4,5,6}}
=> ? => ? => ? = 3
[8,7,6,5,3,4,2,1] => {{1,8},{2,7},{3,4,5,6}}
=> ? => ? => ? = 3
[6,7,8,5,3,4,2,1] => ?
=> ? => ? => ? = 2
[7,5,6,8,3,4,2,1] => ?
=> ? => ? => ? = 1
[8,7,6,3,4,5,2,1] => {{1,8},{2,7},{3,4,5,6}}
=> ? => ? => ? = 3
[5,6,4,7,3,8,2,1] => ?
=> ? => ? => ? = 1
[5,6,7,3,4,8,2,1] => ?
=> ? => ? => ? = 1
[7,4,5,3,6,8,2,1] => ?
=> ? => ? => ? = 1
[7,8,5,6,4,2,3,1] => ?
=> ? => ? => ? = 1
[8,5,6,7,4,2,3,1] => ?
=> ? => ? => ? = 2
[8,6,4,5,7,2,3,1] => {{1,8},{2,6},{3,4,5,7}}
=> ? => ? => ? = 3
[7,5,4,6,8,2,3,1] => ?
=> ? => ? => ? = 1
[6,4,5,7,8,2,3,1] => ?
=> ? => ? => ? = 1
[8,6,7,5,3,2,4,1] => {{1,8},{2,6},{3,4,5,7}}
=> ? => ? => ? = 3
[8,5,6,7,3,2,4,1] => {{1,8},{2,3,5,6},{4,7}}
=> ? => ? => ? = 3
[6,5,7,8,3,2,4,1] => ?
=> ? => ? => ? = 1
[8,7,6,5,2,3,4,1] => {{1,8},{2,4,5,7},{3,6}}
=> ? => ? => ? = 3
[8,6,5,7,2,3,4,1] => {{1,8},{2,3,5,6},{4,7}}
=> ? => ? => ? = 3
[5,6,4,3,7,2,8,1] => ?
=> ? => ? => ? = 3
[6,4,5,7,2,3,8,1] => ?
=> ? => ? => ? = 1
[5,3,4,6,2,7,8,1] => ?
=> ? => ? => ? = 1
[5,6,2,3,4,7,8,1] => ?
=> ? => ? => ? = 1
[7,8,5,6,4,3,1,2] => {{1,7},{2,8},{3,4,5,6}}
=> ? => ? => ? = 3
[8,5,6,7,4,3,1,2] => ?
=> ? => ? => ? = 2
[7,5,6,8,4,3,1,2] => ?
=> ? => ? => ? = 3
[7,8,4,5,6,3,1,2] => {{1,7},{2,8},{3,4,5,6}}
=> ? => ? => ? = 3
[7,4,5,6,8,3,1,2] => ?
=> ? => ? => ? = 2
[7,8,6,5,3,4,1,2] => {{1,7},{2,8},{3,4,5,6}}
=> ? => ? => ? = 3
[8,6,7,5,3,4,1,2] => ?
=> ? => ? => ? = 1
[7,8,6,3,4,5,1,2] => {{1,7},{2,8},{3,4,5,6}}
=> ? => ? => ? = 3
[7,6,8,3,4,5,1,2] => ?
=> ? => ? => ? = 2
[6,7,8,3,4,5,1,2] => ?
=> ? => ? => ? = 1
[8,5,6,3,4,7,1,2] => ?
=> ? => ? => ? = 1
[7,5,6,3,4,8,1,2] => ?
=> ? => ? => ? = 2
[6,5,7,3,4,8,1,2] => ?
=> ? => ? => ? = 1
[8,7,4,5,6,2,1,3] => ?
=> ? => ? => ? = 1
[7,8,4,5,6,2,1,3] => ?
=> ? => ? => ? = 2
[6,7,4,5,8,2,1,3] => ?
=> ? => ? => ? = 2
[7,5,4,6,8,2,1,3] => ?
=> ? => ? => ? = 2
[6,5,4,7,8,2,1,3] => ?
=> ? => ? => ? = 1
[6,7,5,8,4,1,2,3] => {{1,6},{2,7},{3,4,5,8}}
=> ? => ? => ? = 3
[7,6,4,5,8,1,2,3] => ?
=> ? => ? => ? = 2
[6,7,4,5,8,1,2,3] => {{1,6},{2,7},{3,4,5,8}}
=> ? => ? => ? = 3
[6,7,8,5,3,1,2,4] => ?
=> ? => ? => ? = 3
[8,5,6,7,3,1,2,4] => ?
=> ? => ? => ? = 1
[6,5,7,8,3,1,2,4] => {{1,6},{2,3,5,7},{4,8}}
=> ? => ? => ? = 3
[8,6,7,5,2,1,3,4] => ?
=> ? => ? => ? = 2
[7,8,6,2,3,4,1,5] => ?
=> ? => ? => ? = 2
Description
The number of descents of a binary word.
Matching statistic: St000251
(load all 67 compositions to match this statistic)
(load all 67 compositions to match this statistic)
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000251: Set partitions ⟶ ℤResult quality: 57% ●values known / values provided: 78%●distinct values known / distinct values provided: 57%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000251: Set partitions ⟶ ℤResult quality: 57% ●values known / values provided: 78%●distinct values known / distinct values provided: 57%
Values
[1] => [1] => [1,0]
=> {{1}}
=> ? = 0
[1,2] => [1,2] => [1,0,1,0]
=> {{1},{2}}
=> 0
[2,1] => [2,1] => [1,1,0,0]
=> {{1,2}}
=> 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 0
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 1
[2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 1
[3,1,2] => [3,2,1] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 1
[3,2,1] => [2,3,1] => [1,1,0,1,0,0]
=> {{1,3},{2}}
=> 1
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 0
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1
[1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
[1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
[1,4,3,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 1
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 1
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2
[2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 1
[2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
[2,4,1,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
[2,4,3,1] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 1
[3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 1
[3,1,4,2] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
[3,2,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 1
[3,2,4,1] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 1
[3,4,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 2
[3,4,2,1] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
[4,1,2,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
[4,1,3,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 1
[4,2,1,3] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 1
[4,2,3,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> 1
[4,3,1,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
[4,3,2,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 2
[1,2,3,4,5] => [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] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 1
[1,2,4,5,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 1
[1,2,5,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 1
[1,2,5,4,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 2
[1,3,4,2,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 1
[1,3,4,5,2] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1
[1,3,5,2,4] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1
[1,3,5,4,2] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> 1
[1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 1
[1,4,2,5,3] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1
[1,4,3,2,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> 1
[1,4,3,5,2] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> 1
[1,4,5,2,3] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> 2
[1,4,5,3,2] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1
[8,7,6,5,4,3,2,1] => [5,4,6,3,7,2,8,1] => [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> {{1,8},{2,7},{3,6},{4,5}}
=> ? = 4
[8,7,5,6,4,3,2,1] => [6,3,5,4,7,2,8,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> {{1,8},{2,7},{3,4,5,6}}
=> ? = 3
[8,5,6,7,4,3,2,1] => [6,3,7,2,5,4,8,1] => [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> {{1,8},{2,3,4,7},{5,6}}
=> ? = 3
[5,6,7,8,4,3,2,1] => [7,2,6,3,8,1,5,4] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> {{1,2,3,8},{4,5,6,7}}
=> ? = 2
[8,7,4,5,6,3,2,1] => [6,3,4,5,7,2,8,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> {{1,8},{2,7},{3,4,5,6}}
=> ? = 3
[8,7,6,5,3,4,2,1] => [6,4,5,3,7,2,8,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> {{1,8},{2,7},{3,4,5,6}}
=> ? = 3
[8,7,5,6,3,4,2,1] => [5,3,6,4,7,2,8,1] => [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> {{1,8},{2,7},{3,6},{4,5}}
=> ? = 4
[8,7,6,3,4,5,2,1] => [6,5,4,3,7,2,8,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> {{1,8},{2,7},{3,4,5,6}}
=> ? = 3
[4,3,5,6,7,8,2,1] => [7,2,3,5,8,1,4,6] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> {{1,2,3,8},{4,5,6,7}}
=> ? = 2
