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Your data matches 50 different statistics following compositions of up to 3 maps.
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Matching statistic: St000668
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [1,1,1]
=> [1,1]
=> [2]
=> 2
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [3]
=> 3
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [2]
=> 2
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [2]
=> 2
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [2]
=> 2
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [2]
=> 2
[2,1,4,3] => [2,2]
=> [2]
=> [1,1]
=> 1
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [2]
=> 2
[3,4,1,2] => [2,2]
=> [2]
=> [1,1]
=> 1
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [2]
=> 2
[4,3,2,1] => [2,2]
=> [2]
=> [1,1]
=> 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 4
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [3]
=> 3
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [3]
=> 3
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> [2]
=> 2
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> [2]
=> 2
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> [3]
=> 3
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [3]
=> 3
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [2,1]
=> 2
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> [2]
=> 2
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [2]
=> 2
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> [2]
=> 2
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> [3]
=> 3
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [2]
=> 2
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> [2,1]
=> 2
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [2]
=> 2
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [2]
=> 2
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> [3]
=> 3
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> [2,1]
=> 2
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> [3]
=> 3
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> [2,1]
=> 2
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> [2,1]
=> 2
[2,1,4,5,3] => [3,2]
=> [2]
=> [1,1]
=> 1
[2,1,5,3,4] => [3,2]
=> [2]
=> [1,1]
=> 1
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> [2,1]
=> 2
[2,3,1,4,5] => [3,1,1]
=> [1,1]
=> [2]
=> 2
[2,3,1,5,4] => [3,2]
=> [2]
=> [1,1]
=> 1
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> [2]
=> 2
[2,4,5,1,3] => [3,2]
=> [2]
=> [1,1]
=> 1
[2,5,3,4,1] => [3,1,1]
=> [1,1]
=> [2]
=> 2
[2,5,4,3,1] => [3,2]
=> [2]
=> [1,1]
=> 1
[3,1,2,4,5] => [3,1,1]
=> [1,1]
=> [2]
=> 2
[3,1,2,5,4] => [3,2]
=> [2]
=> [1,1]
=> 1
[3,2,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> [3]
=> 3
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> [2,1]
=> 2
[3,2,4,1,5] => [3,1,1]
=> [1,1]
=> [2]
=> 2
[3,2,5,4,1] => [3,1,1]
=> [1,1]
=> [2]
=> 2
[3,4,1,2,5] => [2,2,1]
=> [2,1]
=> [2,1]
=> 2
[3,4,1,5,2] => [3,2]
=> [2]
=> [1,1]
=> 1
[3,4,5,2,1] => [3,2]
=> [2]
=> [1,1]
=> 1
Description
The least common multiple of the parts of the partition.
Matching statistic: St000292
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 31%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 31%
Values
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 101010 => 2
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 10101010 => 3
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 10101100 => 2
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 10110010 => 2
[1,4,3,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 10110100 => 2
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 11001010 => 2
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 11001100 => 1
[3,2,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 11010010 => 2
[3,4,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 11100100 => 1
[4,2,3,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 11010100 => 2
[4,3,2,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 11100100 => 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => 4
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => 3
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 1010110010 => 3
[1,2,4,5,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => 2
[1,2,5,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => 2
[1,2,5,4,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => 3
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 1011001010 => 3
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 1011001100 => 2
[1,3,4,2,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 1011100010 => 2
[1,3,5,4,2] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => 2
[1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 1011100010 => 2
[1,4,3,2,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 1011010010 => 3
[1,4,3,5,2] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => 2
[1,4,5,2,3] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => 2
[1,5,2,4,3] => [1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => 2
[1,5,3,2,4] => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => 2
[1,5,3,4,2] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => 3
[1,5,4,3,2] => [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => 2
[2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 1100101010 => 3
[2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> 1100101100 => 2
[2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> 1100110010 => 2
[2,1,4,5,3] => [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => 1
[2,1,5,3,4] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => 1
[2,1,5,4,3] => [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> 1100110100 => 2
[2,3,1,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 1110001010 => 2
[2,3,1,5,4] => [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => 1
[2,4,3,1,5] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => 2
[2,4,5,1,3] => [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> 1111000100 => 1
[2,5,3,4,1] => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 1110101000 => 2
[2,5,4,3,1] => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> 1111001000 => 1
[3,1,2,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 1110001010 => 2
[3,1,2,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => 1
[3,2,1,4,5] => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> 1101001010 => 3
[3,2,1,5,4] => [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> 1101001100 => 2
[3,2,4,1,5] => [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => 2
[3,2,5,4,1] => [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> 1101101000 => 2
[3,4,1,2,5] => [3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => 2
[3,4,1,5,2] => [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> 1110011000 => 1
[3,4,5,2,1] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 1111001000 => 1
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 10101010101010 => ? = 6
[1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 10101010101100 => ? = 5
[1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> 10101010110010 => ? = 5
[1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 10101010111000 => ? = 4
[1,2,3,4,7,5,6] => [1,2,3,4,7,6,5] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 10101010111000 => ? = 4
[1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 10101010110100 => ? = 5
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> 10101011001010 => ? = 5
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> 10101011001100 => ? = 4
[1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> 10101011100010 => ? = 4
[1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> 10101011110000 => ? = 3
[1,2,3,5,7,4,6] => [1,2,3,7,6,4,5] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> 10101011110000 => ? = 3
[1,2,3,5,7,6,4] => [1,2,3,6,7,4,5] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> 10101011101000 => ? = 4
[1,2,3,6,4,5,7] => [1,2,3,6,5,4,7] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> 10101011100010 => ? = 4
[1,2,3,6,4,7,5] => [1,2,3,7,5,4,6] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> 10101011110000 => ? = 3
[1,2,3,6,5,4,7] => [1,2,3,5,6,4,7] => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> 10101011010010 => ? = 5
[1,2,3,6,5,7,4] => [1,2,3,5,7,4,6] => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> 10101011011000 => ? = 4
[1,2,3,6,7,4,5] => [1,2,3,6,4,7,5] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> 10101011100100 => ? = 4
[1,2,3,6,7,5,4] => [1,2,3,7,4,6,5] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> 10101011110000 => ? = 3
[1,2,3,7,4,5,6] => [1,2,3,7,6,5,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> 10101011110000 => ? = 3
[1,2,3,7,4,6,5] => [1,2,3,6,7,5,4] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> 10101011101000 => ? = 4
[1,2,3,7,5,4,6] => [1,2,3,5,7,6,4] => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> 10101011011000 => ? = 4
[1,2,3,7,5,6,4] => [1,2,3,5,6,7,4] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> 10101011010100 => ? = 5
[1,2,3,7,6,4,5] => [1,2,3,7,5,6,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> 10101011110000 => ? = 3
[1,2,3,7,6,5,4] => [1,2,3,6,5,7,4] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> 10101011100100 => ? = 4
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> 10101100101010 => ? = 5
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> 10101100101100 => ? = 4
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> 10101100110010 => ? = 4
[1,2,4,3,6,7,5] => [1,2,4,3,7,5,6] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> 10101100111000 => ? = 3
[1,2,4,3,7,5,6] => [1,2,4,3,7,6,5] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> 10101100111000 => ? = 3
[1,2,4,3,7,6,5] => [1,2,4,3,6,7,5] => [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> 10101100110100 => ? = 4
[1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> 10101110001010 => ? = 4
[1,2,4,5,3,7,6] => [1,2,5,3,4,7,6] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> 10101110001100 => ? = 3
[1,2,4,5,6,3,7] => [1,2,6,3,4,5,7] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> 10101111000010 => ? = 3
[1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> 10101111100000 => ? = 2
[1,2,4,5,7,3,6] => [1,2,7,6,3,4,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> 10101111100000 => ? = 2
[1,2,4,5,7,6,3] => [1,2,6,7,3,4,5] => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> 10101111010000 => ? = 3
[1,2,4,6,3,5,7] => [1,2,6,5,3,4,7] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> 10101111000010 => ? = 3
[1,2,4,6,3,7,5] => [1,2,7,5,3,4,6] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> 10101111100000 => ? = 2
[1,2,4,6,5,3,7] => [1,2,5,6,3,4,7] => [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> 10101110100010 => ? = 4
[1,2,4,6,5,7,3] => [1,2,5,7,3,4,6] => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> 10101110110000 => ? = 3
[1,2,4,6,7,3,5] => [1,2,6,3,4,7,5] => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> 10101111000100 => ? = 3
[1,2,4,6,7,5,3] => [1,2,7,3,4,6,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> 10101111100000 => ? = 2
[1,2,4,7,3,5,6] => [1,2,7,6,5,3,4] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> 10101111100000 => ? = 2
[1,2,4,7,3,6,5] => [1,2,6,7,5,3,4] => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> 10101111010000 => ? = 3
[1,2,4,7,5,3,6] => [1,2,5,7,6,3,4] => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> 10101110110000 => ? = 3
[1,2,4,7,5,6,3] => [1,2,5,6,7,3,4] => [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> 10101110101000 => ? = 4
[1,2,4,7,6,3,5] => [1,2,7,5,6,3,4] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> 10101111100000 => ? = 2
[1,2,4,7,6,5,3] => [1,2,6,5,7,3,4] => [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> 10101111001000 => ? = 3
[1,2,5,3,4,6,7] => [1,2,5,4,3,6,7] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> 10101110001010 => ? = 4
[1,2,5,3,4,7,6] => [1,2,5,4,3,7,6] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> 10101110001100 => ? = 3
Description
The number of ascents of a binary word.
