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Your data matches 37 different statistics following compositions of up to 3 maps.
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Matching statistic: St001280
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00079: Set partitions —shape⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00079: Set partitions —shape⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => {{1}}
=> [1]
=> 0
[1,2] => {{1},{2}}
=> [1,1]
=> 0
[2,1] => {{1,2}}
=> [2]
=> 1
[1,2,3] => {{1},{2},{3}}
=> [1,1,1]
=> 0
[1,3,2] => {{1},{2,3}}
=> [2,1]
=> 1
[2,1,3] => {{1,2},{3}}
=> [2,1]
=> 1
[2,3,1] => {{1,2,3}}
=> [3]
=> 1
[3,1,2] => {{1,3},{2}}
=> [2,1]
=> 1
[3,2,1] => {{1,3},{2}}
=> [2,1]
=> 1
[1,2,3,4] => {{1},{2},{3},{4}}
=> [1,1,1,1]
=> 0
[1,2,4,3] => {{1},{2},{3,4}}
=> [2,1,1]
=> 1
[1,3,2,4] => {{1},{2,3},{4}}
=> [2,1,1]
=> 1
[1,3,4,2] => {{1},{2,3,4}}
=> [3,1]
=> 1
[1,4,2,3] => {{1},{2,4},{3}}
=> [2,1,1]
=> 1
[1,4,3,2] => {{1},{2,4},{3}}
=> [2,1,1]
=> 1
[2,1,3,4] => {{1,2},{3},{4}}
=> [2,1,1]
=> 1
[2,1,4,3] => {{1,2},{3,4}}
=> [2,2]
=> 2
[2,3,1,4] => {{1,2,3},{4}}
=> [3,1]
=> 1
[2,3,4,1] => {{1,2,3,4}}
=> [4]
=> 1
[2,4,1,3] => {{1,2,4},{3}}
=> [3,1]
=> 1
[2,4,3,1] => {{1,2,4},{3}}
=> [3,1]
=> 1
[3,1,2,4] => {{1,3},{2},{4}}
=> [2,1,1]
=> 1
[3,1,4,2] => {{1,3,4},{2}}
=> [3,1]
=> 1
[3,2,1,4] => {{1,3},{2},{4}}
=> [2,1,1]
=> 1
[3,2,4,1] => {{1,3,4},{2}}
=> [3,1]
=> 1
[3,4,1,2] => {{1,3},{2,4}}
=> [2,2]
=> 2
[3,4,2,1] => {{1,3},{2,4}}
=> [2,2]
=> 2
[4,1,2,3] => {{1,4},{2},{3}}
=> [2,1,1]
=> 1
[4,1,3,2] => {{1,4},{2},{3}}
=> [2,1,1]
=> 1
[4,2,1,3] => {{1,4},{2},{3}}
=> [2,1,1]
=> 1
[4,2,3,1] => {{1,4},{2},{3}}
=> [2,1,1]
=> 1
[4,3,1,2] => {{1,4},{2,3}}
=> [2,2]
=> 2
[4,3,2,1] => {{1,4},{2,3}}
=> [2,2]
=> 2
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> 0
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> 1
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> 1
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> [3,1,1]
=> 1
[1,2,5,3,4] => {{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> 1
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> 1
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> 1
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> [2,2,1]
=> 2
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> [3,1,1]
=> 1
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> [4,1]
=> 1
[1,3,5,2,4] => {{1},{2,3,5},{4}}
=> [3,1,1]
=> 1
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> [3,1,1]
=> 1
[1,4,2,3,5] => {{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> 1
[1,4,2,5,3] => {{1},{2,4,5},{3}}
=> [3,1,1]
=> 1
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> 1
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> [3,1,1]
=> 1
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> [2,2,1]
=> 2
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St000291
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => {{1}}
=> [1] => 1 => 0
[1,2] => {{1},{2}}
=> [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] => {{1,3},{2}}
=> [2,1] => 101 => 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},{2,4},{3}}
=> [1,2,1] => 1101 => 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] => {{1,2,4},{3}}
=> [3,1] => 1001 => 1
[2,4,3,1] => {{1,2,4},{3}}
=> [3,1] => 1001 => 1
[3,1,2,4] => {{1,3},{2},{4}}
=> [2,1,1] => 1011 => 1
[3,1,4,2] => {{1,3,4},{2}}
=> [3,1] => 1001 => 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] => {{1,3},{2,4}}
=> [2,2] => 1010 => 2
[4,1,2,3] => {{1,4},{2},{3}}
=> [2,1,1] => 1011 => 1
[4,1,3,2] => {{1,4},{2},{3}}
=> [2,1,1] => 1011 => 1
[4,2,1,3] => {{1,4},{2},{3}}
=> [2,1,1] => 1011 => 1
[4,2,3,1] => {{1,4},{2},{3}}
=> [2,1,1] => 1011 => 1
[4,3,1,2] => {{1,4},{2,3}}
=> [2,2] => 1010 => 2
[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},{3,5},{4}}
=> [1,1,2,1] => 11101 => 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},{2,3,5},{4}}
=> [1,3,1] => 11001 => 1
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> [1,3,1] => 11001 => 1
[1,4,2,3,5] => {{1},{2,4},{3},{5}}
=> [1,2,1,1] => 11011 => 1
[1,4,2,5,3] => {{1},{2,4,5},{3}}
=> [1,3,1] => 11001 => 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
Description
The number of descents of a binary word.
Matching statistic: St001011
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1,0]
=> 0
[1,2] => [1,2] => [2] => [1,1,0,0]
=> 0
[2,1] => [2,1] => [1,1] => [1,0,1,0]
=> 1
[1,2,3] => [1,2,3] => [3] => [1,1,1,0,0,0]
=> 0
[1,3,2] => [1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[2,1,3] => [2,1,3] => [1,2] => [1,0,1,1,0,0]
=> 1
[2,3,1] => [3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 1
[3,1,2] => [3,1,2] => [1,2] => [1,0,1,1,0,0]
=> 1
[3,2,1] => [2,3,1] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,2,3,4] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,2,4,3] => [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,3,2,4] => [1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[1,3,4,2] => [1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,4,2,3] => [1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[1,4,3,2] => [1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[2,1,3,4] => [2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[2,1,4,3] => [2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[2,3,1,4] => [3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[2,3,4,1] => [4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1
[2,4,1,3] => [4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[2,4,3,1] => [3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[3,1,2,4] => [3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[3,1,4,2] => [4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[3,2,1,4] => [2,3,1,4] => [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
[3,4,1,2] => [4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[3,4,2,1] => [4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[4,1,2,3] => [4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[4,1,3,2] => [3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[4,2,1,3] => [2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[4,2,3,1] => [2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[4,3,1,2] => [3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[4,3,2,1] => [3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,2,3,4,5] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,2,4,5,3] => [1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,2,5,3,4] => [1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,2,5,4,3] => [1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,3,4,2,5] => [1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,3,4,5,2] => [1,5,4,3,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,3,5,2,4] => [1,5,3,2,4] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,3,5,4,2] => [1,4,5,3,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,4,2,3,5] => [1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,4,2,5,3] => [1,5,4,2,3] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,4,3,2,5] => [1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,4,3,5,2] => [1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,4,5,2,3] => [1,5,2,4,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
Description
Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000251
(load all 73 compositions to match this statistic)
(load all 73 compositions to match this statistic)
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000251: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000251: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => {{1}}
=> ? = 0
[1,2] => {{1},{2}}
=> 0
[2,1] => {{1,2}}
=> 1
[1,2,3] => {{1},{2},{3}}
=> 0
[1,3,2] => {{1},{2,3}}
=> 1
[2,1,3] => {{1,2},{3}}
=> 1
[2,3,1] => {{1,2,3}}
=> 1
[3,1,2] => {{1,3},{2}}
=> 1
[3,2,1] => {{1,3},{2}}
=> 1
[1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[1,2,4,3] => {{1},{2},{3,4}}
=> 1
[1,3,2,4] => {{1},{2,3},{4}}
=> 1
[1,3,4,2] => {{1},{2,3,4}}
=> 1
[1,4,2,3] => {{1},{2,4},{3}}
=> 1
[1,4,3,2] => {{1},{2,4},{3}}
=> 1
[2,1,3,4] => {{1,2},{3},{4}}
=> 1
[2,1,4,3] => {{1,2},{3,4}}
=> 2
[2,3,1,4] => {{1,2,3},{4}}
=> 1
[2,3,4,1] => {{1,2,3,4}}
=> 1
[2,4,1,3] => {{1,2,4},{3}}
=> 1
[2,4,3,1] => {{1,2,4},{3}}
=> 1
[3,1,2,4] => {{1,3},{2},{4}}
=> 1
[3,1,4,2] => {{1,3,4},{2}}
=> 1
[3,2,1,4] => {{1,3},{2},{4}}
=> 1
[3,2,4,1] => {{1,3,4},{2}}
=> 1
[3,4,1,2] => {{1,3},{2,4}}
=> 2
[3,4,2,1] => {{1,3},{2,4}}
=> 2
[4,1,2,3] => {{1,4},{2},{3}}
=> 1
[4,1,3,2] => {{1,4},{2},{3}}
=> 1
[4,2,1,3] => {{1,4},{2},{3}}
=> 1
[4,2,3,1] => {{1,4},{2},{3}}
=> 1
[4,3,1,2] => {{1,4},{2,3}}
=> 2
[4,3,2,1] => {{1,4},{2,3}}
=> 2
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 0
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 1
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 1
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 1
[1,2,5,3,4] => {{1},{2},{3,5},{4}}
=> 1
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 1
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 1
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 2
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> 1
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> 1
[1,3,5,2,4] => {{1},{2,3,5},{4}}
=> 1
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> 1
[1,4,2,3,5] => {{1},{2,4},{3},{5}}
=> 1
[1,4,2,5,3] => {{1},{2,4,5},{3}}
=> 1
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 1
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> 1
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> 2
[1,4,5,3,2] => {{1},{2,4},{3,5}}
=> 2
Description
The number of nonsingleton blocks of a set partition.