[8,6,7,5,4,2,3,1] => [5,4,6,2,7,3,8,1] => [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> {{1,8},{2,7},{3,6},{4,5}}
=> ? = 4
[5,6,7,8,4,2,3,1] => [6,2,7,3,8,1,5,4] => [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> {{1,2,3,8},{4,7},{5,6}}
=> ? = 3
[8,6,4,5,7,2,3,1] => [6,2,7,3,4,5,8,1] => [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> {{1,8},{2,3,4,7},{5,6}}
=> ? = 3
[8,6,7,5,3,2,4,1] => [6,2,7,4,5,3,8,1] => [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> {{1,8},{2,3,4,7},{5,6}}
=> ? = 3
[8,6,5,7,3,2,4,1] => [5,3,6,2,7,4,8,1] => [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> {{1,8},{2,7},{3,6},{4,5}}
=> ? = 4
[8,5,6,7,3,2,4,1] => [6,2,5,3,7,4,8,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> {{1,8},{2,7},{3,4,5,6}}
=> ? = 3
[8,7,6,5,2,3,4,1] => [6,3,7,4,5,2,8,1] => [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> {{1,8},{2,3,4,7},{5,6}}
=> ? = 3
[6,7,8,5,2,3,4,1] => [7,4,5,2,8,1,6,3] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> {{1,2,3,8},{4,5,6,7}}
=> ? = 2
[8,6,5,7,2,3,4,1] => [6,3,5,2,7,4,8,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> {{1,8},{2,7},{3,4,5,6}}
=> ? = 3
[8,5,6,7,2,3,4,1] => [5,2,6,3,7,4,8,1] => [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> {{1,8},{2,7},{3,6},{4,5}}
=> ? = 4
[8,5,3,4,2,6,7,1] => [3,4,5,2,6,7,8,1] => [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> {{1,8},{2,5},{3},{4},{6},{7}}
=> ? = 2
[8,4,3,2,5,6,7,1] => [3,4,2,5,6,7,8,1] => [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> {{1,8},{2,4},{3},{5},{6},{7}}
=> ? = 2
[8,3,4,2,5,6,7,1] => [4,2,3,5,6,7,8,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> {{1,8},{2,3,4},{5},{6},{7}}
=> ? = 2
[8,4,2,3,5,6,7,1] => [4,3,2,5,6,7,8,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> {{1,8},{2,3,4},{5},{6},{7}}
=> ? = 2
[8,3,2,4,5,6,7,1] => [3,2,4,5,6,7,8,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,8},{2,3},{4},{5},{6},{7}}
=> ? = 2
[8,2,3,4,5,6,7,1] => [2,3,4,5,6,7,8,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,8},{2},{3},{4},{5},{6},{7}}
=> ? = 1
[5,6,4,3,7,2,8,1] => [4,3,6,2,8,1,5,7] => [1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> {{1,2,7,8},{3,4},{5,6}}
=> ? = 3
[7,8,6,5,4,3,1,2] => [5,4,6,3,7,1,8,2] => [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> {{1,8},{2,7},{3,6},{4,5}}
=> ? = 4
[7,8,5,6,4,3,1,2] => [6,3,5,4,7,1,8,2] => [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> {{1,8},{2,7},{3,4,5,6}}
=> ? = 3
[7,5,6,8,4,3,1,2] => [6,3,7,1,8,2,5,4] => [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> {{1,2,3,8},{4,7},{5,6}}
=> ? = 3
[7,8,4,5,6,3,1,2] => [6,3,4,5,7,1,8,2] => [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> {{1,8},{2,7},{3,4,5,6}}
=> ? = 3
[7,8,6,5,3,4,1,2] => [6,4,5,3,7,1,8,2] => [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> {{1,8},{2,7},{3,4,5,6}}
=> ? = 3
[7,8,5,6,3,4,1,2] => [5,3,6,4,7,1,8,2] => [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> {{1,8},{2,7},{3,6},{4,5}}
=> ? = 4
[5,6,7,8,3,4,1,2] => [7,1,5,3,8,2,6,4] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> {{1,2,3,8},{4,5,6,7}}
=> ? = 2
[7,8,6,3,4,5,1,2] => [6,5,4,3,7,1,8,2] => [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> {{1,8},{2,7},{3,4,5,6}}
=> ? = 3
[3,4,5,6,7,8,1,2] => [7,1,3,5,8,2,4,6] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> {{1,2,3,8},{4,5,6,7}}
=> ? = 2
[7,6,8,5,4,2,1,3] => [5,4,6,2,7,1,8,3] => [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> {{1,8},{2,7},{3,6},{4,5}}
=> ? = 4
[7,6,5,8,4,2,1,3] => [6,2,7,1,8,3,5,4] => [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> {{1,2,3,8},{4,7},{5,6}}
=> ? = 3
[6,7,5,8,4,2,1,3] => [7,1,6,2,8,3,5,4] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> {{1,2,3,8},{4,5,6,7}}
=> ? = 2
[7,6,4,5,8,2,1,3] => [6,2,7,1,8,3,4,5] => [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> {{1,2,3,8},{4,7},{5,6}}
=> ? = 3
[6,7,4,5,8,2,1,3] => [7,1,6,2,8,3,4,5] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> {{1,2,3,8},{4,5,6,7}}
=> ? = 2
[6,7,8,5,4,1,2,3] => [5,4,6,1,7,2,8,3] => [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> {{1,8},{2,7},{3,6},{4,5}}
=> ? = 4
[8,5,6,7,4,1,2,3] => [7,2,5,4,8,3,6,1] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> {{1,2,3,8},{4,5,6,7}}
=> ? = 2
[6,7,5,8,4,1,2,3] => [6,1,7,2,8,3,5,4] => [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> {{1,2,3,8},{4,7},{5,6}}
=> ? = 3
[7,6,4,5,8,1,2,3] => [7,2,6,1,8,3,4,5] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> {{1,2,3,8},{4,5,6,7}}
=> ? = 2
[6,7,4,5,8,1,2,3] => [6,1,7,2,8,3,4,5] => [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> {{1,2,3,8},{4,7},{5,6}}
=> ? = 3
[7,6,8,5,3,2,1,4] => [6,2,7,1,8,4,5,3] => [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> {{1,2,3,8},{4,7},{5,6}}
=> ? = 3
[6,7,8,5,3,2,1,4] => [7,1,6,2,8,4,5,3] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> {{1,2,3,8},{4,5,6,7}}
=> ? = 2
[7,6,5,8,3,2,1,4] => [5,3,6,2,7,1,8,4] => [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> {{1,8},{2,7},{3,6},{4,5}}
=> ? = 4
[7,5,6,8,3,2,1,4] => [6,2,5,3,7,1,8,4] => [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> {{1,8},{2,7},{3,4,5,6}}
=> ? = 3
Description
The number of nonsingleton blocks of a set partition.
Matching statistic: St001280
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 100%
St001280: Integer partitions ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 0
[1,2] => [1,1]
=> 0
[2,1] => [2]
=> 1
[1,2,3] => [1,1,1]
=> 0
[1,3,2] => [2,1]
=> 1
[2,1,3] => [2,1]
=> 1
[2,3,1] => [3]
=> 1
[3,1,2] => [3]
=> 1
[3,2,1] => [2,1]
=> 1
[1,2,3,4] => [1,1,1,1]
=> 0
[1,2,4,3] => [2,1,1]
=> 1
[1,3,2,4] => [2,1,1]
=> 1
[1,3,4,2] => [3,1]
=> 1
[1,4,2,3] => [3,1]
=> 1
[1,4,3,2] => [2,1,1]
=> 1
[2,1,3,4] => [2,1,1]
=> 1
[2,1,4,3] => [2,2]
=> 2
[2,3,1,4] => [3,1]
=> 1
[2,3,4,1] => [4]
=> 1
[2,4,1,3] => [4]
=> 1
[2,4,3,1] => [3,1]
=> 1
[3,1,2,4] => [3,1]
=> 1
[3,1,4,2] => [4]
=> 1
[3,2,1,4] => [2,1,1]
=> 1
[3,2,4,1] => [3,1]
=> 1
[3,4,1,2] => [2,2]
=> 2
[3,4,2,1] => [4]
=> 1
[4,1,2,3] => [4]
=> 1
[4,1,3,2] => [3,1]
=> 1
[4,2,1,3] => [3,1]
=> 1
[4,2,3,1] => [2,1,1]
=> 1
[4,3,1,2] => [4]
=> 1
[4,3,2,1] => [2,2]
=> 2
[1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,2,3,5,4] => [2,1,1,1]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> 1
[1,2,4,5,3] => [3,1,1]
=> 1
[1,2,5,3,4] => [3,1,1]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> 1
[1,3,2,5,4] => [2,2,1]
=> 2
[1,3,4,2,5] => [3,1,1]
=> 1
[1,3,4,5,2] => [4,1]
=> 1
[1,3,5,2,4] => [4,1]
=> 1
[1,3,5,4,2] => [3,1,1]
=> 1
[1,4,2,3,5] => [3,1,1]
=> 1
[1,4,2,5,3] => [4,1]
=> 1
[1,4,3,2,5] => [2,1,1,1]
=> 1
[1,4,3,5,2] => [3,1,1]
=> 1
[1,4,5,2,3] => [2,2,1]
=> 2
[6,7,8,5,3,4,2,1] => ?
=> ? = 2
[7,5,6,8,3,4,2,1] => ?
=> ? = 1
[5,6,4,7,3,8,2,1] => ?
=> ? = 1
[5,6,7,3,4,8,2,1] => ?
=> ? = 1
[7,4,5,3,6,8,2,1] => ?
=> ? = 1
[7,8,5,6,4,2,3,1] => ?
=> ? = 1
[8,5,6,7,4,2,3,1] => ?