Matching statistic: St000157
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 31%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 31%
Values
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 3 = 2 + 1
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 4 = 3 + 1
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 3 = 2 + 1
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 3 = 2 + 1
[1,4,3,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 3 = 2 + 1
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 3 = 2 + 1
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> 2 = 1 + 1
[3,2,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 3 = 2 + 1
[3,4,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> 2 = 1 + 1
[4,2,3,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> 3 = 2 + 1
[4,3,2,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> 2 = 1 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> 5 = 4 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,8],[2,4,6,9,10]]
=> 4 = 3 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [[1,3,5,6,9],[2,4,7,8,10]]
=> 4 = 3 + 1
[1,2,4,5,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> 3 = 2 + 1
[1,2,5,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> 3 = 2 + 1
[1,2,5,4,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,6,8],[2,4,7,9,10]]
=> 4 = 3 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [[1,3,4,7,9],[2,5,6,8,10]]
=> 4 = 3 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [[1,3,4,7,8],[2,5,6,9,10]]
=> 3 = 2 + 1
[1,3,4,2,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [[1,3,4,5,9],[2,6,7,8,10]]
=> 3 = 2 + 1
[1,3,5,4,2] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> 3 = 2 + 1
[1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [[1,3,4,5,9],[2,6,7,8,10]]
=> 3 = 2 + 1
[1,4,3,2,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [[1,3,4,6,9],[2,5,7,8,10]]
=> 4 = 3 + 1
[1,4,3,5,2] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [[1,3,4,6,7],[2,5,8,9,10]]
=> 3 = 2 + 1
[1,4,5,2,3] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> 3 = 2 + 1
[1,5,2,4,3] => [1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> 3 = 2 + 1
[1,5,3,2,4] => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [[1,3,4,6,7],[2,5,8,9,10]]
=> 3 = 2 + 1
[1,5,3,4,2] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [[1,3,4,6,8],[2,5,7,9,10]]
=> 4 = 3 + 1
[1,5,4,3,2] => [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> 3 = 2 + 1
[2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [[1,2,5,7,9],[3,4,6,8,10]]
=> 4 = 3 + 1
[2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [[1,2,5,7,8],[3,4,6,9,10]]
=> 3 = 2 + 1
[2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [[1,2,5,6,9],[3,4,7,8,10]]
=> 3 = 2 + 1
[2,1,4,5,3] => [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [[1,2,5,6,7],[3,4,8,9,10]]
=> 2 = 1 + 1
[2,1,5,3,4] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [[1,2,5,6,7],[3,4,8,9,10]]
=> 2 = 1 + 1
[2,1,5,4,3] => [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [[1,2,5,6,8],[3,4,7,9,10]]
=> 3 = 2 + 1
[2,3,1,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [[1,2,3,7,9],[4,5,6,8,10]]
=> 3 = 2 + 1
[2,3,1,5,4] => [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> 2 = 1 + 1
[2,4,3,1,5] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> 3 = 2 + 1
[2,4,5,1,3] => [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> [[1,2,3,4,8],[5,6,7,9,10]]
=> 2 = 1 + 1
[2,5,3,4,1] => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> [[1,2,3,5,7],[4,6,8,9,10]]
=> 3 = 2 + 1
[2,5,4,3,1] => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [[1,2,3,4,7],[5,6,8,9,10]]
=> 2 = 1 + 1
[3,1,2,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [[1,2,3,7,9],[4,5,6,8,10]]
=> 3 = 2 + 1
[3,1,2,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> 2 = 1 + 1
[3,2,1,4,5] => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [[1,2,4,7,9],[3,5,6,8,10]]
=> 4 = 3 + 1
[3,2,1,5,4] => [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [[1,2,4,7,8],[3,5,6,9,10]]
=> 3 = 2 + 1
[3,2,4,1,5] => [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> 3 = 2 + 1
[3,2,5,4,1] => [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> [[1,2,4,5,7],[3,6,8,9,10]]
=> 3 = 2 + 1
[3,4,1,2,5] => [3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> 3 = 2 + 1
[3,4,1,5,2] => [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> [[1,2,3,6,7],[4,5,8,9,10]]
=> 2 = 1 + 1
[3,4,5,2,1] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> [[1,2,3,4,7],[5,6,8,9,10]]
=> 2 = 1 + 1
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9,11,13],[2,4,6,8,10,12,14]]
=> ? = 6 + 1
[1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,9,11,12],[2,4,6,8,10,13,14]]
=> ? = 5 + 1
[1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [[1,3,5,7,9,10,13],[2,4,6,8,11,12,14]]
=> ? = 5 + 1
[1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,7,9,10,11],[2,4,6,8,12,13,14]]
=> ? = 4 + 1
[1,2,3,4,7,5,6] => [1,2,3,4,7,6,5] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,7,9,10,11],[2,4,6,8,12,13,14]]
=> ? = 4 + 1
[1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,7,9,10,12],[2,4,6,8,11,13,14]]
=> ? = 5 + 1
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [[1,3,5,7,8,11,13],[2,4,6,9,10,12,14]]
=> ? = 5 + 1
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [[1,3,5,7,8,11,12],[2,4,6,9,10,13,14]]
=> ? = 4 + 1
[1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [[1,3,5,7,8,9,13],[2,4,6,10,11,12,14]]
=> ? = 4 + 1
[1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [[1,3,5,7,8,9,10],[2,4,6,11,12,13,14]]
=> ? = 3 + 1
[1,2,3,5,7,4,6] => [1,2,3,7,6,4,5] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [[1,3,5,7,8,9,10],[2,4,6,11,12,13,14]]
=> ? = 3 + 1
[1,2,3,5,7,6,4] => [1,2,3,6,7,4,5] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [[1,3,5,7,8,9,11],[2,4,6,10,12,13,14]]
=> ? = 4 + 1
[1,2,3,6,4,5,7] => [1,2,3,6,5,4,7] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [[1,3,5,7,8,9,13],[2,4,6,10,11,12,14]]
=> ? = 4 + 1
[1,2,3,6,4,7,5] => [1,2,3,7,5,4,6] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [[1,3,5,7,8,9,10],[2,4,6,11,12,13,14]]
=> ? = 3 + 1
[1,2,3,6,5,4,7] => [1,2,3,5,6,4,7] => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [[1,3,5,7,8,10,13],[2,4,6,9,11,12,14]]
=> ? = 5 + 1
[1,2,3,6,5,7,4] => [1,2,3,5,7,4,6] => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [[1,3,5,7,8,10,11],[2,4,6,9,12,13,14]]
=> ? = 4 + 1
[1,2,3,6,7,4,5] => [1,2,3,6,4,7,5] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [[1,3,5,7,8,9,12],[2,4,6,10,11,13,14]]
=> ? = 4 + 1
[1,2,3,6,7,5,4] => [1,2,3,7,4,6,5] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [[1,3,5,7,8,9,10],[2,4,6,11,12,13,14]]
=> ? = 3 + 1
[1,2,3,7,4,5,6] => [1,2,3,7,6,5,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [[1,3,5,7,8,9,10],[2,4,6,11,12,13,14]]
=> ? = 3 + 1
[1,2,3,7,4,6,5] => [1,2,3,6,7,5,4] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [[1,3,5,7,8,9,11],[2,4,6,10,12,13,14]]
=> ? = 4 + 1
[1,2,3,7,5,4,6] => [1,2,3,5,7,6,4] => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [[1,3,5,7,8,10,11],[2,4,6,9,12,13,14]]
=> ? = 4 + 1
[1,2,3,7,5,6,4] => [1,2,3,5,6,7,4] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [[1,3,5,7,8,10,12],[2,4,6,9,11,13,14]]
=> ? = 5 + 1
[1,2,3,7,6,4,5] => [1,2,3,7,5,6,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [[1,3,5,7,8,9,10],[2,4,6,11,12,13,14]]
=> ? = 3 + 1
[1,2,3,7,6,5,4] => [1,2,3,6,5,7,4] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [[1,3,5,7,8,9,12],[2,4,6,10,11,13,14]]
=> ? = 4 + 1
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [[1,3,5,6,9,11,13],[2,4,7,8,10,12,14]]
=> ? = 5 + 1
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [[1,3,5,6,9,11,12],[2,4,7,8,10,13,14]]
=> ? = 4 + 1
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [[1,3,5,6,9,10,13],[2,4,7,8,11,12,14]]
=> ? = 4 + 1
[1,2,4,3,6,7,5] => [1,2,4,3,7,5,6] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [[1,3,5,6,9,10,11],[2,4,7,8,12,13,14]]
=> ? = 3 + 1
[1,2,4,3,7,5,6] => [1,2,4,3,7,6,5] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [[1,3,5,6,9,10,11],[2,4,7,8,12,13,14]]
=> ? = 3 + 1
[1,2,4,3,7,6,5] => [1,2,4,3,6,7,5] => [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [[1,3,5,6,9,10,12],[2,4,7,8,11,13,14]]
=> ? = 4 + 1
[1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [[1,3,5,6,7,11,13],[2,4,8,9,10,12,14]]
=> ? = 4 + 1
[1,2,4,5,3,7,6] => [1,2,5,3,4,7,6] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [[1,3,5,6,7,11,12],[2,4,8,9,10,13,14]]
=> ? = 3 + 1
[1,2,4,5,6,3,7] => [1,2,6,3,4,5,7] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [[1,3,5,6,7,8,13],[2,4,9,10,11,12,14]]
=> ? = 3 + 1
[1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,3,5,6,7,8,9],[2,4,10,11,12,13,14]]
=> ? = 2 + 1
[1,2,4,5,7,3,6] => [1,2,7,6,3,4,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,3,5,6,7,8,9],[2,4,10,11,12,13,14]]
=> ? = 2 + 1
[1,2,4,5,7,6,3] => [1,2,6,7,3,4,5] => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [[1,3,5,6,7,8,10],[2,4,9,11,12,13,14]]
=> ? = 3 + 1
[1,2,4,6,3,5,7] => [1,2,6,5,3,4,7] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [[1,3,5,6,7,8,13],[2,4,9,10,11,12,14]]
=> ? = 3 + 1
[1,2,4,6,3,7,5] => [1,2,7,5,3,4,6] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,3,5,6,7,8,9],[2,4,10,11,12,13,14]]
=> ? = 2 + 1
[1,2,4,6,5,3,7] => [1,2,5,6,3,4,7] => [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [[1,3,5,6,7,9,13],[2,4,8,10,11,12,14]]
=> ? = 4 + 1
[1,2,4,6,5,7,3] => [1,2,5,7,3,4,6] => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [[1,3,5,6,7,9,10],[2,4,8,11,12,13,14]]
=> ? = 3 + 1
[1,2,4,6,7,3,5] => [1,2,6,3,4,7,5] => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [[1,3,5,6,7,8,12],[2,4,9,10,11,13,14]]
=> ? = 3 + 1
[1,2,4,6,7,5,3] => [1,2,7,3,4,6,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,3,5,6,7,8,9],[2,4,10,11,12,13,14]]
=> ? = 2 + 1
[1,2,4,7,3,5,6] => [1,2,7,6,5,3,4] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,3,5,6,7,8,9],[2,4,10,11,12,13,14]]
=> ? = 2 + 1
[1,2,4,7,3,6,5] => [1,2,6,7,5,3,4] => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [[1,3,5,6,7,8,10],[2,4,9,11,12,13,14]]
=> ? = 3 + 1
[1,2,4,7,5,3,6] => [1,2,5,7,6,3,4] => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [[1,3,5,6,7,9,10],[2,4,8,11,12,13,14]]
=> ? = 3 + 1
[1,2,4,7,5,6,3] => [1,2,5,6,7,3,4] => [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [[1,3,5,6,7,9,11],[2,4,8,10,12,13,14]]
=> ? = 4 + 1
[1,2,4,7,6,3,5] => [1,2,7,5,6,3,4] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,3,5,6,7,8,9],[2,4,10,11,12,13,14]]
=> ? = 2 + 1
[1,2,4,7,6,5,3] => [1,2,6,5,7,3,4] => [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [[1,3,5,6,7,8,11],[2,4,9,10,12,13,14]]
=> ? = 3 + 1
[1,2,5,3,4,6,7] => [1,2,5,4,3,6,7] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [[1,3,5,6,7,11,13],[2,4,8,9,10,12,14]]
=> ? = 4 + 1
[1,2,5,3,4,7,6] => [1,2,5,4,3,7,6] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [[1,3,5,6,7,11,12],[2,4,8,9,10,13,14]]
=> ? = 3 + 1
Description
The number of descents of a standard tableau.