Matching statistic: St000390
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00109: Permutations —descent word⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => => ? = 0
[1,2] => [1,2] => 0 => 0
[2,1] => [2,1] => 1 => 1
[1,2,3] => [1,2,3] => 00 => 0
[1,3,2] => [1,3,2] => 01 => 1
[2,1,3] => [2,1,3] => 10 => 1
[2,3,1] => [3,2,1] => 11 => 1
[3,1,2] => [3,1,2] => 10 => 1
[3,2,1] => [2,3,1] => 01 => 1
[1,2,3,4] => [1,2,3,4] => 000 => 0
[1,2,4,3] => [1,2,4,3] => 001 => 1
[1,3,2,4] => [1,3,2,4] => 010 => 1
[1,3,4,2] => [1,4,3,2] => 011 => 1
[1,4,2,3] => [1,4,2,3] => 010 => 1
[1,4,3,2] => [1,3,4,2] => 001 => 1
[2,1,3,4] => [2,1,3,4] => 100 => 1
[2,1,4,3] => [2,1,4,3] => 101 => 2
[2,3,1,4] => [3,2,1,4] => 110 => 1
[2,3,4,1] => [4,3,2,1] => 111 => 1
[2,4,1,3] => [4,2,1,3] => 110 => 1
[2,4,3,1] => [3,4,2,1] => 011 => 1
[3,1,2,4] => [3,1,2,4] => 100 => 1
[3,1,4,2] => [4,3,1,2] => 110 => 1
[3,2,1,4] => [2,3,1,4] => 010 => 1
[3,2,4,1] => [2,4,3,1] => 011 => 1
[3,4,1,2] => [4,1,3,2] => 101 => 2
[3,4,2,1] => [4,2,3,1] => 101 => 2
[4,1,2,3] => [4,1,2,3] => 100 => 1
[4,1,3,2] => [3,4,1,2] => 010 => 1
[4,2,1,3] => [2,4,1,3] => 010 => 1
[4,2,3,1] => [2,3,4,1] => 001 => 1
[4,3,1,2] => [3,1,4,2] => 101 => 2
[4,3,2,1] => [3,2,4,1] => 101 => 2
[1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,2,3,5,4] => [1,2,3,5,4] => 0001 => 1
[1,2,4,3,5] => [1,2,4,3,5] => 0010 => 1
[1,2,4,5,3] => [1,2,5,4,3] => 0011 => 1
[1,2,5,3,4] => [1,2,5,3,4] => 0010 => 1
[1,2,5,4,3] => [1,2,4,5,3] => 0001 => 1
[1,3,2,4,5] => [1,3,2,4,5] => 0100 => 1
[1,3,2,5,4] => [1,3,2,5,4] => 0101 => 2
[1,3,4,2,5] => [1,4,3,2,5] => 0110 => 1
[1,3,4,5,2] => [1,5,4,3,2] => 0111 => 1
[1,3,5,2,4] => [1,5,3,2,4] => 0110 => 1
[1,3,5,4,2] => [1,4,5,3,2] => 0011 => 1
[1,4,2,3,5] => [1,4,2,3,5] => 0100 => 1
[1,4,2,5,3] => [1,5,4,2,3] => 0110 => 1
[1,4,3,2,5] => [1,3,4,2,5] => 0010 => 1
[1,4,3,5,2] => [1,3,5,4,2] => 0011 => 1
[1,4,5,2,3] => [1,5,2,4,3] => 0101 => 2
[1,4,5,3,2] => [1,5,3,4,2] => 0101 => 2
Description
The number of runs of ones in a binary word.
Matching statistic: St000985
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000985: Graphs ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000985: Graphs ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ([],1)
=> 0
[1,2] => [1,2] => [2] => ([],2)
=> 0
[2,1] => [2,1] => [1,1] => ([(0,1)],2)
=> 1
[1,2,3] => [1,2,3] => [3] => ([],3)
=> 0
[1,3,2] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[2,1,3] => [2,1,3] => [1,2] => ([(1,2)],3)
=> 1
[2,3,1] => [3,1,2] => [1,2] => ([(1,2)],3)
=> 1
[3,1,2] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[3,2,1] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,2,3,4] => [1,2,3,4] => [4] => ([],4)
=> 0
[1,2,4,3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,3,2,4] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[1,3,4,2] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[1,4,2,3] => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,4,3,2] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[2,1,3,4] => [2,1,3,4] => [1,3] => ([(2,3)],4)
=> 1
[2,1,4,3] => [2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,3,1,4] => [3,1,2,4] => [1,3] => ([(2,3)],4)
=> 1
[2,3,4,1] => [4,1,2,3] => [1,3] => ([(2,3)],4)
=> 1
[2,4,1,3] => [4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 1
[2,4,3,1] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[3,1,2,4] => [3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 1
[3,1,4,2] => [4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 1
[3,2,1,4] => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[3,2,4,1] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[3,4,1,2] => [3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,4,2,1] => [4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,1,2,3] => [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,1,3,2] => [3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,2,1,3] => [2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,2,3,1] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4,3,1,2] => [4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,3,2,1] => [3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,2,3,4,5] => [1,2,3,4,5] => [5] => ([],5)
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 1
[1,2,4,5,3] => [1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 1
[1,2,5,3,4] => [1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,2,5,4,3] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,4,2,5] => [1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[1,3,4,5,2] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[1,3,5,2,4] => [1,5,4,2,3] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,5,4,2] => [1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 1
[1,4,2,3,5] => [1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,2,5,3] => [1,5,3,2,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,3,2,5] => [1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 1
[1,4,3,5,2] => [1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 1
[1,4,5,2,3] => [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,4,1,3,7,5,6] => [4,3,1,2,7,6,5] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[2,5,1,7,3,4,6] => [5,3,1,2,7,6,4] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[2,5,7,1,4,3,6] => [5,4,1,2,7,6,3] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[2,5,7,3,4,1,6] => [7,6,1,2,5,4,3] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[2,6,1,7,4,3,5] => [6,3,1,2,7,5,4] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[2,6,1,7,4,5,3] => [7,3,1,2,6,5,4] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[2,6,7,1,3,4,5] => [6,4,1,2,7,5,3] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[2,6,7,1,3,5,4] => [7,4,1,2,6,5,3] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[2,6,7,3,1,4,5] => [7,5,1,2,6,4,3] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[2,6,7,3,1,5,4] => [6,5,1,2,7,4,3] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[3,1,2,4,7,5,6] => [3,2,1,4,7,6,5] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[3,1,2,7,4,6,5] => [3,2,1,6,7,5,4] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[3,1,2,7,5,4,6] => [3,2,1,5,7,6,4] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[3,1,4,2,7,5,6] => [4,2,1,3,7,6,5] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[3,1,5,7,2,4,6] => [5,2,1,3,7,6,4] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[3,1,6,7,4,2,5] => [6,2,1,3,7,5,4] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[3,1,6,7,4,5,2] => [7,2,1,3,6,5,4] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[3,7,5,1,4,2,6] => [5,4,1,3,7,6,2] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[3,7,5,2,4,1,6] => [7,6,1,3,5,4,2] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[3,7,6,1,2,4,5] => [6,4,1,3,7,5,2] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[3,7,6,1,2,5,4] => [7,4,1,3,6,5,2] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[3,7,6,2,1,4,5] => [7,5,1,3,6,4,2] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[3,7,6,2,1,5,4] => [6,5,1,3,7,4,2] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[4,1,7,2,3,6,5] => [4,2,1,6,7,5,3] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[4,1,7,2,5,3,6] => [4,2,1,5,7,6,3] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[4,1,7,5,2,3,6] => [5,2,1,4,7,6,3] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[4,1,7,6,3,2,5] => [6,2,1,4,7,5,3] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[4,1,7,6,3,5,2] => [7,2,1,4,6,5,3] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[4,7,1,3,2,6,5] => [4,3,1,6,7,5,2] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[4,7,1,3,5,2,6] => [4,3,1,5,7,6,2] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[4,7,1,5,3,2,6] => [5,3,1,4,7,6,2] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[4,7,1,6,2,3,5] => [6,3,1,4,7,5,2] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[4,7,1,6,2,5,3] => [7,3,1,4,6,5,2] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[4,7,2,5,3,1,6] => [7,6,1,4,5,3,2] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[4,7,2,6,1,3,5] => [7,5,1,4,6,3,2] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[4,7,2,6,1,5,3] => [6,5,1,4,7,3,2] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[5,1,7,3,2,6,4] => [5,2,1,6,7,4,3] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[5,1,7,3,6,2,4] => [6,2,1,5,7,4,3] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[5,1,7,3,6,4,2] => [7,2,1,5,6,4,3] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[5,7,1,2,3,6,4] => [5,3,1,6,7,4,2] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[5,7,1,2,6,3,4] => [6,3,1,5,7,4,2] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[5,7,1,2,6,4,3] => [7,3,1,5,6,4,2] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[5,7,2,1,4,6,3] => [5,4,1,6,7,3,2] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[5,7,2,1,6,3,4] => [7,4,1,5,6,3,2] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[5,7,2,1,6,4,3] => [6,4,1,5,7,3,2] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[7,1,4,5,2,3,6] => [7,6,3,4,5,2,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[7,1,4,6,3,2,5] => [7,5,3,4,6,2,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[7,1,4,6,3,5,2] => [6,5,3,4,7,2,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[7,1,5,3,4,6,2] => [5,4,3,6,7,2,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[7,1,5,3,6,2,4] => [7,4,3,5,6,2,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