=> ? = 2
[6,7,5,8,4,2,3,1] => ?
=> ? = 1
[6,7,4,5,8,2,3,1] => ?
=> ? = 1
[7,5,4,6,8,2,3,1] => ?
=> ? = 1
[6,4,5,7,8,2,3,1] => ?
=> ? = 1
[6,5,7,8,3,2,4,1] => ?
=> ? = 1
[5,6,4,3,7,2,8,1] => ?
=> ? = 3
[6,4,5,7,2,3,8,1] => ?
=> ? = 1
[5,3,4,6,2,7,8,1] => ?
=> ? = 1
[5,6,4,2,3,7,8,1] => ?
=> ? = 1
[5,6,2,3,4,7,8,1] => ?
=> ? = 1
[8,5,6,7,4,3,1,2] => ?
=> ? = 2
[7,5,6,8,4,3,1,2] => ?
=> ? = 3
[7,8,4,5,6,3,1,2] => ?
=> ? = 3
[6,7,4,5,8,3,1,2] => ?
=> ? = 1
[7,4,5,6,8,3,1,2] => ?
=> ? = 2
[8,6,7,5,3,4,1,2] => ?
=> ? = 1
[7,6,8,3,4,5,1,2] => ?
=> ? = 2
[6,7,8,3,4,5,1,2] => ?
=> ? = 1
[8,5,6,3,4,7,1,2] => ?
=> ? = 1
[7,5,6,3,4,8,1,2] => ?
=> ? = 2
[6,5,7,3,4,8,1,2] => ?
=> ? = 1
[7,5,6,8,4,2,1,3] => ?
=> ? = 2
[8,7,4,5,6,2,1,3] => ?
=> ? = 1
[7,8,4,5,6,2,1,3] => ?
=> ? = 2
[7,6,4,5,8,2,1,3] => ?
=> ? = 3
[6,7,4,5,8,2,1,3] => ?
=> ? = 2
[7,5,4,6,8,2,1,3] => ?
=> ? = 2
[6,5,4,7,8,2,1,3] => ?
=> ? = 1
[7,6,4,5,8,1,2,3] => ?
=> ? = 2
[8,7,5,6,2,3,1,4] => ?
=> ? = 1
[8,6,7,5,3,1,2,4] => ?
=> ? = 1
[6,7,8,5,3,1,2,4] => ?
=> ? = 3
[8,5,6,7,3,1,2,4] => ?
=> ? = 1
[8,6,7,5,2,1,3,4] => ?
=> ? = 2
[6,7,8,3,4,2,1,5] => ?
=> ? = 2
[7,8,6,2,3,4,1,5] => ?
=> ? = 2
[7,8,6,2,3,1,4,5] => ?
=> ? = 1
[7,8,6,2,1,3,4,5] => ?
=> ? = 2
[7,6,8,2,1,3,4,5] => ?
=> ? = 1
[8,7,4,5,2,3,1,6] => ?
=> ? = 1
[7,8,5,3,2,4,1,6] => ?
=> ? = 2
[8,7,4,2,3,5,1,6] => ?
=> ? = 1
[7,8,4,5,2,1,3,6] => ?
=> ? = 1
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St000147
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [1]
=> []
=> 0
[1,2] => [1,1]
=> [2]
=> []
=> 0
[2,1] => [2]
=> [1,1]
=> [1]
=> 1
[1,2,3] => [1,1,1]
=> [3]
=> []
=> 0
[1,3,2] => [2,1]
=> [2,1]
=> [1]
=> 1
[2,1,3] => [2,1]
=> [2,1]
=> [1]
=> 1
[2,3,1] => [3]
=> [1,1,1]
=> [1,1]
=> 1
[3,1,2] => [3]
=> [1,1,1]
=> [1,1]
=> 1
[3,2,1] => [2,1]
=> [2,1]
=> [1]
=> 1
[1,2,3,4] => [1,1,1,1]
=> [4]
=> []
=> 0
[1,2,4,3] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[1,3,2,4] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[1,3,4,2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[1,4,2,3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[1,4,3,2] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[2,1,3,4] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[2,1,4,3] => [2,2]
=> [2,2]
=> [2]
=> 2
[2,3,1,4] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[2,3,4,1] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[2,4,1,3] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[2,4,3,1] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[3,1,2,4] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[3,1,4,2] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[3,2,1,4] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[3,2,4,1] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[3,4,1,2] => [2,2]
=> [2,2]
=> [2]
=> 2
[3,4,2,1] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[4,1,2,3] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[4,1,3,2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[4,2,1,3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[4,2,3,1] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[4,3,1,2] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[4,3,2,1] => [2,2]
=> [2,2]
=> [2]
=> 2
[1,2,3,4,5] => [1,1,1,1,1]
=> [5]
=> []
=> 0
[1,2,3,5,4] => [2,1,1,1]
=> [4,1]
=> [1]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> [4,1]
=> [1]
=> 1
[1,2,4,5,3] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,2,5,3,4] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [4,1]
=> [1]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> [4,1]
=> [1]
=> 1
[1,3,2,5,4] => [2,2,1]
=> [3,2]
=> [2]
=> 2
[1,3,4,2,5] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,3,4,5,2] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[1,3,5,2,4] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[1,3,5,4,2] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,4,2,3,5] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,4,2,5,3] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[1,4,3,2,5] => [2,1,1,1]
=> [4,1]
=> [1]
=> 1
[1,4,3,5,2] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [3,2]
=> [2]
=> 2
[6,7,8,5,3,4,2,1] => ?
=> ?
=> ?
=> ? = 2
[7,5,6,8,3,4,2,1] => ?
=> ?
=> ?
=> ? = 1
[5,6,4,7,3,8,2,1] => ?
=> ?
=> ?
=> ? = 1
[5,6,7,3,4,8,2,1] => ?
=> ?
=> ?
=> ? = 1
[7,4,5,3,6,8,2,1] => ?
=> ?
=> ?
=> ? = 1
[7,8,5,6,4,2,3,1] => ?
=> ?
=> ?
=> ? = 1
[8,5,6,7,4,2,3,1] => ?
=> ?
=> ?
=> ? = 2
[6,7,5,8,4,2,3,1] => ?
=> ?
=> ?
=> ? = 1
[6,7,4,5,8,2,3,1] => ?
=> ?
=> ?
=> ? = 1
[7,5,4,6,8,2,3,1] => ?
=> ?
=> ?
=> ? = 1
[6,4,5,7,8,2,3,1] => ?
=> ?
=> ?
=> ? = 1
[6,5,7,8,3,2,4,1] => ?
=> ?
=> ?
=> ? = 1
[5,6,4,3,7,2,8,1] => ?
=> ?
=> ?
=> ? = 3
[6,4,5,7,2,3,8,1] => ?
=> ?
=> ?
=> ? = 1
[5,3,4,6,2,7,8,1] => ?
=> ?
=> ?
=> ? = 1
[5,6,4,2,3,7,8,1] => ?
=> ?
=> ?
=> ? = 1
[5,6,2,3,4,7,8,1] => ?
=> ?
=> ?
=> ? = 1
[8,5,6,7,4,3,1,2] => ?
=> ?
=> ?
=> ? = 2
[7,5,6,8,4,3,1,2] => ?
=> ?
=> ?
=> ? = 3
[7,8,4,5,6,3,1,2] => ?
=> ?
=> ?
=> ? = 3
[6,7,4,5,8,3,1,2] => ?
=> ?
=> ?
=> ? = 1
[7,4,5,6,8,3,1,2] => ?
=> ?
=> ?
=> ? = 2
[8,6,7,5,3,4,1,2] => ?
=> ?
=> ?
=> ? = 1
[7,6,8,3,4,5,1,2] => ?
=> ?
=> ?
=> ? = 2
[6,7,8,3,4,5,1,2] => ?
=> ?
=> ?
=> ? = 1
[8,5,6,3,4,7,1,2] => ?
=> ?
=> ?
=> ? = 1
[7,5,6,3,4,8,1,2] => ?
=> ?
=> ?
=> ? = 2
[6,5,7,3,4,8,1,2] => ?
=> ?
=> ?
=> ? = 1
[7,5,6,8,4,2,1,3] => ?
=> ?
=> ?
=> ? = 2
[8,7,4,5,6,2,1,3] => ?
=> ?
=> ?
=> ? = 1
[7,8,4,5,6,2,1,3] => ?
=> ?
=> ?
=> ? = 2
[7,6,4,5,8,2,1,3] => ?
=> ?
=> ?
=> ? = 3
[6,7,4,5,8,2,1,3] => ?
=> ?
=> ?
=> ? = 2
[7,5,4,6,8,2,1,3] => ?
=> ?
=> ?
=> ? = 2
[6,5,4,7,8,2,1,3] => ?
=> ?
=> ?
=> ? = 1
[7,6,4,5,8,1,2,3] => ?
=> ?
=> ?
=> ? = 2
[8,7,5,6,2,3,1,4] => ?