Entry i of a standard Young tableau is a descent if i+1 appears in a row below the row of i.
Matching statistic: St000164
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000164: Perfect matchings ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 31%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000164: Perfect matchings ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 31%
Values
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 3 = 2 + 1
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 4 = 3 + 1
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> 3 = 2 + 1
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> 3 = 2 + 1
[1,4,3,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> 3 = 2 + 1
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> 3 = 2 + 1
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> 2 = 1 + 1
[3,2,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> 3 = 2 + 1
[3,4,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> 2 = 1 + 1
[4,2,3,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> 3 = 2 + 1
[4,3,2,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> 2 = 1 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> 5 = 4 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> 4 = 3 + 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),(5,8),(6,7),(9,10)]
=> 4 = 3 + 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),(5,10),(6,9),(7,8)]
=> 3 = 2 + 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),(5,10),(6,9),(7,8)]
=> 3 = 2 + 1
[1,2,5,4,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> 4 = 3 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10)]
=> 4 = 3 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 3 = 2 + 1
[1,3,4,2,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> 3 = 2 + 1
[1,3,5,4,2] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 3 = 2 + 1
[1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> 3 = 2 + 1
[1,4,3,2,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10)]
=> 4 = 3 + 1
[1,4,3,5,2] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> 3 = 2 + 1
[1,4,5,2,3] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> 3 = 2 + 1
[1,5,2,4,3] => [1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 3 = 2 + 1
[1,5,3,2,4] => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> 3 = 2 + 1
[1,5,3,4,2] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> 4 = 3 + 1
[1,5,4,3,2] => [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> 3 = 2 + 1
[2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10)]
=> 4 = 3 + 1
[2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9)]
=> 3 = 2 + 1
[2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 3 = 2 + 1
[2,1,4,5,3] => [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> 2 = 1 + 1
[2,1,5,3,4] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> 2 = 1 + 1
[2,1,5,4,3] => [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9)]
=> 3 = 2 + 1
[2,3,1,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> 3 = 2 + 1
[2,3,1,5,4] => [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> 2 = 1 + 1
[2,4,3,1,5] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> 3 = 2 + 1
[2,4,5,1,3] => [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> [(1,10),(2,7),(3,6),(4,5),(8,9)]
=> 2 = 1 + 1
[2,5,3,4,1] => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> [(1,10),(2,9),(3,4),(5,6),(7,8)]
=> 3 = 2 + 1
[2,5,4,3,1] => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8)]
=> 2 = 1 + 1
[3,1,2,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> 3 = 2 + 1
[3,1,2,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> 2 = 1 + 1
[3,2,1,4,5] => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10)]
=> 4 = 3 + 1
[3,2,1,5,4] => [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9)]
=> 3 = 2 + 1
[3,2,4,1,5] => [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> 3 = 2 + 1
[3,2,5,4,1] => [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> [(1,10),(2,3),(4,9),(5,6),(7,8)]
=> 3 = 2 + 1
[3,4,1,2,5] => [3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10)]
=> 3 = 2 + 1
[3,4,1,5,2] => [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> [(1,10),(2,5),(3,4),(6,9),(7,8)]
=> 2 = 1 + 1
[3,4,5,2,1] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8)]
=> 2 = 1 + 1
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14)]
=> ? = 6 + 1
[1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,14),(12,13)]
=> ? = 5 + 1
[1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11),(13,14)]
=> ? = 5 + 1
[1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,14),(10,13),(11,12)]
=> ? = 4 + 1
[1,2,3,4,7,5,6] => [1,2,3,4,7,6,5] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,14),(10,13),(11,12)]
=> ? = 4 + 1
[1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,14),(10,11),(12,13)]
=> ? = 5 + 1
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9),(11,12),(13,14)]
=> ? = 5 + 1
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9),(11,14),(12,13)]
=> ? = 4 + 1
[1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,11),(9,10),(13,14)]
=> ? = 4 + 1
[1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,6),(7,14),(8,13),(9,12),(10,11)]
=> ? = 3 + 1
[1,2,3,5,7,4,6] => [1,2,3,7,6,4,5] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,6),(7,14),(8,13),(9,12),(10,11)]
=> ? = 3 + 1
[1,2,3,5,7,6,4] => [1,2,3,6,7,4,5] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,14),(8,13),(9,10),(11,12)]
=> ? = 4 + 1
[1,2,3,6,4,5,7] => [1,2,3,6,5,4,7] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,11),(9,10),(13,14)]
=> ? = 4 + 1
[1,2,3,6,4,7,5] => [1,2,3,7,5,4,6] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,6),(7,14),(8,13),(9,12),(10,11)]
=> ? = 3 + 1
[1,2,3,6,5,4,7] => [1,2,3,5,6,4,7] => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11),(13,14)]
=> ? = 5 + 1
[1,2,3,6,5,7,4] => [1,2,3,5,7,4,6] => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,14),(8,9),(10,13),(11,12)]
=> ? = 4 + 1
[1,2,3,6,7,4,5] => [1,2,3,6,4,7,5] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,14),(8,11),(9,10),(12,13)]
=> ? = 4 + 1
[1,2,3,6,7,5,4] => [1,2,3,7,4,6,5] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,6),(7,14),(8,13),(9,12),(10,11)]
=> ? = 3 + 1
[1,2,3,7,4,5,6] => [1,2,3,7,6,5,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,6),(7,14),(8,13),(9,12),(10,11)]
=> ? = 3 + 1
[1,2,3,7,4,6,5] => [1,2,3,6,7,5,4] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,14),(8,13),(9,10),(11,12)]
=> ? = 4 + 1
[1,2,3,7,5,4,6] => [1,2,3,5,7,6,4] => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,14),(8,9),(10,13),(11,12)]
=> ? = 4 + 1
[1,2,3,7,5,6,4] => [1,2,3,5,6,7,4] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,14),(8,9),(10,11),(12,13)]
=> ? = 5 + 1
[1,2,3,7,6,4,5] => [1,2,3,7,5,6,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,6),(7,14),(8,13),(9,12),(10,11)]
=> ? = 3 + 1
[1,2,3,7,6,5,4] => [1,2,3,6,5,7,4] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,14),(8,11),(9,10),(12,13)]
=> ? = 4 + 1
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12),(13,14)]
=> ? = 5 + 1
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,14),(12,13)]
=> ? = 4 + 1
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11),(13,14)]
=> ? = 4 + 1
[1,2,4,3,6,7,5] => [1,2,4,3,7,5,6] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,14),(10,13),(11,12)]
=> ? = 3 + 1
[1,2,4,3,7,5,6] => [1,2,4,3,7,6,5] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,14),(10,13),(11,12)]
=> ? = 3 + 1
[1,2,4,3,7,6,5] => [1,2,4,3,6,7,5] => [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,14),(10,11),(12,13)]
=> ? = 4 + 1
[1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8),(11,12),(13,14)]
=> ? = 4 + 1
[1,2,4,5,3,7,6] => [1,2,5,3,4,7,6] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8),(11,14),(12,13)]
=> ? = 3 + 1
[1,2,4,5,6,3,7] => [1,2,6,3,4,5,7] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9),(13,14)]
=> ? = 3 + 1
[1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,4),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> ? = 2 + 1
[1,2,4,5,7,3,6] => [1,2,7,6,3,4,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,4),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> ? = 2 + 1
[1,2,4,5,7,6,3] => [1,2,6,7,3,4,5] => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [(1,2),(3,4),(5,14),(6,13),(7,12),(8,9),(10,11)]
=> ? = 3 + 1
[1,2,4,6,3,5,7] => [1,2,6,5,3,4,7] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9),(13,14)]
=> ? = 3 + 1
[1,2,4,6,3,7,5] => [1,2,7,5,3,4,6] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,4),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> ? = 2 + 1
[1,2,4,6,5,3,7] => [1,2,5,6,3,4,7] => [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,8),(9,10),(13,14)]
=> ? = 4 + 1
[1,2,4,6,5,7,3] => [1,2,5,7,3,4,6] => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,14),(6,13),(7,8),(9,12),(10,11)]
=> ? = 3 + 1
[1,2,4,6,7,3,5] => [1,2,6,3,4,7,5] => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [(1,2),(3,4),(5,14),(6,11),(7,10),(8,9),(12,13)]
=> ? = 3 + 1
[1,2,4,6,7,5,3] => [1,2,7,3,4,6,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,4),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> ? = 2 + 1
[1,2,4,7,3,5,6] => [1,2,7,6,5,3,4] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,4),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> ? = 2 + 1
[1,2,4,7,3,6,5] => [1,2,6,7,5,3,4] => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [(1,2),(3,4),(5,14),(6,13),(7,12),(8,9),(10,11)]
=> ? = 3 + 1
[1,2,4,7,5,3,6] => [1,2,5,7,6,3,4] => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,14),(6,13),(7,8),(9,12),(10,11)]
=> ? = 3 + 1
[1,2,4,7,5,6,3] => [1,2,5,6,7,3,4] => [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [(1,2),(3,4),(5,14),(6,13),(7,8),(9,10),(11,12)]
=> ? = 4 + 1
[1,2,4,7,6,3,5] => [1,2,7,5,6,3,4] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,4),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> ? = 2 + 1
[1,2,4,7,6,5,3] => [1,2,6,5,7,3,4] => [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [(1,2),(3,4),(5,14),(6,13),(7,10),(8,9),(11,12)]
=> ? = 3 + 1
[1,2,5,3,4,6,7] => [1,2,5,4,3,6,7] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8),(11,12),(13,14)]
=> ? = 4 + 1
[1,2,5,3,4,7,6] => [1,2,5,4,3,7,6] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8),(11,14),(12,13)]
=> ? = 3 + 1
Description
The number of short pairs.