Description
The number of positive eigenvalues of the adjacency matrix of the graph.
Matching statistic: St001354
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001354: Graphs ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001354: Graphs ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ([],1)
=> 0
[1,2] => [1,2] => [2] => ([],2)
=> 0
[2,1] => [2,1] => [1,1] => ([(0,1)],2)
=> 1
[1,2,3] => [1,2,3] => [3] => ([],3)
=> 0
[1,3,2] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[2,1,3] => [2,1,3] => [1,2] => ([(1,2)],3)
=> 1
[2,3,1] => [3,1,2] => [1,2] => ([(1,2)],3)
=> 1
[3,1,2] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[3,2,1] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,2,3,4] => [1,2,3,4] => [4] => ([],4)
=> 0
[1,2,4,3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,3,2,4] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[1,3,4,2] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[1,4,2,3] => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,4,3,2] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[2,1,3,4] => [2,1,3,4] => [1,3] => ([(2,3)],4)
=> 1
[2,1,4,3] => [2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,3,1,4] => [3,1,2,4] => [1,3] => ([(2,3)],4)
=> 1
[2,3,4,1] => [4,1,2,3] => [1,3] => ([(2,3)],4)
=> 1
[2,4,1,3] => [4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 1
[2,4,3,1] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[3,1,2,4] => [3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 1
[3,1,4,2] => [4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 1
[3,2,1,4] => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[3,2,4,1] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[3,4,1,2] => [3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,4,2,1] => [4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,1,2,3] => [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,1,3,2] => [3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,2,1,3] => [2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,2,3,1] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4,3,1,2] => [4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,3,2,1] => [3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,2,3,4,5] => [1,2,3,4,5] => [5] => ([],5)
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 1
[1,2,4,5,3] => [1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 1
[1,2,5,3,4] => [1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,2,5,4,3] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,4,2,5] => [1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[1,3,4,5,2] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[1,3,5,2,4] => [1,5,4,2,3] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,5,4,2] => [1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 1
[1,4,2,3,5] => [1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,2,5,3] => [1,5,3,2,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,3,2,5] => [1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 1
[1,4,3,5,2] => [1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 1
[1,4,5,2,3] => [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,7,2,3,4,5,6] => [1,7,6,5,4,3,2] => [2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[2,1,7,3,4,5,6] => [2,1,7,6,5,4,3] => [1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[2,7,1,3,4,5,6] => [7,6,5,4,3,1,2] => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[3,1,2,7,4,5,6] => [3,2,1,7,6,5,4] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[3,1,7,2,4,5,6] => [7,6,5,4,2,1,3] => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[3,7,1,2,4,5,6] => [3,1,7,6,5,4,2] => [1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[3,7,2,1,4,5,6] => [7,6,5,4,1,3,2] => [1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[4,1,2,3,7,5,6] => [4,3,2,1,7,6,5] => [1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[4,1,2,7,3,5,6] => [7,6,5,3,2,1,4] => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[4,1,7,2,3,5,6] => [4,2,1,7,6,5,3] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[4,1,7,3,2,5,6] => [7,6,5,2,1,4,3] => [1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[4,7,1,2,3,5,6] => [7,6,5,3,1,4,2] => [1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[4,7,1,3,2,5,6] => [4,3,1,7,6,5,2] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[4,7,2,1,3,5,6] => [4,1,7,6,5,3,2] => [1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[4,7,2,3,1,5,6] => [7,6,5,1,4,3,2] => [1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[5,1,2,3,4,7,6] => [5,4,3,2,1,7,6] => [1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[5,1,2,3,7,4,6] => [7,6,4,3,2,1,5] => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[5,1,2,7,3,4,6] => [5,3,2,1,7,6,4] => [1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[5,1,2,7,4,3,6] => [7,6,3,2,1,5,4] => [1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[5,1,7,2,3,4,6] => [7,6,4,2,1,5,3] => [1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[5,1,7,2,4,3,6] => [5,4,2,1,7,6,3] => [1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[5,1,7,3,2,4,6] => [5,2,1,7,6,4,3] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[5,1,7,3,4,2,6] => [7,6,2,1,5,4,3] => [1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[5,7,1,2,3,4,6] => [5,3,1,7,6,4,2] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[5,7,1,2,4,3,6] => [7,6,3,1,5,4,2] => [1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[5,7,1,3,2,4,6] => [7,6,4,3,1,5,2] => [1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[5,7,1,3,4,2,6] => [5,4,3,1,7,6,2] => [1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[5,7,2,1,3,4,6] => [7,6,4,1,5,3,2] => [1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[5,7,2,1,4,3,6] => [5,4,1,7,6,3,2] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[5,7,2,3,1,4,6] => [5,1,7,6,4,3,2] => [1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[5,7,2,3,4,1,6] => [7,6,1,5,4,3,2] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[6,1,2,3,4,5,7] => [6,5,4,3,2,1,7] => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[6,1,2,3,4,7,5] => [7,5,4,3,2,1,6] => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[6,1,2,3,7,4,5] => [6,4,3,2,1,7,5] => [1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[6,1,2,3,7,5,4] => [7,4,3,2,1,6,5] => [1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[6,1,2,7,3,4,5] => [7,5,3,2,1,6,4] => [1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[6,1,2,7,3,5,4] => [6,5,3,2,1,7,4] => [1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[6,1,2,7,4,3,5] => [6,3,2,1,7,5,4] => [1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[6,1,2,7,4,5,3] => [7,3,2,1,6,5,4] => [1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[6,1,7,2,3,4,5] => [6,4,2,1,7,5,3] => [1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[6,1,7,2,3,5,4] => [7,4,2,1,6,5,3] => [1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[6,1,7,2,4,3,5] => [7,5,4,2,1,6,3] => [1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[6,1,7,2,4,5,3] => [6,5,4,2,1,7,3] => [1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[6,1,7,3,2,4,5] => [7,5,2,1,6,4,3] => [1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[6,1,7,3,2,5,4] => [6,5,2,1,7,4,3] => [1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[6,1,7,3,4,2,5] => [6,2,1,7,5,4,3] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[6,1,7,3,4,5,2] => [7,2,1,6,5,4,3] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[6,7,1,2,3,4,5] => [7,5,3,1,6,4,2] => [1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[6,7,1,2,3,5,4] => [6,5,3,1,7,4,2] => [1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[6,7,1,2,4,3,5] => [6,3,1,7,5,4,2] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
Description
The number of series nodes in the modular decomposition of a graph.