=> ?
=> ?
=> ? = 1
[8,6,7,5,3,1,2,4] => ?
=> ?
=> ?
=> ? = 1
[6,7,8,5,3,1,2,4] => ?
=> ?
=> ?
=> ? = 3
[8,5,6,7,3,1,2,4] => ?
=> ?
=> ?
=> ? = 1
[8,6,7,5,2,1,3,4] => ?
=> ?
=> ?
=> ? = 2
[6,7,8,3,4,2,1,5] => ?
=> ?
=> ?
=> ? = 2
[7,8,6,2,3,4,1,5] => ?
=> ?
=> ?
=> ? = 2
[7,8,6,2,3,1,4,5] => ?
=> ?
=> ?
=> ? = 1
[7,8,6,2,1,3,4,5] => ?
=> ?
=> ?
=> ? = 2
[7,6,8,2,1,3,4,5] => ?
=> ?
=> ?
=> ? = 1
[8,7,4,5,2,3,1,6] => ?
=> ?
=> ?
=> ? = 1
[7,8,5,3,2,4,1,6] => ?
=> ?
=> ?
=> ? = 2
[8,7,4,2,3,5,1,6] => ?
=> ?
=> ?
=> ? = 1
[7,8,4,5,2,1,3,6] => ?
=> ?
=> ?
=> ? = 1
Description
The largest part of an integer partition.
Matching statistic: St000319
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 86%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 86%
Values
[1] => [1]
=> [1]
=> []
=> ? = 0 - 1
[1,2] => [1,1]
=> [2]
=> []
=> ? = 0 - 1
[2,1] => [2]
=> [1,1]
=> [1]
=> 0 = 1 - 1
[1,2,3] => [1,1,1]
=> [3]
=> []
=> ? = 0 - 1
[1,3,2] => [2,1]
=> [2,1]
=> [1]
=> 0 = 1 - 1
[2,1,3] => [2,1]
=> [2,1]
=> [1]
=> 0 = 1 - 1
[2,3,1] => [3]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[3,1,2] => [3]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[3,2,1] => [2,1]
=> [2,1]
=> [1]
=> 0 = 1 - 1
[1,2,3,4] => [1,1,1,1]
=> [4]
=> []
=> ? = 0 - 1
[1,2,4,3] => [2,1,1]
=> [3,1]
=> [1]
=> 0 = 1 - 1
[1,3,2,4] => [2,1,1]
=> [3,1]
=> [1]
=> 0 = 1 - 1
[1,3,4,2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,4,2,3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,4,3,2] => [2,1,1]
=> [3,1]
=> [1]
=> 0 = 1 - 1
[2,1,3,4] => [2,1,1]
=> [3,1]
=> [1]
=> 0 = 1 - 1
[2,1,4,3] => [2,2]
=> [2,2]
=> [2]
=> 1 = 2 - 1
[2,3,1,4] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[2,3,4,1] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[2,4,1,3] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[2,4,3,1] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[3,1,2,4] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[3,1,4,2] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[3,2,1,4] => [2,1,1]
=> [3,1]
=> [1]
=> 0 = 1 - 1
[3,2,4,1] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[3,4,1,2] => [2,2]
=> [2,2]
=> [2]
=> 1 = 2 - 1
[3,4,2,1] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[4,1,2,3] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[4,1,3,2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[4,2,1,3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[4,2,3,1] => [2,1,1]
=> [3,1]
=> [1]
=> 0 = 1 - 1
[4,3,1,2] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[4,3,2,1] => [2,2]
=> [2,2]
=> [2]
=> 1 = 2 - 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [5]
=> []
=> ? = 0 - 1
[1,2,3,5,4] => [2,1,1,1]
=> [4,1]
=> [1]
=> 0 = 1 - 1
[1,2,4,3,5] => [2,1,1,1]
=> [4,1]
=> [1]
=> 0 = 1 - 1
[1,2,4,5,3] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,2,5,3,4] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,2,5,4,3] => [2,1,1,1]
=> [4,1]
=> [1]
=> 0 = 1 - 1
[1,3,2,4,5] => [2,1,1,1]
=> [4,1]
=> [1]
=> 0 = 1 - 1
[1,3,2,5,4] => [2,2,1]
=> [3,2]
=> [2]
=> 1 = 2 - 1
[1,3,4,2,5] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,3,4,5,2] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[1,3,5,2,4] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[1,3,5,4,2] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,4,2,3,5] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,4,2,5,3] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[1,4,3,2,5] => [2,1,1,1]
=> [4,1]
=> [1]
=> 0 = 1 - 1
[1,4,3,5,2] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,4,5,2,3] => [2,2,1]
=> [3,2]
=> [2]
=> 1 = 2 - 1
[1,4,5,3,2] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[1,5,2,3,4] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[1,5,2,4,3] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,5,3,2,4] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,5,3,4,2] => [2,1,1,1]
=> [4,1]
=> [1]
=> 0 = 1 - 1
[1,2,3,4,5,6] => [1,1,1,1,1,1]
=> [6]
=> []
=> ? = 0 - 1
[1,2,3,4,5,6,7] => [1,1,1,1,1,1,1]
=> [7]
=> []
=> ? = 0 - 1
[6,7,8,5,3,4,2,1] => ?
=> ?
=> ?
=> ? = 2 - 1
[7,5,6,8,3,4,2,1] => ?
=> ?
=> ?
=> ? = 1 - 1
[5,6,4,7,3,8,2,1] => ?
=> ?
=> ?
=> ? = 1 - 1
[5,6,7,3,4,8,2,1] => ?
=> ?
=> ?
=> ? = 1 - 1
[7,4,5,3,6,8,2,1] => ?
=> ?
=> ?
=> ? = 1 - 1
[7,8,5,6,4,2,3,1] => ?
=> ?
=> ?
=> ? = 1 - 1
[8,5,6,7,4,2,3,1] => ?
=> ?
=> ?
=> ? = 2 - 1
[6,7,5,8,4,2,3,1] => ?
=> ?
=> ?
=> ? = 1 - 1
[6,7,4,5,8,2,3,1] => ?
=> ?
=> ?
=> ? = 1 - 1
[7,5,4,6,8,2,3,1] => ?
=> ?
=> ?
=> ? = 1 - 1
[6,4,5,7,8,2,3,1] => ?
=> ?
=> ?
=> ? = 1 - 1
[6,5,7,8,3,2,4,1] => ?
=> ?
=> ?
=> ? = 1 - 1
[5,6,4,3,7,2,8,1] => ?
=> ?
=> ?
=> ? = 3 - 1
[6,4,5,7,2,3,8,1] => ?
=> ?
=> ?
=> ? = 1 - 1
[5,3,4,6,2,7,8,1] => ?
=> ?
=> ?
=> ? = 1 - 1
[5,6,4,2,3,7,8,1] => ?
=> ?
=> ?
=> ? = 1 - 1
[5,6,2,3,4,7,8,1] => ?
=> ?
=> ?
=> ? = 1 - 1
[8,5,6,7,4,3,1,2] => ?
=> ?
=> ?
=> ? = 2 - 1
[7,5,6,8,4,3,1,2] => ?
=> ?
=> ?
=> ? = 3 - 1
[7,8,4,5,6,3,1,2] => ?
=> ?
=> ?
=> ? = 3 - 1
[6,7,4,5,8,3,1,2] => ?
=> ?
=> ?
=> ? = 1 - 1
[7,4,5,6,8,3,1,2] => ?
=> ?
=> ?
=> ? = 2 - 1
[8,6,7,5,3,4,1,2] => ?
=> ?
=> ?
=> ? = 1 - 1
[7,6,8,3,4,5,1,2] => ?
=> ?
=> ?
=> ? = 2 - 1
[6,7,8,3,4,5,1,2] => ?
=> ?
=> ?
=> ? = 1 - 1
[8,5,6,3,4,7,1,2] => ?
=> ?
=> ?
=> ? = 1 - 1
[7,5,6,3,4,8,1,2] => ?
=> ?
=> ?
=> ? = 2 - 1
[6,5,7,3,4,8,1,2] => ?
=> ?
=> ?
=> ? = 1 - 1
[7,5,6,8,4,2,1,3] => ?
=> ?
=> ?
=> ? = 2 - 1
[8,7,4,5,6,2,1,3] => ?
=> ?
=> ?
=> ? = 1 - 1
[7,8,4,5,6,2,1,3] => ?
=> ?
=> ?
=> ? = 2 - 1
[7,6,4,5,8,2,1,3] => ?
=> ?
=> ?
=> ? = 3 - 1
[6,7,4,5,8,2,1,3] => ?
=> ?
=> ?
=> ? = 2 - 1
[7,5,4,6,8,2,1,3] => ?
=> ?
=> ?
=> ? = 2 - 1
[6,5,4,7,8,2,1,3] => ?
=> ?
=> ?
=> ? = 1 - 1
[7,6,4,5,8,1,2,3] => ?
=> ?