A short pair is a matching pair of the form (i,i+1).
Matching statistic: St000291
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 31%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 31%
Values
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 101010 => 3 = 2 + 1
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 10101010 => 4 = 3 + 1
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 10101100 => 3 = 2 + 1
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 10110010 => 3 = 2 + 1
[1,4,3,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 10110100 => 3 = 2 + 1
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 11001010 => 3 = 2 + 1
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 11001100 => 2 = 1 + 1
[3,2,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 11010010 => 3 = 2 + 1
[3,4,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 11100100 => 2 = 1 + 1
[4,2,3,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 11010100 => 3 = 2 + 1
[4,3,2,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 11100100 => 2 = 1 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => 5 = 4 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => 4 = 3 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 1010110010 => 4 = 3 + 1
[1,2,4,5,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => 3 = 2 + 1
[1,2,5,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => 3 = 2 + 1
[1,2,5,4,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => 4 = 3 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 1011001010 => 4 = 3 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 1011001100 => 3 = 2 + 1
[1,3,4,2,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 1011100010 => 3 = 2 + 1
[1,3,5,4,2] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => 3 = 2 + 1
[1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 1011100010 => 3 = 2 + 1
[1,4,3,2,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 1011010010 => 4 = 3 + 1
[1,4,3,5,2] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => 3 = 2 + 1
[1,4,5,2,3] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => 3 = 2 + 1
[1,5,2,4,3] => [1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => 3 = 2 + 1
[1,5,3,2,4] => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => 3 = 2 + 1
[1,5,3,4,2] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => 4 = 3 + 1
[1,5,4,3,2] => [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => 3 = 2 + 1
[2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 1100101010 => 4 = 3 + 1
[2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> 1100101100 => 3 = 2 + 1
[2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> 1100110010 => 3 = 2 + 1
[2,1,4,5,3] => [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => 2 = 1 + 1
[2,1,5,3,4] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => 2 = 1 + 1
[2,1,5,4,3] => [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> 1100110100 => 3 = 2 + 1
[2,3,1,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 1110001010 => 3 = 2 + 1
[2,3,1,5,4] => [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => 2 = 1 + 1
[2,4,3,1,5] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => 3 = 2 + 1
[2,4,5,1,3] => [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> 1111000100 => 2 = 1 + 1
[2,5,3,4,1] => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 1110101000 => 3 = 2 + 1
[2,5,4,3,1] => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> 1111001000 => 2 = 1 + 1
[3,1,2,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 1110001010 => 3 = 2 + 1
[3,1,2,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => 2 = 1 + 1
[3,2,1,4,5] => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> 1101001010 => 4 = 3 + 1
[3,2,1,5,4] => [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> 1101001100 => 3 = 2 + 1
[3,2,4,1,5] => [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => 3 = 2 + 1
[3,2,5,4,1] => [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> 1101101000 => 3 = 2 + 1
[3,4,1,2,5] => [3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => 3 = 2 + 1
[3,4,1,5,2] => [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> 1110011000 => 2 = 1 + 1
[3,4,5,2,1] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 1111001000 => 2 = 1 + 1
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 10101010101010 => ? = 6 + 1
[1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 10101010101100 => ? = 5 + 1
[1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> 10101010110010 => ? = 5 + 1
[1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 10101010111000 => ? = 4 + 1
[1,2,3,4,7,5,6] => [1,2,3,4,7,6,5] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 10101010111000 => ? = 4 + 1
[1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 10101010110100 => ? = 5 + 1
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> 10101011001010 => ? = 5 + 1
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> 10101011001100 => ? = 4 + 1
[1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> 10101011100010 => ? = 4 + 1
[1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> 10101011110000 => ? = 3 + 1
[1,2,3,5,7,4,6] => [1,2,3,7,6,4,5] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> 10101011110000 => ? = 3 + 1
[1,2,3,5,7,6,4] => [1,2,3,6,7,4,5] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> 10101011101000 => ? = 4 + 1
[1,2,3,6,4,5,7] => [1,2,3,6,5,4,7] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> 10101011100010 => ? = 4 + 1
[1,2,3,6,4,7,5] => [1,2,3,7,5,4,6] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> 10101011110000 => ? = 3 + 1
[1,2,3,6,5,4,7] => [1,2,3,5,6,4,7] => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> 10101011010010 => ? = 5 + 1
[1,2,3,6,5,7,4] => [1,2,3,5,7,4,6] => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> 10101011011000 => ? = 4 + 1
[1,2,3,6,7,4,5] => [1,2,3,6,4,7,5] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> 10101011100100 => ? = 4 + 1
[1,2,3,6,7,5,4] => [1,2,3,7,4,6,5] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> 10101011110000 => ? = 3 + 1
[1,2,3,7,4,5,6] => [1,2,3,7,6,5,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> 10101011110000 => ? = 3 + 1
[1,2,3,7,4,6,5] => [1,2,3,6,7,5,4] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> 10101011101000 => ? = 4 + 1
[1,2,3,7,5,4,6] => [1,2,3,5,7,6,4] => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> 10101011011000 => ? = 4 + 1
[1,2,3,7,5,6,4] => [1,2,3,5,6,7,4] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> 10101011010100 => ? = 5 + 1
[1,2,3,7,6,4,5] => [1,2,3,7,5,6,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> 10101011110000 => ? = 3 + 1
[1,2,3,7,6,5,4] => [1,2,3,6,5,7,4] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> 10101011100100 => ? = 4 + 1
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> 10101100101010 => ? = 5 + 1
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> 10101100101100 => ? = 4 + 1
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> 10101100110010 => ? = 4 + 1
[1,2,4,3,6,7,5] => [1,2,4,3,7,5,6] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> 10101100111000 => ? = 3 + 1
[1,2,4,3,7,5,6] => [1,2,4,3,7,6,5] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> 10101100111000 => ? = 3 + 1
[1,2,4,3,7,6,5] => [1,2,4,3,6,7,5] => [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> 10101100110100 => ? = 4 + 1
[1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> 10101110001010 => ? = 4 + 1
[1,2,4,5,3,7,6] => [1,2,5,3,4,7,6] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> 10101110001100 => ? = 3 + 1
[1,2,4,5,6,3,7] => [1,2,6,3,4,5,7] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> 10101111000010 => ? = 3 + 1
[1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> 10101111100000 => ? = 2 + 1
[1,2,4,5,7,3,6] => [1,2,7,6,3,4,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> 10101111100000 => ? = 2 + 1
[1,2,4,5,7,6,3] => [1,2,6,7,3,4,5] => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> 10101111010000 => ? = 3 + 1
[1,2,4,6,3,5,7] => [1,2,6,5,3,4,7] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> 10101111000010 => ? = 3 + 1
[1,2,4,6,3,7,5] => [1,2,7,5,3,4,6] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> 10101111100000 => ? = 2 + 1
[1,2,4,6,5,3,7] => [1,2,5,6,3,4,7] => [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> 10101110100010 => ? = 4 + 1
[1,2,4,6,5,7,3] => [1,2,5,7,3,4,6] => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> 10101110110000 => ? = 3 + 1
[1,2,4,6,7,3,5] => [1,2,6,3,4,7,5] => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> 10101111000100 => ? = 3 + 1
[1,2,4,6,7,5,3] => [1,2,7,3,4,6,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> 10101111100000 => ? = 2 + 1
[1,2,4,7,3,5,6] => [1,2,7,6,5,3,4] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> 10101111100000 => ? = 2 + 1
[1,2,4,7,3,6,5] => [1,2,6,7,5,3,4] => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> 10101111010000 => ? = 3 + 1
[1,2,4,7,5,3,6] => [1,2,5,7,6,3,4] => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> 10101110110000 => ? = 3 + 1
[1,2,4,7,5,6,3] => [1,2,5,6,7,3,4] => [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> 10101110101000 => ? = 4 + 1
[1,2,4,7,6,3,5] => [1,2,7,5,6,3,4] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> 10101111100000 => ? = 2 + 1
[1,2,4,7,6,5,3] => [1,2,6,5,7,3,4] => [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> 10101111001000 => ? = 3 + 1
[1,2,5,3,4,6,7] => [1,2,5,4,3,6,7] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> 10101110001010 => ? = 4 + 1
[1,2,5,3,4,7,6] => [1,2,5,4,3,7,6] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> 10101110001100 => ? = 3 + 1
Description
The number of descents of a binary word.