Matching statistic: St000994
(load all 95 compositions to match this statistic)
(load all 95 compositions to match this statistic)
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000994: Permutations ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 100%
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000994: Permutations ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => [2,1,3] => [2,1,3] => 1
[2,3,1] => [3,1,2] => [3,1,2] => [3,1,2] => 1
[3,1,2] => [3,2,1] => [3,2,1] => [2,3,1] => 1
[3,2,1] => [2,3,1] => [1,3,2] => [1,3,2] => 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2] => [1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 1
[1,4,2,3] => [1,4,3,2] => [1,4,3,2] => [1,3,4,2] => 1
[1,4,3,2] => [1,3,4,2] => [1,2,4,3] => [1,2,4,3] => 1
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[2,3,1,4] => [3,1,2,4] => [3,1,2,4] => [3,1,2,4] => 1
[2,3,4,1] => [4,1,2,3] => [4,1,2,3] => [4,1,2,3] => 1
[2,4,1,3] => [4,3,1,2] => [4,3,1,2] => [3,1,4,2] => 1
[2,4,3,1] => [3,4,1,2] => [2,4,1,3] => [4,2,1,3] => 1
[3,1,2,4] => [3,2,1,4] => [3,2,1,4] => [2,3,1,4] => 1
[3,1,4,2] => [4,2,1,3] => [4,2,1,3] => [2,4,1,3] => 1
[3,2,1,4] => [2,3,1,4] => [1,3,2,4] => [1,3,2,4] => 1
[3,2,4,1] => [2,4,1,3] => [2,4,1,3] => [4,2,1,3] => 1
[3,4,1,2] => [3,1,4,2] => [2,1,4,3] => [2,1,4,3] => 2
[3,4,2,1] => [4,1,3,2] => [4,1,3,2] => [3,4,1,2] => 2
[4,1,2,3] => [4,3,2,1] => [4,3,2,1] => [2,3,4,1] => 1
[4,1,3,2] => [3,4,2,1] => [1,4,3,2] => [1,3,4,2] => 1
[4,2,1,3] => [2,4,3,1] => [1,4,3,2] => [1,3,4,2] => 1
[4,2,3,1] => [2,3,4,1] => [1,2,4,3] => [1,2,4,3] => 1
[4,3,1,2] => [4,2,3,1] => [4,1,3,2] => [3,4,1,2] => 2
[4,3,2,1] => [3,2,4,1] => [2,1,4,3] => [2,1,4,3] => 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] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => 1
[1,2,5,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,4,5,3] => 1
[1,2,5,4,3] => [1,2,4,5,3] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,3,4,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => 1
[1,3,4,5,2] => [1,5,2,3,4] => [1,5,2,3,4] => [1,5,2,3,4] => 1
[1,3,5,2,4] => [1,5,4,2,3] => [1,5,4,2,3] => [1,4,2,5,3] => 1
[1,3,5,4,2] => [1,4,5,2,3] => [1,3,5,2,4] => [1,5,3,2,4] => 1
[1,4,2,3,5] => [1,4,3,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => 1
[1,4,2,5,3] => [1,5,3,2,4] => [1,5,3,2,4] => [1,3,5,2,4] => 1
[1,4,3,2,5] => [1,3,4,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,4,3,5,2] => [1,3,5,2,4] => [1,3,5,2,4] => [1,5,3,2,4] => 1
[1,4,5,2,3] => [1,4,2,5,3] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,3,4,5,2,6,7] => [1,5,2,3,4,6,7] => [1,5,2,3,4,6,7] => [1,5,2,3,4,6,7] => ? = 1
[1,3,4,5,6,2,7] => [1,6,2,3,4,5,7] => [1,6,2,3,4,5,7] => [1,6,2,3,4,5,7] => ? = 1
[1,3,4,5,7,6,2] => [1,6,7,2,3,4,5] => [1,5,7,2,3,4,6] => [1,7,2,3,5,4,6] => ? = 1
[1,3,4,6,2,5,7] => [1,6,5,2,3,4,7] => [1,6,5,2,3,4,7] => [1,5,2,3,6,4,7] => ? = 1
[1,3,4,6,2,7,5] => [1,7,5,2,3,4,6] => [1,7,5,2,3,4,6] => [1,5,2,3,7,4,6] => ? = 1
[1,3,4,6,5,2,7] => [1,5,6,2,3,4,7] => [1,4,6,2,3,5,7] => [1,6,2,4,3,5,7] => ? = 1
[1,3,4,6,5,7,2] => [1,5,7,2,3,4,6] => [1,5,7,2,3,4,6] => [1,7,2,3,5,4,6] => ? = 1
[1,3,4,7,2,5,6] => [1,7,6,5,2,3,4] => [1,7,6,5,2,3,4] => [1,5,2,3,6,7,4] => ? = 1
[1,3,4,7,2,6,5] => [1,6,7,5,2,3,4] => [1,4,7,6,2,3,5] => [1,6,2,4,3,7,5] => ? = 1
[1,3,4,7,5,2,6] => [1,5,7,6,2,3,4] => [1,4,7,6,2,3,5] => [1,6,2,4,3,7,5] => ? = 1
[1,3,4,7,5,6,2] => [1,5,6,7,2,3,4] => [1,3,5,7,2,4,6] => [1,7,3,2,5,4,6] => ? = 1
[1,3,4,7,6,2,5] => [1,7,5,6,2,3,4] => [1,7,4,6,2,3,5] => [1,6,2,7,4,3,5] => ? = 2
[1,3,4,7,6,5,2] => [1,6,5,7,2,3,4] => [1,5,4,7,2,3,6] => [1,7,2,5,4,3,6] => ? = 2
[1,3,5,4,2,6,7] => [1,4,5,2,3,6,7] => [1,3,5,2,4,6,7] => [1,5,3,2,4,6,7] => ? = 1
[1,3,5,4,2,7,6] => [1,4,5,2,3,7,6] => [1,3,5,2,4,7,6] => [1,5,3,2,4,7,6] => ? = 2
[1,3,5,4,6,2,7] => [1,4,6,2,3,5,7] => [1,4,6,2,3,5,7] => [1,6,2,4,3,5,7] => ? = 1
[1,3,5,4,6,7,2] => [1,4,7,2,3,5,6] => [1,4,7,2,3,5,6] => [1,7,2,4,3,5,6] => ? = 1
[1,3,5,4,7,2,6] => [1,4,7,6,2,3,5] => [1,4,7,6,2,3,5] => [1,6,2,4,3,7,5] => ? = 1
[1,3,5,4,7,6,2] => [1,4,6,7,2,3,5] => [1,3,5,7,2,4,6] => [1,7,3,2,5,4,6] => ? = 1
[1,3,5,6,2,7,4] => [1,5,2,3,7,4,6] => [1,5,2,3,7,4,6] => [1,7,2,5,3,4,6] => ? = 2
[1,3,5,6,4,2,7] => [1,6,2,3,5,4,7] => [1,6,2,3,5,4,7] => [1,5,2,6,3,4,7] => ? = 2
[1,3,5,6,7,2,4] => [1,7,4,6,2,3,5] => [1,7,4,6,2,3,5] => [1,6,2,7,4,3,5] => ? = 2
[1,3,5,6,7,4,2] => [1,6,4,7,2,3,5] => [1,5,4,7,2,3,6] => [1,7,2,5,4,3,6] => ? = 2
[1,3,5,7,4,2,6] => [1,7,6,2,3,5,4] => [1,7,6,2,3,5,4] => [1,5,2,6,3,7,4] => ? = 2
[1,3,5,7,4,6,2] => [1,6,7,2,3,5,4] => [1,4,7,2,3,6,5] => [1,6,2,4,7,3,5] => ? = 2
[1,3,6,2,5,4,7] => [1,5,6,4,2,3,7] => [1,3,6,5,2,4,7] => [1,5,3,2,6,4,7] => ? = 1
[1,3,6,4,2,5,7] => [1,4,6,5,2,3,7] => [1,3,6,5,2,4,7] => [1,5,3,2,6,4,7] => ? = 1