=> ?
=> ? = 2 - 1
[8,7,5,6,2,3,1,4] => ?
=> ?
=> ?
=> ? = 1 - 1
[8,6,7,5,3,1,2,4] => ?
=> ?
=> ?
=> ? = 1 - 1
[6,7,8,5,3,1,2,4] => ?
=> ?
=> ?
=> ? = 3 - 1
[8,5,6,7,3,1,2,4] => ?
=> ?
=> ?
=> ? = 1 - 1
[8,6,7,5,2,1,3,4] => ?
=> ?
=> ?
=> ? = 2 - 1
[6,7,8,3,4,2,1,5] => ?
=> ?
=> ?
=> ? = 2 - 1
[7,8,6,2,3,4,1,5] => ?
=> ?
=> ?
=> ? = 2 - 1
Description
The spin of an integer partition.
The Ferrers shape of an integer partition $\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of $\lambda$ with the vertical lines in the Ferrers shape.
The following example is taken from Appendix B in [1]: Let $\lambda = (5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$$
The first strip $(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses $4$ times, the second strip $(4,3,3,1) \setminus (2,2)$ crosses $3$ times, the strip $(2,2) \setminus (1)$ crosses $1$ time, and the remaining strip $(1) \setminus ()$ does not cross.
This yields the spin of $(5,5,4,4,2,1)$ to be $4+3+1 = 8$.
Matching statistic: St000320
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 86%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 86%
Values
[1] => [1]
=> [1]
=> []
=> ? = 0 - 1
[1,2] => [1,1]
=> [2]
=> []
=> ? = 0 - 1
[2,1] => [2]
=> [1,1]
=> [1]
=> 0 = 1 - 1
[1,2,3] => [1,1,1]
=> [3]
=> []
=> ? = 0 - 1
[1,3,2] => [2,1]
=> [2,1]
=> [1]
=> 0 = 1 - 1
[2,1,3] => [2,1]
=> [2,1]
=> [1]
=> 0 = 1 - 1
[2,3,1] => [3]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[3,1,2] => [3]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[3,2,1] => [2,1]
=> [2,1]
=> [1]
=> 0 = 1 - 1
[1,2,3,4] => [1,1,1,1]
=> [4]
=> []
=> ? = 0 - 1
[1,2,4,3] => [2,1,1]
=> [3,1]
=> [1]
=> 0 = 1 - 1
[1,3,2,4] => [2,1,1]
=> [3,1]
=> [1]
=> 0 = 1 - 1
[1,3,4,2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,4,2,3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,4,3,2] => [2,1,1]
=> [3,1]
=> [1]
=> 0 = 1 - 1
[2,1,3,4] => [2,1,1]
=> [3,1]
=> [1]
=> 0 = 1 - 1
[2,1,4,3] => [2,2]
=> [2,2]
=> [2]
=> 1 = 2 - 1
[2,3,1,4] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[2,3,4,1] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[2,4,1,3] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[2,4,3,1] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[3,1,2,4] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[3,1,4,2] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[3,2,1,4] => [2,1,1]
=> [3,1]
=> [1]
=> 0 = 1 - 1
[3,2,4,1] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[3,4,1,2] => [2,2]
=> [2,2]
=> [2]
=> 1 = 2 - 1
[3,4,2,1] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[4,1,2,3] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[4,1,3,2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[4,2,1,3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[4,2,3,1] => [2,1,1]
=> [3,1]
=> [1]
=> 0 = 1 - 1
[4,3,1,2] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[4,3,2,1] => [2,2]
=> [2,2]
=> [2]
=> 1 = 2 - 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [5]
=> []
=> ? = 0 - 1
[1,2,3,5,4] => [2,1,1,1]
=> [4,1]
=> [1]
=> 0 = 1 - 1
[1,2,4,3,5] => [2,1,1,1]
=> [4,1]
=> [1]
=> 0 = 1 - 1
[1,2,4,5,3] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,2,5,3,4] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,2,5,4,3] => [2,1,1,1]
=> [4,1]
=> [1]
=> 0 = 1 - 1
[1,3,2,4,5] => [2,1,1,1]
=> [4,1]
=> [1]
=> 0 = 1 - 1
[1,3,2,5,4] => [2,2,1]
=> [3,2]
=> [2]
=> 1 = 2 - 1
[1,3,4,2,5] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,3,4,5,2] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[1,3,5,2,4] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[1,3,5,4,2] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,4,2,3,5] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,4,2,5,3] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[1,4,3,2,5] => [2,1,1,1]
=> [4,1]
=> [1]
=> 0 = 1 - 1
[1,4,3,5,2] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,4,5,2,3] => [2,2,1]
=> [3,2]
=> [2]
=> 1 = 2 - 1
[1,4,5,3,2] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[1,5,2,3,4] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[1,5,2,4,3] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,5,3,2,4] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,5,3,4,2] => [2,1,1,1]
=> [4,1]
=> [1]
=> 0 = 1 - 1
[1,2,3,4,5,6] => [1,1,1,1,1,1]
=> [6]
=> []
=> ? = 0 - 1
[1,2,3,4,5,6,7] => [1,1,1,1,1,1,1]
=> [7]
=> []
=> ? = 0 - 1
[6,7,8,5,3,4,2,1] => ?
=> ?
=> ?
=> ? = 2 - 1
[7,5,6,8,3,4,2,1] => ?
=> ?
=> ?
=> ? = 1 - 1
[5,6,4,7,3,8,2,1] => ?
=> ?
=> ?
=> ? = 1 - 1
[5,6,7,3,4,8,2,1] => ?
=> ?
=> ?
=> ? = 1 - 1
[7,4,5,3,6,8,2,1] => ?
=> ?
=> ?
=> ? = 1 - 1
[7,8,5,6,4,2,3,1] => ?
=> ?
=> ?
=> ? = 1 - 1
[8,5,6,7,4,2,3,1] => ?
=> ?
=> ?
=> ? = 2 - 1
[6,7,5,8,4,2,3,1] => ?
=> ?
=> ?
=> ? = 1 - 1
[6,7,4,5,8,2,3,1] => ?
=> ?
=> ?
=> ? = 1 - 1
[7,5,4,6,8,2,3,1] => ?
=> ?
=> ?
=> ? = 1 - 1
[6,4,5,7,8,2,3,1] => ?
=> ?
=> ?
=> ? = 1 - 1
[6,5,7,8,3,2,4,1] => ?
=> ?
=> ?
=> ? = 1 - 1
[5,6,4,3,7,2,8,1] => ?
=> ?
=> ?
=> ? = 3 - 1
[6,4,5,7,2,3,8,1] => ?
=> ?
=> ?
=> ? = 1 - 1
[5,3,4,6,2,7,8,1] => ?
=> ?
=> ?
=> ? = 1 - 1
[5,6,4,2,3,7,8,1] => ?
=> ?
=> ?
=> ? = 1 - 1
[5,6,2,3,4,7,8,1] => ?
=> ?
=> ?
=> ? = 1 - 1
[8,5,6,7,4,3,1,2] => ?
=> ?
=> ?
=> ? = 2 - 1
[7,5,6,8,4,3,1,2] => ?
=> ?
=> ?
=> ? = 3 - 1
[7,8,4,5,6,3,1,2] => ?
=> ?
=> ?
=> ? = 3 - 1
[6,7,4,5,8,3,1,2] => ?
=> ?
=> ?
=> ? = 1 - 1
[7,4,5,6,8,3,1,2] => ?
=> ?
=> ?
=> ? = 2 - 1
[8,6,7,5,3,4,1,2] => ?
=> ?
=> ?
=> ? = 1 - 1
[7,6,8,3,4,5,1,2] => ?
=> ?
=> ?
=> ? = 2 - 1
[6,7,8,3,4,5,1,2] => ?
=> ?
=> ?
=> ? = 1 - 1
[8,5,6,3,4,7,1,2] => ?
=> ?
=> ?
=> ? = 1 - 1
[7,5,6,3,4,8,1,2] => ?
=> ?
=> ?
=> ? = 2 - 1
[6,5,7,3,4,8,1,2] => ?
=> ?
=> ?
=> ? = 1 - 1
[7,5,6,8,4,2,1,3] => ?
=> ?
=> ?
=> ? = 2 - 1
[8,7,4,5,6,2,1,3] => ?
=> ?
=> ?
=> ? = 1 - 1
[7,8,4,5,6,2,1,3] => ?
=> ?
=> ?
=> ? = 2 - 1
[7,6,4,5,8,2,1,3] => ?
=> ?
=> ?
=> ? = 3 - 1
[6,7,4,5,8,2,1,3] => ?
=> ?
=> ?
=> ? = 2 - 1
[7,5,4,6,8,2,1,3] => ?
=> ?
=> ?
=> ? = 2 - 1
[6,5,4,7,8,2,1,3] => ?
=> ?
=> ?
=> ? = 1 - 1
[7,6,4,5,8,1,2,3] => ?
=> ?