Matching statistic: St000390
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 31%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 31%
Values
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 101010 => 3 = 2 + 1
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 10101010 => 4 = 3 + 1
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 10101100 => 3 = 2 + 1
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 10110010 => 3 = 2 + 1
[1,4,3,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 10110100 => 3 = 2 + 1
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 11001010 => 3 = 2 + 1
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 11001100 => 2 = 1 + 1
[3,2,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 11010010 => 3 = 2 + 1
[3,4,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 11100100 => 2 = 1 + 1
[4,2,3,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 11010100 => 3 = 2 + 1
[4,3,2,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 11100100 => 2 = 1 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => 5 = 4 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => 4 = 3 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 1010110010 => 4 = 3 + 1
[1,2,4,5,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => 3 = 2 + 1
[1,2,5,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => 3 = 2 + 1
[1,2,5,4,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => 4 = 3 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 1011001010 => 4 = 3 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 1011001100 => 3 = 2 + 1
[1,3,4,2,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 1011100010 => 3 = 2 + 1
[1,3,5,4,2] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => 3 = 2 + 1
[1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 1011100010 => 3 = 2 + 1
[1,4,3,2,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 1011010010 => 4 = 3 + 1
[1,4,3,5,2] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => 3 = 2 + 1
[1,4,5,2,3] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => 3 = 2 + 1
[1,5,2,4,3] => [1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => 3 = 2 + 1
[1,5,3,2,4] => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => 3 = 2 + 1
[1,5,3,4,2] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => 4 = 3 + 1
[1,5,4,3,2] => [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => 3 = 2 + 1
[2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 1100101010 => 4 = 3 + 1
[2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> 1100101100 => 3 = 2 + 1
[2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> 1100110010 => 3 = 2 + 1
[2,1,4,5,3] => [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => 2 = 1 + 1
[2,1,5,3,4] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => 2 = 1 + 1
[2,1,5,4,3] => [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> 1100110100 => 3 = 2 + 1
[2,3,1,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 1110001010 => 3 = 2 + 1
[2,3,1,5,4] => [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => 2 = 1 + 1
[2,4,3,1,5] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => 3 = 2 + 1
[2,4,5,1,3] => [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> 1111000100 => 2 = 1 + 1
[2,5,3,4,1] => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 1110101000 => 3 = 2 + 1
[2,5,4,3,1] => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> 1111001000 => 2 = 1 + 1
[3,1,2,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 1110001010 => 3 = 2 + 1
[3,1,2,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => 2 = 1 + 1
[3,2,1,4,5] => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> 1101001010 => 4 = 3 + 1
[3,2,1,5,4] => [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> 1101001100 => 3 = 2 + 1
[3,2,4,1,5] => [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => 3 = 2 + 1
[3,2,5,4,1] => [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> 1101101000 => 3 = 2 + 1
[3,4,1,2,5] => [3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => 3 = 2 + 1
[3,4,1,5,2] => [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> 1110011000 => 2 = 1 + 1
[3,4,5,2,1] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 1111001000 => 2 = 1 + 1
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 10101010101010 => ? = 6 + 1
[1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 10101010101100 => ? = 5 + 1
[1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> 10101010110010 => ? = 5 + 1
[1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 10101010111000 => ? = 4 + 1
[1,2,3,4,7,5,6] => [1,2,3,4,7,6,5] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 10101010111000 => ? = 4 + 1
[1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 10101010110100 => ? = 5 + 1
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> 10101011001010 => ? = 5 + 1
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> 10101011001100 => ? = 4 + 1
[1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> 10101011100010 => ? = 4 + 1
[1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> 10101011110000 => ? = 3 + 1
[1,2,3,5,7,4,6] => [1,2,3,7,6,4,5] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> 10101011110000 => ? = 3 + 1
[1,2,3,5,7,6,4] => [1,2,3,6,7,4,5] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> 10101011101000 => ? = 4 + 1
[1,2,3,6,4,5,7] => [1,2,3,6,5,4,7] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> 10101011100010 => ? = 4 + 1
[1,2,3,6,4,7,5] => [1,2,3,7,5,4,6] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> 10101011110000 => ? = 3 + 1
[1,2,3,6,5,4,7] => [1,2,3,5,6,4,7] => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> 10101011010010 => ? = 5 + 1
[1,2,3,6,5,7,4] => [1,2,3,5,7,4,6] => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> 10101011011000 => ? = 4 + 1
[1,2,3,6,7,4,5] => [1,2,3,6,4,7,5] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> 10101011100100 => ? = 4 + 1
[1,2,3,6,7,5,4] => [1,2,3,7,4,6,5] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> 10101011110000 => ? = 3 + 1
[1,2,3,7,4,5,6] => [1,2,3,7,6,5,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> 10101011110000 => ? = 3 + 1
[1,2,3,7,4,6,5] => [1,2,3,6,7,5,4] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> 10101011101000 => ? = 4 + 1
[1,2,3,7,5,4,6] => [1,2,3,5,7,6,4] => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> 10101011011000 => ? = 4 + 1
[1,2,3,7,5,6,4] => [1,2,3,5,6,7,4] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> 10101011010100 => ? = 5 + 1
[1,2,3,7,6,4,5] => [1,2,3,7,5,6,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> 10101011110000 => ? = 3 + 1
[1,2,3,7,6,5,4] => [1,2,3,6,5,7,4] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> 10101011100100 => ? = 4 + 1
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> 10101100101010 => ? = 5 + 1
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> 10101100101100 => ? = 4 + 1
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> 10101100110010 => ? = 4 + 1
[1,2,4,3,6,7,5] => [1,2,4,3,7,5,6] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> 10101100111000 => ? = 3 + 1
[1,2,4,3,7,5,6] => [1,2,4,3,7,6,5] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> 10101100111000 => ? = 3 + 1
[1,2,4,3,7,6,5] => [1,2,4,3,6,7,5] => [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> 10101100110100 => ? = 4 + 1
[1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> 10101110001010 => ? = 4 + 1
[1,2,4,5,3,7,6] => [1,2,5,3,4,7,6] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> 10101110001100 => ? = 3 + 1
[1,2,4,5,6,3,7] => [1,2,6,3,4,5,7] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> 10101111000010 => ? = 3 + 1
[1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> 10101111100000 => ? = 2 + 1
[1,2,4,5,7,3,6] => [1,2,7,6,3,4,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> 10101111100000 => ? = 2 + 1
[1,2,4,5,7,6,3] => [1,2,6,7,3,4,5] => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> 10101111010000 => ? = 3 + 1
[1,2,4,6,3,5,7] => [1,2,6,5,3,4,7] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> 10101111000010 => ? = 3 + 1
[1,2,4,6,3,7,5] => [1,2,7,5,3,4,6] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> 10101111100000 => ? = 2 + 1
[1,2,4,6,5,3,7] => [1,2,5,6,3,4,7] => [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> 10101110100010 => ? = 4 + 1
[1,2,4,6,5,7,3] => [1,2,5,7,3,4,6] => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> 10101110110000 => ? = 3 + 1
[1,2,4,6,7,3,5] => [1,2,6,3,4,7,5] => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> 10101111000100 => ? = 3 + 1
[1,2,4,6,7,5,3] => [1,2,7,3,4,6,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> 10101111100000 => ? = 2 + 1
[1,2,4,7,3,5,6] => [1,2,7,6,5,3,4] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> 10101111100000 => ? = 2 + 1
[1,2,4,7,3,6,5] => [1,2,6,7,5,3,4] => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> 10101111010000 => ? = 3 + 1
[1,2,4,7,5,3,6] => [1,2,5,7,6,3,4] => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> 10101110110000 => ? = 3 + 1
[1,2,4,7,5,6,3] => [1,2,5,6,7,3,4] => [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> 10101110101000 => ? = 4 + 1
[1,2,4,7,6,3,5] => [1,2,7,5,6,3,4] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> 10101111100000 => ? = 2 + 1
[1,2,4,7,6,5,3] => [1,2,6,5,7,3,4] => [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> 10101111001000 => ? = 3 + 1
[1,2,5,3,4,6,7] => [1,2,5,4,3,6,7] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> 10101110001010 => ? = 4 + 1
[1,2,5,3,4,7,6] => [1,2,5,4,3,7,6] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> 10101110001100 => ? = 3 + 1
Description
The number of runs of ones in a binary word.
Matching statistic: St001232
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 31%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 31%
Values
[1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,1,3,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,1,4,3] => [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[3,4,1,2] => [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[4,3,2,1] => [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,2,3,5,4] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[1,2,4,3,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[1,2,4,5,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,2,5,3,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,2,5,4,3] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[1,3,2,4,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[1,3,2,5,4] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 2
[1,3,4,2,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,4,2,3,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,4,3,2,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,4,5,2,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 2
[1,5,2,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,5,3,2,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,5,3,4,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[1,5,4,3,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 2
[2,1,3,4,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[2,1,3,5,4] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 2
[2,1,4,3,5] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 2
[2,1,4,5,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
[2,1,5,3,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
[2,1,5,4,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 2
[2,3,1,4,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[2,3,1,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
[2,4,3,1,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[2,4,5,1,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
[2,5,3,4,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[2,5,4,3,1] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
[3,1,2,4,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[3,1,2,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