[1,3,6,4,2,7,5] => [1,4,7,5,2,3,6] => [1,3,7,5,2,4,6] => [1,5,3,2,7,4,6] => ? = 1
[1,3,6,4,5,7,2] => [1,4,5,7,2,3,6] => [1,3,5,7,2,4,6] => [1,7,3,2,5,4,6] => ? = 1
[1,3,6,4,7,2,5] => [1,4,6,2,3,7,5] => [1,3,5,2,4,7,6] => [1,5,3,2,4,7,6] => ? = 2
[1,3,6,4,7,5,2] => [1,4,7,2,3,6,5] => [1,4,7,2,3,6,5] => [1,6,2,4,7,3,5] => ? = 2
[1,3,6,5,2,4,7] => [1,6,4,5,2,3,7] => [1,6,3,5,2,4,7] => [1,5,6,3,2,4,7] => ? = 2
[1,3,6,5,2,7,4] => [1,7,4,5,2,3,6] => [1,7,3,5,2,4,6] => [1,5,7,3,2,4,6] => ? = 2
[1,3,6,5,4,2,7] => [1,5,4,6,2,3,7] => [1,4,3,6,2,5,7] => [1,6,4,3,2,5,7] => ? = 2
[1,3,6,5,4,7,2] => [1,5,4,7,2,3,6] => [1,5,4,7,2,3,6] => [1,7,2,5,4,3,6] => ? = 2
[1,3,6,5,7,2,4] => [1,6,2,3,7,4,5] => [1,5,2,3,7,4,6] => [1,7,2,5,3,4,6] => ? = 2
[1,3,6,7,4,5,2] => [1,7,2,3,6,5,4] => [1,7,2,3,6,5,4] => [1,5,2,6,7,3,4] => ? = 2
[1,3,6,7,5,2,4] => [1,5,6,2,3,7,4] => [1,3,5,2,4,7,6] => [1,5,3,2,4,7,6] => ? = 2
[1,3,6,7,5,4,2] => [1,5,7,2,3,6,4] => [1,4,7,2,3,6,5] => [1,6,2,4,7,3,5] => ? = 2
[1,3,7,2,4,6,5] => [1,6,7,5,4,2,3] => [1,3,7,6,5,2,4] => [1,5,3,2,6,7,4] => ? = 1
[1,3,7,2,5,4,6] => [1,5,7,6,4,2,3] => [1,3,7,6,5,2,4] => [1,5,3,2,6,7,4] => ? = 1
[1,3,7,2,6,4,5] => [1,7,5,6,4,2,3] => [1,7,3,6,5,2,4] => [1,5,6,3,7,2,4] => ? = 2
[1,3,7,2,6,5,4] => [1,6,5,7,4,2,3] => [1,4,3,7,6,2,5] => [1,6,4,3,2,7,5] => ? = 2
[1,3,7,4,2,5,6] => [1,4,7,6,5,2,3] => [1,3,7,6,5,2,4] => [1,5,3,2,6,7,4] => ? = 1
[1,3,7,5,2,4,6] => [1,7,6,4,5,2,3] => [1,7,6,3,5,2,4] => [1,5,6,3,2,7,4] => ? = 2
[1,3,7,5,2,6,4] => [1,6,7,4,5,2,3] => [1,4,7,3,6,2,5] => [1,6,7,4,3,2,5] => ? = 2
[1,3,7,5,4,2,6] => [1,5,4,7,6,2,3] => [1,4,3,7,6,2,5] => [1,6,4,3,2,7,5] => ? = 2
[1,3,7,5,6,2,4] => [1,7,4,5,6,2,3] => [1,7,2,4,6,3,5] => [1,6,7,2,4,3,5] => ? = 2
[1,3,7,5,6,4,2] => [1,6,4,5,7,2,3] => [1,5,2,4,7,3,6] => [1,7,5,2,4,3,6] => ? = 2
[1,3,7,6,2,4,5] => [1,6,4,7,5,2,3] => [1,4,3,7,6,2,5] => [1,6,4,3,2,7,5] => ? = 2
Description
The number of cycle peaks and the number of cycle valleys of a permutation.
A '''cycle peak''' of a permutation $\pi$ is an index $i$ such that $\pi^{-1}(i) < i > \pi(i)$. Analogously, a '''cycle valley''' is an index $i$ such that $\pi^{-1}(i) > i < \pi(i)$.
Clearly, every cycle of $\pi$ contains as many peaks as valleys.
Matching statistic: St000834
(load all 36 compositions to match this statistic)
(load all 36 compositions to match this statistic)
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => [2,1] => 0
[2,1] => [2,1] => [1,2] => [1,2] => 1
[1,2,3] => [1,2,3] => [3,2,1] => [3,2,1] => 0
[1,3,2] => [1,3,2] => [2,3,1] => [1,3,2] => 1
[2,1,3] => [2,1,3] => [3,1,2] => [3,1,2] => 1
[2,3,1] => [3,2,1] => [1,2,3] => [1,2,3] => 1
[3,1,2] => [3,1,2] => [2,1,3] => [2,1,3] => 1
[3,2,1] => [2,3,1] => [1,3,2] => [1,3,2] => 1
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 0
[1,2,4,3] => [1,2,4,3] => [3,4,2,1] => [1,4,3,2] => 1
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => [4,1,3,2] => 1
[1,3,4,2] => [1,4,3,2] => [2,3,4,1] => [1,2,4,3] => 1
[1,4,2,3] => [1,4,2,3] => [3,2,4,1] => [2,1,4,3] => 1
[1,4,3,2] => [1,3,4,2] => [2,4,3,1] => [1,4,3,2] => 1
[2,1,3,4] => [2,1,3,4] => [4,3,1,2] => [4,3,1,2] => 1
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => [2,4,1,3] => 2
[2,3,1,4] => [3,2,1,4] => [4,1,2,3] => [4,1,2,3] => 1
[2,3,4,1] => [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 1
[2,4,1,3] => [4,2,1,3] => [3,1,2,4] => [3,1,2,4] => 1
[2,4,3,1] => [3,4,2,1] => [1,2,4,3] => [1,2,4,3] => 1
[3,1,2,4] => [3,1,2,4] => [4,2,1,3] => [4,2,1,3] => 1
[3,1,4,2] => [4,3,1,2] => [2,1,3,4] => [2,1,3,4] => 1
[3,2,1,4] => [2,3,1,4] => [4,1,3,2] => [4,1,3,2] => 1
[3,2,4,1] => [2,4,3,1] => [1,3,4,2] => [1,2,4,3] => 1
[3,4,1,2] => [4,1,3,2] => [2,3,1,4] => [1,3,2,4] => 2
[3,4,2,1] => [4,2,3,1] => [1,3,2,4] => [1,3,2,4] => 2
[4,1,2,3] => [4,1,2,3] => [3,2,1,4] => [3,2,1,4] => 1
[4,1,3,2] => [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 1
[4,2,1,3] => [2,4,1,3] => [3,1,4,2] => [2,1,4,3] => 1
[4,2,3,1] => [2,3,4,1] => [1,4,3,2] => [1,4,3,2] => 1
[4,3,1,2] => [3,1,4,2] => [2,4,1,3] => [2,4,1,3] => 2
[4,3,2,1] => [3,2,4,1] => [1,4,2,3] => [1,4,2,3] => 2
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [4,5,3,2,1] => [1,5,4,3,2] => 1
[1,2,4,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => [5,1,4,3,2] => 1
[1,2,4,5,3] => [1,2,5,4,3] => [3,4,5,2,1] => [1,2,5,4,3] => 1
[1,2,5,3,4] => [1,2,5,3,4] => [4,3,5,2,1] => [2,1,5,4,3] => 1
[1,2,5,4,3] => [1,2,4,5,3] => [3,5,4,2,1] => [1,5,4,3,2] => 1
[1,3,2,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => [5,4,1,3,2] => 1
[1,3,2,5,4] => [1,3,2,5,4] => [4,5,2,3,1] => [2,5,1,4,3] => 2
[1,3,4,2,5] => [1,4,3,2,5] => [5,2,3,4,1] => [5,1,2,4,3] => 1
[1,3,4,5,2] => [1,5,4,3,2] => [2,3,4,5,1] => [1,2,3,5,4] => 1
[1,3,5,2,4] => [1,5,3,2,4] => [4,2,3,5,1] => [3,1,2,5,4] => 1