=> ?
=> ? = 2 - 1
[8,7,5,6,2,3,1,4] => ?
=> ?
=> ?
=> ? = 1 - 1
[8,6,7,5,3,1,2,4] => ?
=> ?
=> ?
=> ? = 1 - 1
[6,7,8,5,3,1,2,4] => ?
=> ?
=> ?
=> ? = 3 - 1
[8,5,6,7,3,1,2,4] => ?
=> ?
=> ?
=> ? = 1 - 1
[8,6,7,5,2,1,3,4] => ?
=> ?
=> ?
=> ? = 2 - 1
[6,7,8,3,4,2,1,5] => ?
=> ?
=> ?
=> ? = 2 - 1
[7,8,6,2,3,4,1,5] => ?
=> ?
=> ?
=> ? = 2 - 1
Description
The dinv adjustment of an integer partition.
The Ferrers shape of an integer partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ can be decomposed into border strips. For $0 \leq j < \lambda_1$ let $n_j$ be the length of the border strip starting at $(\lambda_1-j,0)$.
The dinv adjustment is then defined by
$$\sum_{j:n_j > 0}(\lambda_1-1-j).$$
The following example is taken from Appendix B in [2]: Let $\lambda=(5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(),$$
and we obtain $(n_0,\ldots,n_4) = (10,7,0,3,1)$.
The dinv adjustment is thus $4+3+1+0 = 8$.
Matching statistic: St000665
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000665: Permutations ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 86%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000665: Permutations ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 86%
Values
[1] => [1]
=> [[1]]
=> [1] => 0
[1,2] => [1,1]
=> [[1],[2]]
=> [2,1] => 0
[2,1] => [2]
=> [[1,2]]
=> [1,2] => 1
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[1,3,2] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[2,1,3] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[2,3,1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 1
[3,1,2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 1
[3,2,1] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
[1,2,4,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[1,3,2,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[1,3,4,2] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[1,4,2,3] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[1,4,3,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[2,1,3,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[2,3,1,4] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[2,3,4,1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1
[2,4,1,3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1
[2,4,3,1] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[3,1,2,4] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[3,1,4,2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1
[3,2,1,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[3,2,4,1] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[3,4,1,2] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[3,4,2,1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1
[4,1,2,3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1
[4,1,3,2] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[4,2,1,3] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[4,2,3,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[4,3,1,2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1
[4,3,2,1] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[1,2,3,4,5] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 0
[1,2,3,5,4] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[1,2,4,3,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[1,2,4,5,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,2,5,3,4] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,2,5,4,3] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[1,3,2,4,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[1,3,2,5,4] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2
[1,3,4,2,5] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,3,4,5,2] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1
[1,3,5,2,4] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1
[1,3,5,4,2] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,4,2,3,5] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,4,2,5,3] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1
[1,4,3,2,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[1,4,3,5,2] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,4,5,2,3] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2
[6,7,8,5,3,4,2,1] => ?
=> ?
=> ? => ? = 2
[7,5,6,8,3,4,2,1] => ?
=> ?
=> ? => ? = 1
[5,6,4,7,3,8,2,1] => ?
=> ?
=> ? => ? = 1
[5,6,7,3,4,8,2,1] => ?
=> ?
=> ? => ? = 1
[7,4,5,3,6,8,2,1] => ?
=> ?
=> ? => ? = 1
[7,8,5,6,4,2,3,1] => ?
=> ?
=> ? => ? = 1
[8,5,6,7,4,2,3,1] => ?
=> ?
=> ? => ? = 2
[6,7,5,8,4,2,3,1] => ?
=> ?
=> ? => ? = 1
[6,7,4,5,8,2,3,1] => ?
=> ?
=> ? => ? = 1
[7,5,4,6,8,2,3,1] => ?
=> ?
=> ? => ? = 1
[6,4,5,7,8,2,3,1] => ?
=> ?
=> ? => ? = 1
[6,5,7,8,3,2,4,1] => ?
=> ?
=> ? => ? = 1
[5,6,4,3,7,2,8,1] => ?
=> ?
=> ? => ? = 3
[6,4,5,7,2,3,8,1] => ?
=> ?
=> ? => ? = 1
[5,3,4,6,2,7,8,1] => ?
=> ?
=> ? => ? = 1
[5,6,4,2,3,7,8,1] => ?
=> ?
=> ? => ? = 1
[5,6,2,3,4,7,8,1] => ?
=> ?
=> ? => ? = 1
[8,5,6,7,4,3,1,2] => ?
=> ?
=> ? => ? = 2
[7,5,6,8,4,3,1,2] => ?
=> ?
=> ? => ? = 3
[7,8,4,5,6,3,1,2] => ?
=> ?
=> ? => ? = 3
[6,7,4,5,8,3,1,2] => ?
=> ?
=> ? => ? = 1
[7,4,5,6,8,3,1,2] => ?
=> ?
=> ? => ? = 2
[8,6,7,5,3,4,1,2] => ?
=> ?
=> ? => ? = 1
[7,6,8,3,4,5,1,2] => ?
=> ?
=> ? => ? = 2
[6,7,8,3,4,5,1,2] => ?
=> ?
=> ? => ? = 1
[8,5,6,3,4,7,1,2] => ?
=> ?
=> ? => ? = 1
[7,5,6,3,4,8,1,2] => ?
=> ?
=> ? => ? = 2
[6,5,7,3,4,8,1,2] => ?
=> ?
=> ? => ? = 1
[7,5,6,8,4,2,1,3] => ?
=> ?
=> ? => ? = 2
[8,7,4,5,6,2,1,3] => ?
=> ?
=> ? => ? = 1
[7,8,4,5,6,2,1,3] => ?
=> ?
=> ? => ? = 2
[7,6,4,5,8,2,1,3] => ?
=> ?
=> ? => ? = 3
[6,7,4,5,8,2,1,3] => ?
=> ?
=> ? => ? = 2
[7,5,4,6,8,2,1,3] => ?
=> ?
=> ? => ? = 2
[6,5,4,7,8,2,1,3] => ?
=> ?
=> ? => ? = 1
[7,6,4,5,8,1,2,3] => ?
=> ?
=> ? => ? = 2
[8,7,5,6,2,3,1,4] => ?
=> ?
=> ? => ? = 1
[8,6,7,5,3,1,2,4] => ?
=> ?
=> ? => ? = 1
[6,7,8,5,3,1,2,4] => ?
=> ?
=> ? => ? = 3
[8,5,6,7,3,1,2,4] => ?
=> ?
=> ? => ? = 1
[8,6,7,5,2,1,3,4] => ?
=> ?
=> ? => ? = 2
[6,7,8,3,4,2,1,5] => ?
=> ?
=> ? => ? = 2
[7,8,6,2,3,4,1,5] => ?
=> ?
=> ? => ? = 2
[7,8,6,2,3,1,4,5] => ?
=> ?
=> ? => ? = 1
[7,8,6,2,1,3,4,5] => ?
=> ?
=> ? => ? = 2
[7,6,8,2,1,3,4,5] => ?
=> ?
=> ? => ? = 1
[8,7,4,5,2,3,1,6] => ?
=> ?
=> ? => ? = 1
[7,8,5,3,2,4,1,6] => ?
=> ?
=> ? => ? = 2
[8,7,4,2,3,5,1,6] => ?
=> ?
=> ? => ? = 1
[7,8,4,5,2,1,3,6] => ?
=> ?
=> ? => ? = 1
Description
The number of rafts of a permutation.
Let $\pi$ be a permutation of length $n$. A small ascent of $\pi$ is an index $i$ such that $\pi(i+1)= \pi(i)+1$, see [[St000441]], and a raft of $\pi$ is a non-empty maximal sequence of consecutive small ascents.
Matching statistic: St000834
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 86%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 86%
Values
[1] => [1]
=> [[1]]
=> [1] => 0
[1,2] => [1,1]
=> [[1],[2]]
=> [2,1] => 0
[2,1] => [2]
=> [[1,2]]
=> [1,2] => 1
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[1,3,2] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[2,1,3] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[2,3,1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 1
[3,1,2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 1
[3,2,1] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
[1,2,4,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[1,3,2,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[1,3,4,2] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[1,4,2,3] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[1,4,3,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[2,1,3,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[2,3,1,4] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[2,3,4,1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1
[2,4,1,3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1
[2,4,3,1] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[3,1,2,4] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[3,1,4,2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1
[3,2,1,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[3,2,4,1] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[3,4,1,2] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[3,4,2,1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1
[4,1,2,3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1
[4,1,3,2] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[4,2,1,3] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[4,2,3,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[4,3,1,2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1
[4,3,2,1] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[1,2,3,4,5] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 0
[1,2,3,5,4] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[1,2,4,3,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[1,2,4,5,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,2,5,3,4] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,2,5,4,3] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[1,3,2,4,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[1,3,2,5,4] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2
[1,3,4,2,5] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,3,4,5,2] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1
[1,3,5,2,4] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1
[1,3,5,4,2] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,4,2,3,5] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,4,2,5,3] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1
[1,4,3,2,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[1,4,3,5,2] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,4,5,2,3] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2
[6,7,8,5,3,4,2,1] => ?