[3,2,1,4,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[3,2,1,5,4] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 2
[3,2,4,1,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[3,2,5,4,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[3,4,1,2,5] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 2
[3,4,1,5,2] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
[3,4,5,2,1] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
[3,5,1,2,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
[3,5,1,4,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 2
[3,5,4,1,2] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
[4,1,3,2,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[4,1,5,2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
[4,2,1,3,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[4,2,3,1,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[4,2,3,5,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[4,2,5,1,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 2
[4,3,2,1,5] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 2
[4,3,2,5,1] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
[4,3,5,1,2] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
[4,5,1,3,2] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
[4,5,2,1,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
[4,5,3,1,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 2
[5,1,3,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[5,1,4,3,2] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
[5,2,1,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[5,2,3,1,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[5,2,3,4,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[5,2,4,3,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 2
[5,3,2,1,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
[5,3,2,4,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 2
[5,3,4,2,1] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
[5,4,1,2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
[5,4,2,3,1] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
[5,4,3,2,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 2
[1,2,3,4,5,6] => [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[1,2,3,4,6,5] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4
[1,2,3,5,4,6] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4
[1,2,3,5,6,4] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3
[1,2,3,6,4,5] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3
[1,2,3,6,5,4] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4
[1,2,4,3,5,6] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4
[1,2,4,3,6,5] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ? = 3
[1,2,4,5,3,6] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3
[1,2,4,5,6,3] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2
[1,2,4,6,3,5] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2
[1,2,4,6,5,3] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3
[1,2,5,6,3,4] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ? = 3
[1,2,6,5,4,3] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ? = 3
[1,3,2,4,6,5] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ? = 3
[1,3,2,5,4,6] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ? = 3
[1,3,2,5,6,4] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ? = 2
[1,3,2,6,4,5] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ? = 2
[1,3,2,6,5,4] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ? = 3
[1,3,4,2,6,5] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ? = 2
[1,3,5,6,2,4] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ? = 2
[1,3,6,5,4,2] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ? = 2
[1,4,2,3,6,5] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ? = 2
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000031
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000031: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 44%
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000031: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 44%
Values
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 3 = 2 + 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 4 = 3 + 1
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 3 = 2 + 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 3 = 2 + 1
[1,4,3,2] => [1,3,4,2] => [1,2,4,3] => [1,2,4,3] => 3 = 2 + 1
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 3 = 2 + 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2 = 1 + 1
[3,2,1,4] => [2,3,1,4] => [1,3,2,4] => [1,3,2,4] => 3 = 2 + 1
[3,4,1,2] => [3,1,4,2] => [2,1,4,3] => [2,1,4,3] => 2 = 1 + 1
[4,2,3,1] => [2,3,4,1] => [1,2,4,3] => [1,2,4,3] => 3 = 2 + 1
[4,3,2,1] => [3,2,4,1] => [2,1,4,3] => [2,1,4,3] => 2 = 1 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 4 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 4 = 3 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 4 = 3 + 1
[1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => 3 = 2 + 1
[1,2,5,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,4,5,3] => 3 = 2 + 1
[1,2,5,4,3] => [1,2,4,5,3] => [1,2,3,5,4] => [1,2,3,5,4] => 4 = 3 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 4 = 3 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 3 = 2 + 1
[1,3,4,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => 3 = 2 + 1
[1,3,5,4,2] => [1,4,5,2,3] => [1,3,5,2,4] => [1,5,3,2,4] => 3 = 2 + 1
[1,4,2,3,5] => [1,4,3,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => 3 = 2 + 1
[1,4,3,2,5] => [1,3,4,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => 4 = 3 + 1
[1,4,3,5,2] => [1,3,5,2,4] => [1,3,5,2,4] => [1,5,3,2,4] => 3 = 2 + 1
[1,4,5,2,3] => [1,4,2,5,3] => [1,3,2,5,4] => [1,3,2,5,4] => 3 = 2 + 1
[1,5,2,4,3] => [1,4,5,3,2] => [1,2,5,4,3] => [1,2,4,5,3] => 3 = 2 + 1
[1,5,3,2,4] => [1,3,5,4,2] => [1,2,5,4,3] => [1,2,4,5,3] => 3 = 2 + 1
[1,5,3,4,2] => [1,3,4,5,2] => [1,2,3,5,4] => [1,2,3,5,4] => 4 = 3 + 1
[1,5,4,3,2] => [1,4,3,5,2] => [1,3,2,5,4] => [1,3,2,5,4] => 3 = 2 + 1
[2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 4 = 3 + 1
[2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 3 = 2 + 1
[2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 3 = 2 + 1
[2,1,4,5,3] => [2,1,5,3,4] => [2,1,5,3,4] => [2,1,5,3,4] => 2 = 1 + 1
[2,1,5,3,4] => [2,1,5,4,3] => [2,1,5,4,3] => [2,1,4,5,3] => 2 = 1 + 1
[2,1,5,4,3] => [2,1,4,5,3] => [2,1,3,5,4] => [2,1,3,5,4] => 3 = 2 + 1
[2,3,1,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => 3 = 2 + 1
[2,3,1,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => 2 = 1 + 1
[2,4,3,1,5] => [3,4,1,2,5] => [2,4,1,3,5] => [4,2,1,3,5] => 3 = 2 + 1
[2,4,5,1,3] => [4,1,2,5,3] => [3,1,2,5,4] => [3,1,2,5,4] => 2 = 1 + 1
[2,5,3,4,1] => [3,4,5,1,2] => [1,3,5,2,4] => [1,5,3,2,4] => 3 = 2 + 1
[2,5,4,3,1] => [4,3,5,1,2] => [3,2,5,1,4] => [5,3,2,1,4] => 2 = 1 + 1
[3,1,2,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => [2,3,1,4,5] => 3 = 2 + 1
[3,1,2,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => [2,3,1,5,4] => 2 = 1 + 1
[3,2,1,4,5] => [2,3,1,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 4 = 3 + 1
[3,2,1,5,4] => [2,3,1,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 3 = 2 + 1
[3,2,4,1,5] => [2,4,1,3,5] => [2,4,1,3,5] => [4,2,1,3,5] => 3 = 2 + 1
[3,2,5,4,1] => [2,4,5,1,3] => [1,3,5,2,4] => [1,5,3,2,4] => 3 = 2 + 1
[3,4,1,2,5] => [3,1,4,2,5] => [2,1,4,3,5] => [2,1,4,3,5] => 3 = 2 + 1
[3,4,1,5,2] => [3,1,5,2,4] => [3,1,5,2,4] => [3,5,1,2,4] => 2 = 1 + 1
[3,4,5,2,1] => [4,2,5,1,3] => [3,2,5,1,4] => [5,3,2,1,4] => 2 = 1 + 1
[2,6,4,5,3,1] => [5,3,4,6,1,2] => [4,1,3,6,2,5] => [4,6,1,3,2,5] => ? = 1 + 1
[3,1,6,5,4,2] => [5,4,6,2,1,3] => [4,3,6,2,1,5] => [2,6,4,3,1,5] => ? = 1 + 1
[3,4,5,6,1,2] => [5,1,3,6,2,4] => [4,1,3,6,2,5] => [4,6,1,3,2,5] => ? = 1 + 1
[3,4,6,5,2,1] => [5,2,4,6,1,3] => [4,1,3,6,2,5] => [4,6,1,3,2,5] => ? = 1 + 1
[3,5,4,1,6,2] => [4,1,3,6,2,5] => [4,1,3,6,2,5] => [4,6,1,3,2,5] => ? = 1 + 1
[4,1,5,6,3,2] => [5,3,6,2,1,4] => [4,3,6,2,1,5] => [2,6,4,3,1,5] => ? = 1 + 1
[4,3,5,6,2,1] => [5,2,3,6,1,4] => [4,1,3,6,2,5] => [4,6,1,3,2,5] => ? = 1 + 1
[4,3,6,5,1,2] => [5,1,4,6,2,3] => [4,1,3,6,2,5] => [4,6,1,3,2,5] => ? = 1 + 1
[5,1,4,3,6,2] => [4,3,6,2,1,5] => [4,3,6,2,1,5] => [2,6,4,3,1,5] => ? = 1 + 1
[5,3,4,2,6,1] => [4,2,3,6,1,5] => [4,1,3,6,2,5] => [4,6,1,3,2,5] => ? = 1 + 1
[1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => ? = 4 + 1
[1,2,3,4,7,5,6] => [1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => ? = 4 + 1
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => ? = 5 + 1
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => ? = 4 + 1
[1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => ? = 4 + 1
[1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => [1,2,3,7,4,5,6] => [1,2,3,7,4,5,6] => ? = 3 + 1
[1,2,3,5,7,6,4] => [1,2,3,6,7,4,5] => [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => ? = 4 + 1
[1,2,3,6,4,5,7] => [1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => [1,2,3,5,6,4,7] => ? = 4 + 1
[1,2,3,6,4,7,5] => [1,2,3,7,5,4,6] => [1,2,3,7,5,4,6] => [1,2,3,5,7,4,6] => ? = 3 + 1
[1,2,3,6,5,7,4] => [1,2,3,5,7,4,6] => [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => ? = 4 + 1
[1,2,3,6,7,4,5] => [1,2,3,6,4,7,5] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => ? = 4 + 1
[1,2,3,6,7,5,4] => [1,2,3,7,4,6,5] => [1,2,3,7,4,6,5] => [1,2,3,7,6,4,5] => ? = 3 + 1
[1,2,3,7,4,5,6] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => [1,2,3,5,6,7,4] => ? = 3 + 1
[1,2,3,7,4,6,5] => [1,2,3,6,7,5,4] => [1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => ? = 4 + 1
[1,2,3,7,5,4,6] => [1,2,3,5,7,6,4] => [1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => ? = 4 + 1
[1,2,3,7,6,4,5] => [1,2,3,7,5,6,4] => [1,2,3,7,4,6,5] => [1,2,3,7,6,4,5] => ? = 3 + 1
[1,2,3,7,6,5,4] => [1,2,3,6,5,7,4] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => ? = 4 + 1
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => ? = 5 + 1
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => ? = 4 + 1
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => ? = 4 + 1
[1,2,4,3,6,7,5] => [1,2,4,3,7,5,6] => [1,2,4,3,7,5,6] => [1,2,4,3,7,5,6] => ? = 3 + 1
[1,2,4,3,7,5,6] => [1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => [1,2,4,3,6,7,5] => ? = 3 + 1
[1,2,4,3,7,6,5] => [1,2,4,3,6,7,5] => [1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => ? = 4 + 1
[1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => [1,2,5,3,4,6,7] => [1,2,5,3,4,6,7] => ? = 4 + 1
[1,2,4,5,3,7,6] => [1,2,5,3,4,7,6] => [1,2,5,3,4,7,6] => [1,2,5,3,4,7,6] => ? = 3 + 1
[1,2,4,5,6,3,7] => [1,2,6,3,4,5,7] => [1,2,6,3,4,5,7] => [1,2,6,3,4,5,7] => ? = 3 + 1
[1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => [1,2,7,3,4,5,6] => [1,2,7,3,4,5,6] => ? = 2 + 1
[1,2,4,5,7,6,3] => [1,2,6,7,3,4,5] => [1,2,5,7,3,4,6] => [1,2,7,3,5,4,6] => ? = 3 + 1
[1,2,4,6,3,5,7] => [1,2,6,5,3,4,7] => [1,2,6,5,3,4,7] => [1,2,5,3,6,4,7] => ? = 3 + 1
[1,2,4,6,3,7,5] => [1,2,7,5,3,4,6] => [1,2,7,5,3,4,6] => [1,2,5,3,7,4,6] => ? = 2 + 1
[1,2,4,6,5,3,7] => [1,2,5,6,3,4,7] => [1,2,4,6,3,5,7] => [1,2,6,4,3,5,7] => ? = 4 + 1
[1,2,4,6,5,7,3] => [1,2,5,7,3,4,6] => [1,2,5,7,3,4,6] => [1,2,7,3,5,4,6] => ? = 3 + 1
[1,2,4,6,7,3,5] => [1,2,6,3,4,7,5] => [1,2,5,3,4,7,6] => [1,2,5,3,4,7,6] => ? = 3 + 1
[1,2,4,6,7,5,3] => [1,2,7,3,4,6,5] => [1,2,7,3,4,6,5] => [1,2,7,3,6,4,5] => ? = 2 + 1
[1,2,4,7,3,5,6] => [1,2,7,6,5,3,4] => [1,2,7,6,5,3,4] => [1,2,5,3,6,7,4] => ? = 2 + 1
[1,2,4,7,3,6,5] => [1,2,6,7,5,3,4] => [1,2,4,7,6,3,5] => [1,2,6,4,3,7,5] => ? = 3 + 1
[1,2,4,7,5,3,6] => [1,2,5,7,6,3,4] => [1,2,4,7,6,3,5] => [1,2,6,4,3,7,5] => ? = 3 + 1
[1,2,4,7,5,6,3] => [1,2,5,6,7,3,4] => [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => ? = 4 + 1
[1,2,4,7,6,3,5] => [1,2,7,5,6,3,4] => [1,2,7,4,6,3,5] => [1,2,6,7,3,4,5] => ? = 2 + 1
[1,2,4,7,6,5,3] => [1,2,6,5,7,3,4] => [1,2,5,4,7,3,6] => [1,2,7,5,4,3,6] => ? = 3 + 1
Description
The number of cycles in the cycle decomposition of a permutation.