[1,3,5,4,2] => [1,4,5,3,2] => [2,3,5,4,1] => [1,2,5,4,3] => 1
[1,4,2,3,5] => [1,4,2,3,5] => [5,3,2,4,1] => [5,2,1,4,3] => 1
[1,4,2,5,3] => [1,5,4,2,3] => [3,2,4,5,1] => [2,1,3,5,4] => 1
[1,4,3,2,5] => [1,3,4,2,5] => [5,2,4,3,1] => [5,1,4,3,2] => 1
[1,4,3,5,2] => [1,3,5,4,2] => [2,4,5,3,1] => [1,2,5,4,3] => 1
[1,4,5,2,3] => [1,5,2,4,3] => [3,4,2,5,1] => [1,3,2,5,4] => 2
[1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [7,1,6,5,4,3,2] => ? = 1
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => [7,6,1,5,4,3,2] => ? = 1
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => [2,7,1,6,5,4,3] => ? = 2
[1,2,3,5,6,4,7] => [1,2,3,6,5,4,7] => [7,4,5,6,3,2,1] => [7,1,2,6,5,4,3] => ? = 1
[1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => [6,4,5,7,3,2,1] => [3,1,2,7,6,5,4] => ? = 1
[1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [7,5,4,6,3,2,1] => [7,2,1,6,5,4,3] => ? = 1
[1,2,3,6,4,7,5] => [1,2,3,7,6,4,5] => [5,4,6,7,3,2,1] => [2,1,3,7,6,5,4] => ? = 1
[1,2,3,6,5,4,7] => [1,2,3,5,6,4,7] => [7,4,6,5,3,2,1] => [7,1,6,5,4,3,2] => ? = 1
[1,2,3,7,4,5,6] => [1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => [3,2,1,7,6,5,4] => ? = 1
[1,2,3,7,6,4,5] => [1,2,3,6,4,7,5] => [5,7,4,6,3,2,1] => [2,7,1,6,5,4,3] => ? = 2
[1,2,3,7,6,5,4] => [1,2,3,6,5,7,4] => [4,7,5,6,3,2,1] => [1,7,2,6,5,4,3] => ? = 2
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [7,6,5,3,4,2,1] => [7,6,5,1,4,3,2] => ? = 1
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [6,7,5,3,4,2,1] => [2,7,6,1,5,4,3] => ? = 2
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [7,5,6,3,4,2,1] => [7,2,6,1,5,4,3] => ? = 2
[1,2,4,3,7,5,6] => [1,2,4,3,7,5,6] => [6,5,7,3,4,2,1] => [3,2,7,1,6,5,4] => ? = 2
[1,2,4,3,7,6,5] => [1,2,4,3,6,7,5] => [5,7,6,3,4,2,1] => [2,7,6,1,5,4,3] => ? = 2
[1,2,4,5,3,6,7] => [1,2,5,4,3,6,7] => [7,6,3,4,5,2,1] => [7,6,1,2,5,4,3] => ? = 1
[1,2,4,5,3,7,6] => [1,2,5,4,3,7,6] => [6,7,3,4,5,2,1] => [3,7,1,2,6,5,4] => ? = 2
[1,2,4,5,6,3,7] => [1,2,6,5,4,3,7] => [7,3,4,5,6,2,1] => [7,1,2,3,6,5,4] => ? = 1
[1,2,4,5,7,3,6] => [1,2,7,5,4,3,6] => [6,3,4,5,7,2,1] => [4,1,2,3,7,6,5] => ? = 1
[1,2,4,6,3,5,7] => [1,2,6,4,3,5,7] => [7,5,3,4,6,2,1] => [7,3,1,2,6,5,4] => ? = 1
[1,2,4,6,3,7,5] => [1,2,7,6,4,3,5] => [5,3,4,6,7,2,1] => [3,1,2,4,7,6,5] => ? = 1
[1,2,4,6,5,3,7] => [1,2,5,6,4,3,7] => [7,3,4,6,5,2,1] => [7,1,2,6,5,4,3] => ? = 1
[1,2,4,6,7,3,5] => [1,2,7,4,3,6,5] => [5,6,3,4,7,2,1] => [2,4,1,3,7,6,5] => ? = 2
[1,2,4,7,3,5,6] => [1,2,7,4,3,5,6] => [6,5,3,4,7,2,1] => [4,3,1,2,7,6,5] => ? = 1
[1,2,4,7,3,6,5] => [1,2,6,7,4,3,5] => [5,3,4,7,6,2,1] => [3,1,2,7,6,5,4] => ? = 1
[1,2,4,7,5,3,6] => [1,2,5,7,4,3,6] => [6,3,4,7,5,2,1] => [3,1,2,7,6,5,4] => ? = 1
[1,2,4,7,6,3,5] => [1,2,6,4,3,7,5] => [5,7,3,4,6,2,1] => [3,7,1,2,6,5,4] => ? = 2
[1,2,5,3,4,6,7] => [1,2,5,3,4,6,7] => [7,6,4,3,5,2,1] => [7,6,2,1,5,4,3] => ? = 1
[1,2,5,3,4,7,6] => [1,2,5,3,4,7,6] => [6,7,4,3,5,2,1] => [3,7,2,1,6,5,4] => ? = 2
[1,2,5,3,6,4,7] => [1,2,6,5,3,4,7] => [7,4,3,5,6,2,1] => [7,2,1,3,6,5,4] => ? = 1
[1,2,5,3,6,7,4] => [1,2,7,6,5,3,4] => [4,3,5,6,7,2,1] => [2,1,3,4,7,6,5] => ? = 1
[1,2,5,3,7,4,6] => [1,2,7,5,3,4,6] => [6,4,3,5,7,2,1] => [4,2,1,3,7,6,5] => ? = 1
[1,2,5,3,7,6,4] => [1,2,6,7,5,3,4] => [4,3,5,7,6,2,1] => [2,1,3,7,6,5,4] => ? = 1
[1,2,5,4,3,6,7] => [1,2,4,5,3,6,7] => [7,6,3,5,4,2,1] => [7,6,1,5,4,3,2] => ? = 1
[1,2,5,4,3,7,6] => [1,2,4,5,3,7,6] => [6,7,3,5,4,2,1] => [2,7,1,6,5,4,3] => ? = 2
[1,2,5,4,6,3,7] => [1,2,4,6,5,3,7] => [7,3,5,6,4,2,1] => [7,1,2,6,5,4,3] => ? = 1
[1,2,5,4,7,3,6] => [1,2,4,7,5,3,6] => [6,3,5,7,4,2,1] => [3,1,2,7,6,5,4] => ? = 1
[1,2,5,6,3,4,7] => [1,2,6,3,5,4,7] => [7,4,5,3,6,2,1] => [7,1,3,2,6,5,4] => ? = 2
[1,2,5,6,4,3,7] => [1,2,6,4,5,3,7] => [7,3,5,4,6,2,1] => [7,1,3,2,6,5,4] => ? = 2
[1,2,5,7,3,4,6] => [1,2,7,3,5,4,6] => [6,4,5,3,7,2,1] => [4,1,3,2,7,6,5] => ? = 2
[1,2,5,7,4,3,6] => [1,2,7,4,5,3,6] => [6,3,5,4,7,2,1] => [4,1,3,2,7,6,5] => ? = 2
[1,2,6,3,4,5,7] => [1,2,6,3,4,5,7] => [7,5,4,3,6,2,1] => [7,3,2,1,6,5,4] => ? = 1
[1,2,6,3,4,7,5] => [1,2,7,6,3,4,5] => [5,4,3,6,7,2,1] => [3,2,1,4,7,6,5] => ? = 1
[1,2,6,3,5,4,7] => [1,2,5,6,3,4,7] => [7,4,3,6,5,2,1] => [7,2,1,6,5,4,3] => ? = 1
[1,2,6,3,5,7,4] => [1,2,5,7,6,3,4] => [4,3,6,7,5,2,1] => [2,1,3,7,6,5,4] => ? = 1
[1,2,6,3,7,5,4] => [1,2,7,5,6,3,4] => [4,3,6,5,7,2,1] => [2,1,4,3,7,6,5] => ? = 2
[1,2,6,4,3,5,7] => [1,2,4,6,3,5,7] => [7,5,3,6,4,2,1] => [7,2,1,6,5,4,3] => ? = 1
[1,2,6,4,3,7,5] => [1,2,4,7,6,3,5] => [5,3,6,7,4,2,1] => [2,1,3,7,6,5,4] => ? = 1
[1,2,6,4,5,3,7] => [1,2,4,5,6,3,7] => [7,3,6,5,4,2,1] => [7,1,6,5,4,3,2] => ? = 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: St000703
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00176: Set partitions —rotate decreasing⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000703: Permutations ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 100%