=> ?
=> ? => ? = 2
[7,5,6,8,3,4,2,1] => ?
=> ?
=> ? => ? = 1
[5,6,4,7,3,8,2,1] => ?
=> ?
=> ? => ? = 1
[5,6,7,3,4,8,2,1] => ?
=> ?
=> ? => ? = 1
[7,4,5,3,6,8,2,1] => ?
=> ?
=> ? => ? = 1
[7,8,5,6,4,2,3,1] => ?
=> ?
=> ? => ? = 1
[8,5,6,7,4,2,3,1] => ?
=> ?
=> ? => ? = 2
[6,7,5,8,4,2,3,1] => ?
=> ?
=> ? => ? = 1
[6,7,4,5,8,2,3,1] => ?
=> ?
=> ? => ? = 1
[7,5,4,6,8,2,3,1] => ?
=> ?
=> ? => ? = 1
[6,4,5,7,8,2,3,1] => ?
=> ?
=> ? => ? = 1
[6,5,7,8,3,2,4,1] => ?
=> ?
=> ? => ? = 1
[5,6,4,3,7,2,8,1] => ?
=> ?
=> ? => ? = 3
[6,4,5,7,2,3,8,1] => ?
=> ?
=> ? => ? = 1
[5,3,4,6,2,7,8,1] => ?
=> ?
=> ? => ? = 1
[5,6,4,2,3,7,8,1] => ?
=> ?
=> ? => ? = 1
[5,6,2,3,4,7,8,1] => ?
=> ?
=> ? => ? = 1
[8,5,6,7,4,3,1,2] => ?
=> ?
=> ? => ? = 2
[7,5,6,8,4,3,1,2] => ?
=> ?
=> ? => ? = 3
[7,8,4,5,6,3,1,2] => ?
=> ?
=> ? => ? = 3
[6,7,4,5,8,3,1,2] => ?
=> ?
=> ? => ? = 1
[7,4,5,6,8,3,1,2] => ?
=> ?
=> ? => ? = 2
[8,6,7,5,3,4,1,2] => ?
=> ?
=> ? => ? = 1
[7,6,8,3,4,5,1,2] => ?
=> ?
=> ? => ? = 2
[6,7,8,3,4,5,1,2] => ?
=> ?
=> ? => ? = 1
[8,5,6,3,4,7,1,2] => ?
=> ?
=> ? => ? = 1
[7,5,6,3,4,8,1,2] => ?
=> ?
=> ? => ? = 2
[6,5,7,3,4,8,1,2] => ?
=> ?
=> ? => ? = 1
[7,5,6,8,4,2,1,3] => ?
=> ?
=> ? => ? = 2
[8,7,4,5,6,2,1,3] => ?
=> ?
=> ? => ? = 1
[7,8,4,5,6,2,1,3] => ?
=> ?
=> ? => ? = 2
[7,6,4,5,8,2,1,3] => ?
=> ?
=> ? => ? = 3
[6,7,4,5,8,2,1,3] => ?
=> ?
=> ? => ? = 2
[7,5,4,6,8,2,1,3] => ?
=> ?
=> ? => ? = 2
[6,5,4,7,8,2,1,3] => ?
=> ?
=> ? => ? = 1
[7,6,4,5,8,1,2,3] => ?
=> ?
=> ? => ? = 2
[8,7,5,6,2,3,1,4] => ?
=> ?
=> ? => ? = 1
[8,6,7,5,3,1,2,4] => ?
=> ?
=> ? => ? = 1
[6,7,8,5,3,1,2,4] => ?
=> ?
=> ? => ? = 3
[8,5,6,7,3,1,2,4] => ?
=> ?
=> ? => ? = 1
[8,6,7,5,2,1,3,4] => ?
=> ?
=> ? => ? = 2
[6,7,8,3,4,2,1,5] => ?
=> ?
=> ? => ? = 2
[7,8,6,2,3,4,1,5] => ?
=> ?
=> ? => ? = 2
[7,8,6,2,3,1,4,5] => ?
=> ?
=> ? => ? = 1
[7,8,6,2,1,3,4,5] => ?
=> ?
=> ? => ? = 2
[7,6,8,2,1,3,4,5] => ?
=> ?
=> ? => ? = 1
[8,7,4,5,2,3,1,6] => ?
=> ?
=> ? => ? = 1
[7,8,5,3,2,4,1,6] => ?
=> ?
=> ? => ? = 2
[8,7,4,2,3,5,1,6] => ?
=> ?
=> ? => ? = 1
[7,8,4,5,2,1,3,6] => ?
=> ?
=> ? => ? = 1
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
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000884: Permutations ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 86%
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000884: Permutations ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 86%
Values
[1] => {{1}}
=> [1] => [1] => 0
[1,2] => {{1},{2}}
=> [1,2] => [1,2] => 0
[2,1] => {{1,2}}
=> [2,1] => [2,1] => 1
[1,2,3] => {{1},{2},{3}}
=> [1,2,3] => [1,2,3] => 0
[1,3,2] => {{1},{2,3}}
=> [1,3,2] => [1,3,2] => 1
[2,1,3] => {{1,2},{3}}
=> [2,1,3] => [2,1,3] => 1
[2,3,1] => {{1,2,3}}
=> [2,3,1] => [3,1,2] => 1
[3,1,2] => {{1,2,3}}
=> [2,3,1] => [3,1,2] => 1
[3,2,1] => {{1,3},{2}}
=> [3,2,1] => [2,3,1] => 1
[1,2,3,4] => {{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => {{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => 1
[1,3,2,4] => {{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2] => {{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => 1
[1,4,2,3] => {{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => 1
[1,4,3,2] => {{1},{2,4},{3}}
=> [1,4,3,2] => [1,3,4,2] => 1
[2,1,3,4] => {{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => {{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => 2
[2,3,1,4] => {{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => 1
[2,3,4,1] => {{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => 1
[2,4,1,3] => {{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => 1
[2,4,3,1] => {{1,2,4},{3}}
=> [2,4,3,1] => [3,4,1,2] => 1
[3,1,2,4] => {{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => 1
[3,1,4,2] => {{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => 1
[3,2,1,4] => {{1,3},{2},{4}}
=> [3,2,1,4] => [2,3,1,4] => 1
[3,2,4,1] => {{1,3,4},{2}}
=> [3,2,4,1] => [2,4,1,3] => 1
[3,4,1,2] => {{1,3},{2,4}}
=> [3,4,1,2] => [3,1,4,2] => 2
[3,4,2,1] => {{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => 1
[4,1,2,3] => {{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => 1
[4,1,3,2] => {{1,2,4},{3}}
=> [2,4,3,1] => [3,4,1,2] => 1
[4,2,1,3] => {{1,3,4},{2}}
=> [3,2,4,1] => [2,4,1,3] => 1
[4,2,3,1] => {{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,4,1] => 1
[4,3,1,2] => {{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => 1
[4,3,2,1] => {{1,4},{2,3}}
=> [4,3,2,1] => [3,2,4,1] => 2
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [1,2,5,3,4] => 1
[1,2,5,3,4] => {{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [1,2,5,3,4] => 1
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [1,2,4,5,3] => 1
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,4,2,3,5] => 1
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,5,2,3,4] => 1
[1,3,5,2,4] => {{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,5,2,3,4] => 1
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,4,5,2,3] => 1
[1,4,2,3,5] => {{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,4,2,3,5] => 1
[1,4,2,5,3] => {{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,5,2,3,4] => 1
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,3,4,2,5] => 1
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,3,5,2,4] => 1
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,4,2,5,3] => 2
[1,3,4,7,6,5,2] => {{1},{2,3,4,7},{5,6}}
=> [1,3,4,7,6,5,2] => [1,6,5,7,2,3,4] => ? = 2
[1,3,5,6,7,4,2] => {{1},{2,3,5,7},{4,6}}
=> [1,3,5,6,7,4,2] => [1,6,4,7,2,3,5] => ? = 2
[1,3,5,7,6,2,4] => {{1},{2,3,5,6},{4,7}}