Matching statistic: St000314
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000314: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 31%
St000314: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 31%
Values
[1,2,3] => [1,2,3] => 3 = 2 + 1
[1,2,3,4] => [1,2,3,4] => 4 = 3 + 1
[1,2,4,3] => [1,2,4,3] => 3 = 2 + 1
[1,3,2,4] => [1,3,2,4] => 3 = 2 + 1
[1,4,3,2] => [1,3,4,2] => 3 = 2 + 1
[2,1,3,4] => [2,1,3,4] => 3 = 2 + 1
[2,1,4,3] => [2,1,4,3] => 2 = 1 + 1
[3,2,1,4] => [2,3,1,4] => 3 = 2 + 1
[3,4,1,2] => [3,1,4,2] => 2 = 1 + 1
[4,2,3,1] => [2,3,4,1] => 3 = 2 + 1
[4,3,2,1] => [3,2,4,1] => 2 = 1 + 1
[1,2,3,4,5] => [1,2,3,4,5] => 5 = 4 + 1
[1,2,3,5,4] => [1,2,3,5,4] => 4 = 3 + 1
[1,2,4,3,5] => [1,2,4,3,5] => 4 = 3 + 1
[1,2,4,5,3] => [1,2,5,3,4] => 3 = 2 + 1
[1,2,5,3,4] => [1,2,5,4,3] => 3 = 2 + 1
[1,2,5,4,3] => [1,2,4,5,3] => 4 = 3 + 1
[1,3,2,4,5] => [1,3,2,4,5] => 4 = 3 + 1
[1,3,2,5,4] => [1,3,2,5,4] => 3 = 2 + 1
[1,3,4,2,5] => [1,4,2,3,5] => 3 = 2 + 1
[1,3,5,4,2] => [1,4,5,2,3] => 3 = 2 + 1
[1,4,2,3,5] => [1,4,3,2,5] => 3 = 2 + 1
[1,4,3,2,5] => [1,3,4,2,5] => 4 = 3 + 1
[1,4,3,5,2] => [1,3,5,2,4] => 3 = 2 + 1
[1,4,5,2,3] => [1,4,2,5,3] => 3 = 2 + 1
[1,5,2,4,3] => [1,4,5,3,2] => 3 = 2 + 1
[1,5,3,2,4] => [1,3,5,4,2] => 3 = 2 + 1
[1,5,3,4,2] => [1,3,4,5,2] => 4 = 3 + 1
[1,5,4,3,2] => [1,4,3,5,2] => 3 = 2 + 1
[2,1,3,4,5] => [2,1,3,4,5] => 4 = 3 + 1
[2,1,3,5,4] => [2,1,3,5,4] => 3 = 2 + 1
[2,1,4,3,5] => [2,1,4,3,5] => 3 = 2 + 1
[2,1,4,5,3] => [2,1,5,3,4] => 2 = 1 + 1
[2,1,5,3,4] => [2,1,5,4,3] => 2 = 1 + 1
[2,1,5,4,3] => [2,1,4,5,3] => 3 = 2 + 1
[2,3,1,4,5] => [3,1,2,4,5] => 3 = 2 + 1
[2,3,1,5,4] => [3,1,2,5,4] => 2 = 1 + 1
[2,4,3,1,5] => [3,4,1,2,5] => 3 = 2 + 1
[2,4,5,1,3] => [4,1,2,5,3] => 2 = 1 + 1
[2,5,3,4,1] => [3,4,5,1,2] => 3 = 2 + 1
[2,5,4,3,1] => [4,3,5,1,2] => 2 = 1 + 1
[3,1,2,4,5] => [3,2,1,4,5] => 3 = 2 + 1
[3,1,2,5,4] => [3,2,1,5,4] => 2 = 1 + 1
[3,2,1,4,5] => [2,3,1,4,5] => 4 = 3 + 1
[3,2,1,5,4] => [2,3,1,5,4] => 3 = 2 + 1
[3,2,4,1,5] => [2,4,1,3,5] => 3 = 2 + 1
[3,2,5,4,1] => [2,4,5,1,3] => 3 = 2 + 1
[3,4,1,2,5] => [3,1,4,2,5] => 3 = 2 + 1
[3,4,1,5,2] => [3,1,5,2,4] => 2 = 1 + 1
[3,4,5,2,1] => [4,2,5,1,3] => 2 = 1 + 1
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => ? = 5 + 1
[1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => ? = 5 + 1
[1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => ? = 4 + 1
[1,2,3,4,7,5,6] => [1,2,3,4,7,6,5] => ? = 4 + 1
[1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => ? = 5 + 1
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => ? = 5 + 1
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => ? = 4 + 1
[1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => ? = 4 + 1
[1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => ? = 3 + 1
[1,2,3,5,7,4,6] => [1,2,3,7,6,4,5] => ? = 3 + 1
[1,2,3,5,7,6,4] => [1,2,3,6,7,4,5] => ? = 4 + 1
[1,2,3,6,4,5,7] => [1,2,3,6,5,4,7] => ? = 4 + 1
[1,2,3,6,4,7,5] => [1,2,3,7,5,4,6] => ? = 3 + 1
[1,2,3,6,5,4,7] => [1,2,3,5,6,4,7] => ? = 5 + 1
[1,2,3,6,5,7,4] => [1,2,3,5,7,4,6] => ? = 4 + 1
[1,2,3,6,7,4,5] => [1,2,3,6,4,7,5] => ? = 4 + 1
[1,2,3,6,7,5,4] => [1,2,3,7,4,6,5] => ? = 3 + 1
[1,2,3,7,4,5,6] => [1,2,3,7,6,5,4] => ? = 3 + 1
[1,2,3,7,4,6,5] => [1,2,3,6,7,5,4] => ? = 4 + 1
[1,2,3,7,5,4,6] => [1,2,3,5,7,6,4] => ? = 4 + 1
[1,2,3,7,5,6,4] => [1,2,3,5,6,7,4] => ? = 5 + 1
[1,2,3,7,6,4,5] => [1,2,3,7,5,6,4] => ? = 3 + 1
[1,2,3,7,6,5,4] => [1,2,3,6,5,7,4] => ? = 4 + 1
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => ? = 5 + 1
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => ? = 4 + 1
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => ? = 4 + 1
[1,2,4,3,6,7,5] => [1,2,4,3,7,5,6] => ? = 3 + 1
[1,2,4,3,7,5,6] => [1,2,4,3,7,6,5] => ? = 3 + 1
[1,2,4,3,7,6,5] => [1,2,4,3,6,7,5] => ? = 4 + 1
[1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => ? = 4 + 1
[1,2,4,5,3,7,6] => [1,2,5,3,4,7,6] => ? = 3 + 1
[1,2,4,5,6,3,7] => [1,2,6,3,4,5,7] => ? = 3 + 1
[1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => ? = 2 + 1
[1,2,4,5,7,3,6] => [1,2,7,6,3,4,5] => ? = 2 + 1
[1,2,4,5,7,6,3] => [1,2,6,7,3,4,5] => ? = 3 + 1
[1,2,4,6,3,5,7] => [1,2,6,5,3,4,7] => ? = 3 + 1
[1,2,4,6,3,7,5] => [1,2,7,5,3,4,6] => ? = 2 + 1
[1,2,4,6,5,3,7] => [1,2,5,6,3,4,7] => ? = 4 + 1
[1,2,4,6,5,7,3] => [1,2,5,7,3,4,6] => ? = 3 + 1
[1,2,4,6,7,3,5] => [1,2,6,3,4,7,5] => ? = 3 + 1
[1,2,4,6,7,5,3] => [1,2,7,3,4,6,5] => ? = 2 + 1
[1,2,4,7,3,5,6] => [1,2,7,6,5,3,4] => ? = 2 + 1
[1,2,4,7,3,6,5] => [1,2,6,7,5,3,4] => ? = 3 + 1
[1,2,4,7,5,3,6] => [1,2,5,7,6,3,4] => ? = 3 + 1
[1,2,4,7,5,6,3] => [1,2,5,6,7,3,4] => ? = 4 + 1
[1,2,4,7,6,3,5] => [1,2,7,5,6,3,4] => ? = 2 + 1
[1,2,4,7,6,5,3] => [1,2,6,5,7,3,4] => ? = 3 + 1
[1,2,5,3,4,6,7] => [1,2,5,4,3,6,7] => ? = 4 + 1
[1,2,5,3,4,7,6] => [1,2,5,4,3,7,6] => ? = 3 + 1
Description
The number of left-to-right-maxima of a permutation.
An integer σi in the one-line notation of a permutation σ is a '''left-to-right-maximum''' if there does not exist a j<i such that σj>σi.
This is also the number of weak exceedences of a permutation that are not mid-points of a decreasing subsequence of length 3, see [1] for more on the later description.