Mp00176: Set partitions —rotate decreasing⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000703: Permutations ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 100%
Values
[1] => {{1}}
=> {{1}}
=> [1] => 0
[1,2] => {{1},{2}}
=> {{1},{2}}
=> [1,2] => 0
[2,1] => {{1,2}}
=> {{1,2}}
=> [2,1] => 1
[1,2,3] => {{1},{2},{3}}
=> {{1},{2},{3}}
=> [1,2,3] => 0
[1,3,2] => {{1},{2,3}}
=> {{1,2},{3}}
=> [2,1,3] => 1
[2,1,3] => {{1,2},{3}}
=> {{1,3},{2}}
=> [3,2,1] => 1
[2,3,1] => {{1,2,3}}
=> {{1,2,3}}
=> [2,3,1] => 1
[3,1,2] => {{1,3},{2}}
=> {{1},{2,3}}
=> [1,3,2] => 1
[3,2,1] => {{1,3},{2}}
=> {{1},{2,3}}
=> [1,3,2] => 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},{2,3},{4}}
=> [1,3,2,4] => 1
[1,3,2,4] => {{1},{2,3},{4}}
=> {{1,2},{3},{4}}
=> [2,1,3,4] => 1
[1,3,4,2] => {{1},{2,3,4}}
=> {{1,2,3},{4}}
=> [2,3,1,4] => 1
[1,4,2,3] => {{1},{2,4},{3}}
=> {{1,3},{2},{4}}
=> [3,2,1,4] => 1
[1,4,3,2] => {{1},{2,4},{3}}
=> {{1,3},{2},{4}}
=> [3,2,1,4] => 1
[2,1,3,4] => {{1,2},{3},{4}}
=> {{1,4},{2},{3}}
=> [4,2,3,1] => 1
[2,1,4,3] => {{1,2},{3,4}}
=> {{1,4},{2,3}}
=> [4,3,2,1] => 2
[2,3,1,4] => {{1,2,3},{4}}
=> {{1,2,4},{3}}
=> [2,4,3,1] => 1
[2,3,4,1] => {{1,2,3,4}}
=> {{1,2,3,4}}
=> [2,3,4,1] => 1
[2,4,1,3] => {{1,2,4},{3}}
=> {{1,3,4},{2}}
=> [3,2,4,1] => 1
[2,4,3,1] => {{1,2,4},{3}}
=> {{1,3,4},{2}}
=> [3,2,4,1] => 1
[3,1,2,4] => {{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> [1,4,3,2] => 1
[3,1,4,2] => {{1,3,4},{2}}
=> {{1},{2,3,4}}
=> [1,3,4,2] => 1
[3,2,1,4] => {{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> [1,4,3,2] => 1
[3,2,4,1] => {{1,3,4},{2}}
=> {{1},{2,3,4}}
=> [1,3,4,2] => 1
[3,4,1,2] => {{1,3},{2,4}}
=> {{1,3},{2,4}}
=> [3,4,1,2] => 2
[3,4,2,1] => {{1,3},{2,4}}
=> {{1,3},{2,4}}
=> [3,4,1,2] => 2
[4,1,2,3] => {{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> [1,2,4,3] => 1
[4,1,3,2] => {{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> [1,2,4,3] => 1
[4,2,1,3] => {{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> [1,2,4,3] => 1
[4,2,3,1] => {{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> [1,2,4,3] => 1
[4,3,1,2] => {{1,4},{2,3}}
=> {{1,2},{3,4}}
=> [2,1,4,3] => 2
[4,3,2,1] => {{1,4},{2,3}}
=> {{1,2},{3,4}}
=> [2,1,4,3] => 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},{3,4},{5}}
=> [1,2,4,3,5] => 1
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> {{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => 1
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> {{1},{2,3,4},{5}}
=> [1,3,4,2,5] => 1
[1,2,5,3,4] => {{1},{2},{3,5},{4}}
=> {{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => 1
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> {{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => 1
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => 1
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> {{1,2},{3,4},{5}}
=> [2,1,4,3,5] => 2
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> {{1,2,3},{4},{5}}
=> [2,3,1,4,5] => 1
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> {{1,2,3,4},{5}}
=> [2,3,4,1,5] => 1
[1,3,5,2,4] => {{1},{2,3,5},{4}}
=> {{1,2,4},{3},{5}}
=> [2,4,3,1,5] => 1
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> {{1,2,4},{3},{5}}
=> [2,4,3,1,5] => 1
[1,4,2,3,5] => {{1},{2,4},{3},{5}}
=> {{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => 1
[1,4,2,5,3] => {{1},{2,4,5},{3}}
=> {{1,3,4},{2},{5}}
=> [3,2,4,1,5] => 1
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> {{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => 1
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> {{1,3,4},{2},{5}}
=> [3,2,4,1,5] => 1
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> {{1,3},{2,4},{5}}
=> [3,4,1,2,5] => 2
[1,2,6,3,4,5,7] => {{1},{2},{3,6},{4},{5},{7}}
=> {{1},{2,5},{3},{4},{6},{7}}
=> [1,5,3,4,2,6,7] => ? = 1
[1,2,6,3,4,7,5] => {{1},{2},{3,6,7},{4},{5}}
=> {{1},{2,5,6},{3},{4},{7}}
=> [1,5,3,4,6,2,7] => ? = 1
[1,2,6,3,5,4,7] => {{1},{2},{3,6},{4},{5},{7}}
=> {{1},{2,5},{3},{4},{6},{7}}
=> [1,5,3,4,2,6,7] => ? = 1
[1,2,6,3,5,7,4] => {{1},{2},{3,6,7},{4},{5}}
=> {{1},{2,5,6},{3},{4},{7}}
=> [1,5,3,4,6,2,7] => ? = 1
[1,2,6,3,7,4,5] => {{1},{2},{3,6},{4},{5,7}}
=> {{1},{2,5},{3},{4,6},{7}}
=> [1,5,3,6,2,4,7] => ? = 2
[1,2,6,3,7,5,4] => {{1},{2},{3,6},{4},{5,7}}
=> {{1},{2,5},{3},{4,6},{7}}
=> [1,5,3,6,2,4,7] => ? = 2
[1,2,6,4,3,5,7] => {{1},{2},{3,6},{4},{5},{7}}
=> {{1},{2,5},{3},{4},{6},{7}}
=> [1,5,3,4,2,6,7] => ? = 1
[1,2,6,4,3,7,5] => {{1},{2},{3,6,7},{4},{5}}
=> {{1},{2,5,6},{3},{4},{7}}
=> [1,5,3,4,6,2,7] => ? = 1
[1,2,6,4,5,3,7] => {{1},{2},{3,6},{4},{5},{7}}
=> {{1},{2,5},{3},{4},{6},{7}}
=> [1,5,3,4,2,6,7] => ? = 1
[1,2,6,4,5,7,3] => {{1},{2},{3,6,7},{4},{5}}
=> {{1},{2,5,6},{3},{4},{7}}