=> [1,3,5,7,6,2,4] => [1,6,2,3,5,7,4] => ? = 2
[1,3,6,5,4,7,2] => {{1},{2,3,6,7},{4,5}}
=> [1,3,6,5,4,7,2] => [1,5,4,7,2,3,6] => ? = 2
[1,3,6,7,2,5,4] => {{1},{2,3,5,6},{4,7}}
=> [1,3,5,7,6,2,4] => [1,6,2,3,5,7,4] => ? = 2
[1,3,7,2,6,5,4] => {{1},{2,3,4,7},{5,6}}
=> [1,3,4,7,6,5,2] => [1,6,5,7,2,3,4] => ? = 2
[1,3,7,5,4,2,6] => {{1},{2,3,6,7},{4,5}}
=> [1,3,6,5,4,7,2] => [1,5,4,7,2,3,6] => ? = 2
[1,3,7,5,6,4,2] => {{1},{2,3,7},{4,5,6}}
=> [1,3,7,5,6,4,2] => [1,6,4,5,7,2,3] => ? = 2
[1,3,7,6,2,4,5] => {{1},{2,3,5,7},{4,6}}
=> [1,3,5,6,7,4,2] => [1,6,4,7,2,3,5] => ? = 2
[1,3,7,6,4,5,2] => {{1},{2,3,7},{4,5,6}}
=> [1,3,7,5,6,4,2] => [1,6,4,5,7,2,3] => ? = 2
[1,3,7,6,5,4,2] => {{1},{2,3,7},{4,6},{5}}
=> [1,3,7,6,5,4,2] => [1,5,6,4,7,2,3] => ? = 2
[1,4,2,7,6,5,3] => {{1},{2,3,4,7},{5,6}}
=> [1,3,4,7,6,5,2] => [1,6,5,7,2,3,4] => ? = 2
[1,4,5,6,3,7,2] => {{1},{2,4,6,7},{3,5}}
=> [1,4,5,6,3,7,2] => [1,5,3,7,2,4,6] => ? = 2
[1,4,5,6,7,2,3] => {{1},{2,4,6},{3,5,7}}
=> [1,4,5,6,7,2,3] => [1,6,2,4,7,3,5] => ? = 2
[1,4,5,7,3,2,6] => {{1},{2,4,6,7},{3,5}}
=> [1,4,5,6,3,7,2] => [1,5,3,7,2,4,6] => ? = 2
[1,4,5,7,6,3,2] => {{1},{2,4,7},{3,5,6}}
=> [1,4,5,7,6,3,2] => [1,6,3,5,7,2,4] => ? = 2
[1,4,6,5,7,3,2] => {{1},{2,4,5,7},{3,6}}
=> [1,4,6,5,7,3,2] => [1,6,3,7,2,4,5] => ? = 2
[1,4,6,7,2,3,5] => {{1},{2,4,5,7},{3,6}}
=> [1,4,6,5,7,3,2] => [1,6,3,7,2,4,5] => ? = 2
[1,4,6,7,3,5,2] => {{1},{2,4,7},{3,5,6}}
=> [1,4,5,7,6,3,2] => [1,6,3,5,7,2,4] => ? = 2
[1,4,6,7,5,3,2] => {{1},{2,4,7},{3,6},{5}}
=> [1,4,6,7,5,3,2] => [1,5,6,3,7,2,4] => ? = 2
[1,4,7,3,6,5,2] => {{1},{2,3,4,7},{5,6}}
=> [1,3,4,7,6,5,2] => [1,6,5,7,2,3,4] => ? = 2
[1,4,7,5,6,2,3] => {{1},{2,4,5,6},{3,7}}
=> [1,4,7,5,6,2,3] => [1,6,2,4,5,7,3] => ? = 2
[1,4,7,6,2,5,3] => {{1},{2,4,5,6},{3,7}}
=> [1,4,7,5,6,2,3] => [1,6,2,4,5,7,3] => ? = 2
[1,4,7,6,3,2,5] => {{1},{2,4,6},{3,5,7}}
=> [1,4,5,6,7,2,3] => [1,6,2,4,7,3,5] => ? = 2
[1,5,2,6,7,4,3] => {{1},{2,3,5,7},{4,6}}
=> [1,3,5,6,7,4,2] => [1,6,4,7,2,3,5] => ? = 2
[1,5,2,7,6,3,4] => {{1},{2,3,5,6},{4,7}}
=> [1,3,5,7,6,2,4] => [1,6,2,3,5,7,4] => ? = 2
[1,5,4,6,7,3,2] => {{1},{2,5,7},{3,4,6}}
=> [1,5,4,6,7,3,2] => [1,6,3,4,7,2,5] => ? = 2
[1,5,4,7,6,2,3] => {{1},{2,5,6},{3,4,7}}
=> [1,5,4,7,6,2,3] => [1,6,2,5,7,3,4] => ? = 2
[1,5,6,2,7,3,4] => {{1},{2,4,5,7},{3,6}}
=> [1,4,6,5,7,3,2] => [1,6,3,7,2,4,5] => ? = 2
[1,5,6,3,7,4,2] => {{1},{2,5,7},{3,4,6}}
=> [1,5,4,6,7,3,2] => [1,6,3,4,7,2,5] => ? = 2
[1,5,6,7,3,2,4] => {{1},{2,3,5,6},{4,7}}
=> [1,3,5,7,6,2,4] => [1,6,2,3,5,7,4] => ? = 2
[1,5,6,7,4,3,2] => {{1},{2,4,5,7},{3,6}}
=> [1,4,6,5,7,3,2] => [1,6,3,7,2,4,5] => ? = 2
[1,5,7,2,6,4,3] => {{1},{2,4,5,6},{3,7}}
=> [1,4,7,5,6,2,3] => [1,6,2,4,5,7,3] => ? = 2
[1,5,7,3,6,2,4] => {{1},{2,5,6},{3,4,7}}
=> [1,5,4,7,6,2,3] => [1,6,2,5,7,3,4] => ? = 2
[1,5,7,6,3,4,2] => {{1},{2,3,5,7},{4,6}}
=> [1,3,5,6,7,4,2] => [1,6,4,7,2,3,5] => ? = 2
[1,5,7,6,4,2,3] => {{1},{2,4,5,6},{3,7}}
=> [1,4,7,5,6,2,3] => [1,6,2,4,5,7,3] => ? = 2
[1,6,2,5,4,7,3] => {{1},{2,3,6,7},{4,5}}
=> [1,3,6,5,4,7,2] => [1,5,4,7,2,3,6] => ? = 2
[1,6,2,7,3,5,4] => {{1},{2,3,5,6},{4,7}}
=> [1,3,5,7,6,2,4] => [1,6,2,3,5,7,4] => ? = 2
[1,6,4,7,2,5,3] => {{1},{2,5,6},{3,4,7}}
=> [1,5,4,7,6,2,3] => [1,6,2,5,7,3,4] => ? = 2
[1,6,5,2,3,7,4] => {{1},{2,4,6,7},{3,5}}
=> [1,4,5,6,3,7,2] => [1,5,3,7,2,4,6] => ? = 2
[1,6,5,2,7,4,3] => {{1},{2,4,6},{3,5,7}}
=> [1,4,5,6,7,2,3] => [1,6,2,4,7,3,5] => ? = 2
[1,6,5,7,2,3,4] => {{1},{2,3,5,6},{4,7}}
=> [1,3,5,7,6,2,4] => [1,6,2,3,5,7,4] => ? = 2
[1,6,5,7,3,4,2] => {{1},{2,4,6,7},{3,5}}
=> [1,4,5,6,3,7,2] => [1,5,3,7,2,4,6] => ? = 2
[1,6,7,2,3,4,5] => {{1},{2,4,6},{3,5,7}}
=> [1,4,5,6,7,2,3] => [1,6,2,4,7,3,5] => ? = 2
[1,6,7,2,4,5,3] => {{1},{2,4,5,6},{3,7}}
=> [1,4,7,5,6,2,3] => [1,6,2,4,5,7,3] => ? = 2
[1,6,7,3,2,5,4] => {{1},{2,5,6},{3,4,7}}
=> [1,5,4,7,6,2,3] => [1,6,2,5,7,3,4] => ? = 2
[1,6,7,5,2,4,3] => {{1},{2,4,5,6},{3,7}}
=> [1,4,7,5,6,2,3] => [1,6,2,4,5,7,3] => ? = 2
[1,6,7,5,4,3,2] => {{1},{2,3,6,7},{4,5}}
=> [1,3,6,5,4,7,2] => [1,5,4,7,2,3,6] => ? = 2
[1,7,2,3,6,5,4] => {{1},{2,3,4,7},{5,6}}
=> [1,3,4,7,6,5,2] => [1,6,5,7,2,3,4] => ? = 2
[1,7,2,5,4,3,6] => {{1},{2,3,6,7},{4,5}}
=> [1,3,6,5,4,7,2] => [1,5,4,7,2,3,6] => ? = 2
Description
The number of isolated descents of a permutation.
A descent $i$ is isolated if neither $i+1$ nor $i-1$ are descents. If a permutation has only isolated descents, then it is called primitive in [1].
The following 31 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000703The number of deficiencies of a permutation. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000245The number of ascents of a permutation. St000659The number of rises of length at least 2 of a Dyck path. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle 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. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000021The number of descents of a permutation. St000155The number of exceedances (also excedences) of a permutation. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St000325The width of the tree associated to 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. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. 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. St001905The number of preferred parking spots in a parking function less than the index of the car. St001597The Frobenius rank of a skew partition. St000264The girth of a graph, which is not a tree. St001624The breadth of a lattice.
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