Matching statistic: St000541
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000541: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 31%
Mp00069: Permutations —complement⟶ Permutations
St000541: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 31%
Values
[1,2,3] => [1,2,3] => [3,2,1] => 2
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 3
[1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 2
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 2
[1,4,3,2] => [1,3,4,2] => [4,2,1,3] => 2
[2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 2
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 1
[3,2,1,4] => [2,3,1,4] => [3,2,4,1] => 2
[3,4,1,2] => [3,1,4,2] => [2,4,1,3] => 1
[4,2,3,1] => [2,3,4,1] => [3,2,1,4] => 2
[4,3,2,1] => [3,2,4,1] => [2,3,1,4] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 4
[1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => 3
[1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => 3
[1,2,4,5,3] => [1,2,5,3,4] => [5,4,1,3,2] => 2
[1,2,5,3,4] => [1,2,5,4,3] => [5,4,1,2,3] => 2
[1,2,5,4,3] => [1,2,4,5,3] => [5,4,2,1,3] => 3
[1,3,2,4,5] => [1,3,2,4,5] => [5,3,4,2,1] => 3
[1,3,2,5,4] => [1,3,2,5,4] => [5,3,4,1,2] => 2
[1,3,4,2,5] => [1,4,2,3,5] => [5,2,4,3,1] => 2
[1,3,5,4,2] => [1,4,5,2,3] => [5,2,1,4,3] => 2
[1,4,2,3,5] => [1,4,3,2,5] => [5,2,3,4,1] => 2
[1,4,3,2,5] => [1,3,4,2,5] => [5,3,2,4,1] => 3
[1,4,3,5,2] => [1,3,5,2,4] => [5,3,1,4,2] => 2
[1,4,5,2,3] => [1,4,2,5,3] => [5,2,4,1,3] => 2
[1,5,2,4,3] => [1,4,5,3,2] => [5,2,1,3,4] => 2
[1,5,3,2,4] => [1,3,5,4,2] => [5,3,1,2,4] => 2
[1,5,3,4,2] => [1,3,4,5,2] => [5,3,2,1,4] => 3
[1,5,4,3,2] => [1,4,3,5,2] => [5,2,3,1,4] => 2
[2,1,3,4,5] => [2,1,3,4,5] => [4,5,3,2,1] => 3
[2,1,3,5,4] => [2,1,3,5,4] => [4,5,3,1,2] => 2
[2,1,4,3,5] => [2,1,4,3,5] => [4,5,2,3,1] => 2
[2,1,4,5,3] => [2,1,5,3,4] => [4,5,1,3,2] => 1
[2,1,5,3,4] => [2,1,5,4,3] => [4,5,1,2,3] => 1
[2,1,5,4,3] => [2,1,4,5,3] => [4,5,2,1,3] => 2
[2,3,1,4,5] => [3,1,2,4,5] => [3,5,4,2,1] => 2
[2,3,1,5,4] => [3,1,2,5,4] => [3,5,4,1,2] => 1
[2,4,3,1,5] => [3,4,1,2,5] => [3,2,5,4,1] => 2
[2,4,5,1,3] => [4,1,2,5,3] => [2,5,4,1,3] => 1
[2,5,3,4,1] => [3,4,5,1,2] => [3,2,1,5,4] => 2
[2,5,4,3,1] => [4,3,5,1,2] => [2,3,1,5,4] => 1
[3,1,2,4,5] => [3,2,1,4,5] => [3,4,5,2,1] => 2
[3,1,2,5,4] => [3,2,1,5,4] => [3,4,5,1,2] => 1
[3,2,1,4,5] => [2,3,1,4,5] => [4,3,5,2,1] => 3
[3,2,1,5,4] => [2,3,1,5,4] => [4,3,5,1,2] => 2
[3,2,4,1,5] => [2,4,1,3,5] => [4,2,5,3,1] => 2
[3,2,5,4,1] => [2,4,5,1,3] => [4,2,1,5,3] => 2
[3,4,1,2,5] => [3,1,4,2,5] => [3,5,2,4,1] => 2
[3,4,1,5,2] => [3,1,5,2,4] => [3,5,1,4,2] => 1
[3,4,5,2,1] => [4,2,5,1,3] => [2,4,1,5,3] => 1
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 6
[1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [7,6,5,4,3,1,2] => ? = 5
[1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => ? = 5
[1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => ? = 4
[1,2,3,4,7,5,6] => [1,2,3,4,7,6,5] => [7,6,5,4,1,2,3] => ? = 4
[1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => [7,6,5,4,2,1,3] => ? = 5
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => ? = 5
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [7,6,5,3,4,1,2] => ? = 4
[1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => [7,6,5,2,4,3,1] => ? = 4
[1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => [7,6,5,1,4,3,2] => ? = 3
[1,2,3,5,7,4,6] => [1,2,3,7,6,4,5] => [7,6,5,1,2,4,3] => ? = 3
[1,2,3,5,7,6,4] => [1,2,3,6,7,4,5] => [7,6,5,2,1,4,3] => ? = 4
[1,2,3,6,4,5,7] => [1,2,3,6,5,4,7] => [7,6,5,2,3,4,1] => ? = 4
[1,2,3,6,4,7,5] => [1,2,3,7,5,4,6] => [7,6,5,1,3,4,2] => ? = 3
[1,2,3,6,5,4,7] => [1,2,3,5,6,4,7] => [7,6,5,3,2,4,1] => ? = 5
[1,2,3,6,5,7,4] => [1,2,3,5,7,4,6] => [7,6,5,3,1,4,2] => ? = 4
[1,2,3,6,7,4,5] => [1,2,3,6,4,7,5] => [7,6,5,2,4,1,3] => ? = 4
[1,2,3,6,7,5,4] => [1,2,3,7,4,6,5] => [7,6,5,1,4,2,3] => ? = 3
[1,2,3,7,4,5,6] => [1,2,3,7,6,5,4] => [7,6,5,1,2,3,4] => ? = 3
[1,2,3,7,4,6,5] => [1,2,3,6,7,5,4] => [7,6,5,2,1,3,4] => ? = 4
[1,2,3,7,5,4,6] => [1,2,3,5,7,6,4] => [7,6,5,3,1,2,4] => ? = 4
[1,2,3,7,5,6,4] => [1,2,3,5,6,7,4] => [7,6,5,3,2,1,4] => ? = 5
[1,2,3,7,6,4,5] => [1,2,3,7,5,6,4] => [7,6,5,1,3,2,4] => ? = 3
[1,2,3,7,6,5,4] => [1,2,3,6,5,7,4] => [7,6,5,2,3,1,4] => ? = 4
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [7,6,4,5,3,2,1] => ? = 5
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [7,6,4,5,3,1,2] => ? = 4
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [7,6,4,5,2,3,1] => ? = 4
[1,2,4,3,6,7,5] => [1,2,4,3,7,5,6] => [7,6,4,5,1,3,2] => ? = 3
[1,2,4,3,7,5,6] => [1,2,4,3,7,6,5] => [7,6,4,5,1,2,3] => ? = 3
[1,2,4,3,7,6,5] => [1,2,4,3,6,7,5] => [7,6,4,5,2,1,3] => ? = 4
[1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => [7,6,3,5,4,2,1] => ? = 4
[1,2,4,5,3,7,6] => [1,2,5,3,4,7,6] => [7,6,3,5,4,1,2] => ? = 3
[1,2,4,5,6,3,7] => [1,2,6,3,4,5,7] => [7,6,2,5,4,3,1] => ? = 3
[1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => [7,6,1,5,4,3,2] => ? = 2
[1,2,4,5,7,3,6] => [1,2,7,6,3,4,5] => [7,6,1,2,5,4,3] => ? = 2
[1,2,4,5,7,6,3] => [1,2,6,7,3,4,5] => [7,6,2,1,5,4,3] => ? = 3
[1,2,4,6,3,5,7] => [1,2,6,5,3,4,7] => [7,6,2,3,5,4,1] => ? = 3
[1,2,4,6,3,7,5] => [1,2,7,5,3,4,6] => [7,6,1,3,5,4,2] => ? = 2
[1,2,4,6,5,3,7] => [1,2,5,6,3,4,7] => [7,6,3,2,5,4,1] => ? = 4
[1,2,4,6,5,7,3] => [1,2,5,7,3,4,6] => [7,6,3,1,5,4,2] => ? = 3
[1,2,4,6,7,3,5] => [1,2,6,3,4,7,5] => [7,6,2,5,4,1,3] => ? = 3
[1,2,4,6,7,5,3] => [1,2,7,3,4,6,5] => [7,6,1,5,4,2,3] => ? = 2
[1,2,4,7,3,5,6] => [1,2,7,6,5,3,4] => [7,6,1,2,3,5,4] => ? = 2
[1,2,4,7,3,6,5] => [1,2,6,7,5,3,4] => [7,6,2,1,3,5,4] => ? = 3
[1,2,4,7,5,3,6] => [1,2,5,7,6,3,4] => [7,6,3,1,2,5,4] => ? = 3
[1,2,4,7,5,6,3] => [1,2,5,6,7,3,4] => [7,6,3,2,1,5,4] => ? = 4
[1,2,4,7,6,3,5] => [1,2,7,5,6,3,4] => [7,6,1,3,2,5,4] => ? = 2
[1,2,4,7,6,5,3] => [1,2,6,5,7,3,4] => [7,6,2,3,1,5,4] => ? = 3
[1,2,5,3,4,6,7] => [1,2,5,4,3,6,7] => [7,6,3,4,5,2,1] => ? = 4
[1,2,5,3,4,7,6] => [1,2,5,4,3,7,6] => [7,6,3,4,5,1,2] => ? = 3
Description
The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right.
For a permutation π of length n, this is the number of indices 2≤j≤n such that for all 1≤i<j, the pair (i,j) is an inversion of π.
The following 40 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St000015The number of peaks of a Dyck path. St000542The number of left-to-right-minima of a permutation. St000991The number of right-to-left minima of a permutation. St000021The number of descents of a permutation. St000053The number of valleys of the Dyck path. St000083The number of left oriented leafs of a binary tree except the first one. St000155The number of exceedances (also excedences) of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000354The number of recoils of a permutation. St000619The number of cyclic descents of a permutation. St000831The number of indices that are either descents or recoils. St001061The number of indices that are both descents and recoils of a permutation. St001205The number of non-simple indecomposable projective-injective modules of the algebra eAe in the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by τΩ1 composed with its inverse in the corresponding Nakayama algebra. St001489The maximum of the number of descents and the number of inverse descents. St000061The number of nodes on the left branch of a binary tree. St000062The length of the longest increasing subsequence of the permutation. St000084The number of subtrees. St000213The number of weak exceedances (also weak excedences) of a permutation. St000239The number of small weak excedances. St000325The width of the tree associated to a permutation. St000443The number of long tunnels of a Dyck path. St000470The number of runs in a permutation. St000822The Hadwiger number of the graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series L=[c0,c1,...,cn−1] such that n=c0<ci for all i>0 a special CNakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001812The biclique partition number of a graph. St001330The hat guessing number of a graph. St000702The number of weak deficiencies of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001769The reflection length of a signed permutation.
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