=> [1,5,3,4,6,2,7] => ? = 1
[1,2,6,4,7,3,5] => {{1},{2},{3,6},{4},{5,7}}
=> {{1},{2,5},{3},{4,6},{7}}
=> [1,5,3,6,2,4,7] => ? = 2
[1,2,6,4,7,5,3] => {{1},{2},{3,6},{4},{5,7}}
=> {{1},{2,5},{3},{4,6},{7}}
=> [1,5,3,6,2,4,7] => ? = 2
[1,2,6,5,3,4,7] => {{1},{2},{3,6},{4,5},{7}}
=> {{1},{2,5},{3,4},{6},{7}}
=> [1,5,4,3,2,6,7] => ? = 2
[1,2,6,5,3,7,4] => {{1},{2},{3,6,7},{4,5}}
=> {{1},{2,5,6},{3,4},{7}}
=> [1,5,4,3,6,2,7] => ? = 2
[1,2,6,5,4,3,7] => {{1},{2},{3,6},{4,5},{7}}
=> {{1},{2,5},{3,4},{6},{7}}
=> [1,5,4,3,2,6,7] => ? = 2
[1,2,6,5,4,7,3] => {{1},{2},{3,6,7},{4,5}}
=> {{1},{2,5,6},{3,4},{7}}
=> [1,5,4,3,6,2,7] => ? = 2
[1,2,6,5,7,3,4] => {{1},{2},{3,6},{4,5,7}}
=> {{1},{2,5},{3,4,6},{7}}
=> [1,5,4,6,2,3,7] => ? = 2
[1,2,6,5,7,4,3] => {{1},{2},{3,6},{4,5,7}}
=> {{1},{2,5},{3,4,6},{7}}
=> [1,5,4,6,2,3,7] => ? = 2
[1,2,6,7,3,4,5] => {{1},{2},{3,6},{4,7},{5}}
=> {{1},{2,5},{3,6},{4},{7}}
=> [1,5,6,4,2,3,7] => ? = 2
[1,2,6,7,3,5,4] => {{1},{2},{3,6},{4,7},{5}}
=> {{1},{2,5},{3,6},{4},{7}}
=> [1,5,6,4,2,3,7] => ? = 2
[1,2,6,7,4,3,5] => {{1},{2},{3,6},{4,7},{5}}
=> {{1},{2,5},{3,6},{4},{7}}
=> [1,5,6,4,2,3,7] => ? = 2
[1,2,6,7,4,5,3] => {{1},{2},{3,6},{4,7},{5}}
=> {{1},{2,5},{3,6},{4},{7}}
=> [1,5,6,4,2,3,7] => ? = 2
[1,2,6,7,5,3,4] => {{1},{2},{3,6},{4,7},{5}}
=> {{1},{2,5},{3,6},{4},{7}}
=> [1,5,6,4,2,3,7] => ? = 2
[1,2,6,7,5,4,3] => {{1},{2},{3,6},{4,7},{5}}
=> {{1},{2,5},{3,6},{4},{7}}
=> [1,5,6,4,2,3,7] => ? = 2
[1,2,7,3,4,5,6] => {{1},{2},{3,7},{4},{5},{6}}
=> {{1},{2,6},{3},{4},{5},{7}}
=> [1,6,3,4,5,2,7] => ? = 1
[1,2,7,3,4,6,5] => {{1},{2},{3,7},{4},{5},{6}}
=> {{1},{2,6},{3},{4},{5},{7}}
=> [1,6,3,4,5,2,7] => ? = 1
[1,2,7,3,5,4,6] => {{1},{2},{3,7},{4},{5},{6}}
=> {{1},{2,6},{3},{4},{5},{7}}
=> [1,6,3,4,5,2,7] => ? = 1
[1,2,7,3,5,6,4] => {{1},{2},{3,7},{4},{5},{6}}
=> {{1},{2,6},{3},{4},{5},{7}}
=> [1,6,3,4,5,2,7] => ? = 1
[1,2,7,3,6,4,5] => {{1},{2},{3,7},{4},{5,6}}
=> {{1},{2,6},{3},{4,5},{7}}
=> [1,6,3,5,4,2,7] => ? = 2
[1,2,7,3,6,5,4] => {{1},{2},{3,7},{4},{5,6}}
=> {{1},{2,6},{3},{4,5},{7}}
=> [1,6,3,5,4,2,7] => ? = 2
[1,2,7,4,3,5,6] => {{1},{2},{3,7},{4},{5},{6}}
=> {{1},{2,6},{3},{4},{5},{7}}
=> [1,6,3,4,5,2,7] => ? = 1
[1,2,7,4,3,6,5] => {{1},{2},{3,7},{4},{5},{6}}
=> {{1},{2,6},{3},{4},{5},{7}}
=> [1,6,3,4,5,2,7] => ? = 1
[1,2,7,4,5,3,6] => {{1},{2},{3,7},{4},{5},{6}}
=> {{1},{2,6},{3},{4},{5},{7}}
=> [1,6,3,4,5,2,7] => ? = 1
[1,2,7,4,5,6,3] => {{1},{2},{3,7},{4},{5},{6}}
=> {{1},{2,6},{3},{4},{5},{7}}
=> [1,6,3,4,5,2,7] => ? = 1
[1,2,7,4,6,3,5] => {{1},{2},{3,7},{4},{5,6}}
=> {{1},{2,6},{3},{4,5},{7}}
=> [1,6,3,5,4,2,7] => ? = 2
[1,2,7,4,6,5,3] => {{1},{2},{3,7},{4},{5,6}}
=> {{1},{2,6},{3},{4,5},{7}}
=> [1,6,3,5,4,2,7] => ? = 2
[1,2,7,5,3,4,6] => {{1},{2},{3,7},{4,5},{6}}
=> {{1},{2,6},{3,4},{5},{7}}
=> [1,6,4,3,5,2,7] => ? = 2
[1,2,7,5,3,6,4] => {{1},{2},{3,7},{4,5},{6}}
=> {{1},{2,6},{3,4},{5},{7}}
=> [1,6,4,3,5,2,7] => ? = 2
[1,2,7,5,4,3,6] => {{1},{2},{3,7},{4,5},{6}}
=> {{1},{2,6},{3,4},{5},{7}}
=> [1,6,4,3,5,2,7] => ? = 2
[1,2,7,5,4,6,3] => {{1},{2},{3,7},{4,5},{6}}
=> {{1},{2,6},{3,4},{5},{7}}
=> [1,6,4,3,5,2,7] => ? = 2
[1,2,7,5,6,3,4] => {{1},{2},{3,7},{4,5,6}}
=> {{1},{2,6},{3,4,5},{7}}
=> [1,6,4,5,3,2,7] => ? = 2
[1,2,7,5,6,4,3] => {{1},{2},{3,7},{4,5,6}}
=> {{1},{2,6},{3,4,5},{7}}
=> [1,6,4,5,3,2,7] => ? = 2
[1,2,7,6,3,4,5] => {{1},{2},{3,7},{4,6},{5}}
=> {{1},{2,6},{3,5},{4},{7}}
=> [1,6,5,4,3,2,7] => ? = 2
[1,2,7,6,3,5,4] => {{1},{2},{3,7},{4,6},{5}}
=> {{1},{2,6},{3,5},{4},{7}}
=> [1,6,5,4,3,2,7] => ? = 2
[1,2,7,6,4,3,5] => {{1},{2},{3,7},{4,6},{5}}
=> {{1},{2,6},{3,5},{4},{7}}
=> [1,6,5,4,3,2,7] => ? = 2
[1,2,7,6,4,5,3] => {{1},{2},{3,7},{4,6},{5}}
=> {{1},{2,6},{3,5},{4},{7}}
=> [1,6,5,4,3,2,7] => ? = 2
[1,2,7,6,5,3,4] => {{1},{2},{3,7},{4,6},{5}}
=> {{1},{2,6},{3,5},{4},{7}}
=> [1,6,5,4,3,2,7] => ? = 2
[1,2,7,6,5,4,3] => {{1},{2},{3,7},{4,6},{5}}
=> {{1},{2,6},{3,5},{4},{7}}
=> [1,6,5,4,3,2,7] => ? = 2
[1,3,2,4,5,7,6] => {{1},{2,3},{4},{5},{6,7}}
=> {{1,2},{3},{4},{5,6},{7}}
=> [2,1,3,4,6,5,7] => ? = 2
[1,3,2,4,6,5,7] => {{1},{2,3},{4},{5,6},{7}}
=> {{1,2},{3},{4,5},{6},{7}}
=> [2,1,3,5,4,6,7] => ? = 2
Description
The number of deficiencies of a permutation.
This is defined as
$$\operatorname{dec}(\sigma)=\#\{i:\sigma(i) < i\}.$$
The number of exceedances is [[St000155]].
The following 27 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000884The number of isolated descents of a permutation. 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. St000035The number of left outer peaks of 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. 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. St000325The width of the tree associated to a permutation. 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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