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Your data matches 3 different statistics following compositions of up to 3 maps.
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Matching statistic: St001669
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(load all 8 compositions to match this statistic)
St001669: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 2
[1,0,1,0]
=> 4
[1,1,0,0]
=> 0
[1,0,1,0,1,0]
=> 6
[1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0]
=> 2
[1,1,1,0,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> 8
[1,0,1,0,1,1,0,0]
=> 4
[1,0,1,1,0,0,1,0]
=> 4
[1,0,1,1,0,1,0,0]
=> 4
[1,0,1,1,1,0,0,0]
=> 2
[1,1,0,0,1,0,1,0]
=> 4
[1,1,0,0,1,1,0,0]
=> 0
[1,1,0,1,0,0,1,0]
=> 4
[1,1,0,1,0,1,0,0]
=> 4
[1,1,0,1,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> 2
[1,1,1,0,0,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> 2
[1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> 10
[1,0,1,0,1,0,1,1,0,0]
=> 6
[1,0,1,0,1,1,0,0,1,0]
=> 6
[1,0,1,0,1,1,0,1,0,0]
=> 6
[1,0,1,0,1,1,1,0,0,0]
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> 6
[1,0,1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> 6
[1,0,1,1,0,1,0,1,0,0]
=> 6
[1,0,1,1,0,1,1,0,0,0]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> 6
[1,1,0,0,1,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> 6
[1,1,0,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> 6
[1,1,0,1,0,1,0,1,0,0]
=> 6
[1,1,0,1,0,1,1,0,0,0]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> 1
Description
The number of single rises in a Dyck path.
A single rise is a step which is neither preceded nor followed by a step of the same kind.
Matching statistic: St000475
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 71%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 71%
Values
[1,0]
=> 10 => [1,1] => [1,1]
=> 2
[1,0,1,0]
=> 1010 => [1,1,1,1] => [1,1,1,1]
=> 4
[1,1,0,0]
=> 1100 => [2,2] => [2,2]
=> 0
[1,0,1,0,1,0]
=> 101010 => [1,1,1,1,1,1] => [1,1,1,1,1,1]
=> 6
[1,0,1,1,0,0]
=> 101100 => [1,1,2,2] => [2,2,1,1]
=> 2
[1,1,0,0,1,0]
=> 110010 => [2,2,1,1] => [2,2,1,1]
=> 2
[1,1,0,1,0,0]
=> 110100 => [2,1,1,2] => [2,2,1,1]
=> 2
[1,1,1,0,0,0]
=> 111000 => [3,3] => [3,3]
=> 0
[1,0,1,0,1,0,1,0]
=> 10101010 => [1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1]
=> 8
[1,0,1,0,1,1,0,0]
=> 10101100 => [1,1,1,1,2,2] => [2,2,1,1,1,1]
=> 4
[1,0,1,1,0,0,1,0]
=> 10110010 => [1,1,2,2,1,1] => [2,2,1,1,1,1]
=> 4
[1,0,1,1,0,1,0,0]
=> 10110100 => [1,1,2,1,1,2] => [2,2,1,1,1,1]
=> 4
[1,0,1,1,1,0,0,0]
=> 10111000 => [1,1,3,3] => [3,3,1,1]
=> 2
[1,1,0,0,1,0,1,0]
=> 11001010 => [2,2,1,1,1,1] => [2,2,1,1,1,1]
=> 4
[1,1,0,0,1,1,0,0]
=> 11001100 => [2,2,2,2] => [2,2,2,2]
=> 0
[1,1,0,1,0,0,1,0]
=> 11010010 => [2,1,1,2,1,1] => [2,2,1,1,1,1]
=> 4
[1,1,0,1,0,1,0,0]
=> 11010100 => [2,1,1,1,1,2] => [2,2,1,1,1,1]
=> 4
[1,1,0,1,1,0,0,0]
=> 11011000 => [2,1,2,3] => [3,2,2,1]
=> 1
[1,1,1,0,0,0,1,0]
=> 11100010 => [3,3,1,1] => [3,3,1,1]
=> 2
[1,1,1,0,0,1,0,0]
=> 11100100 => [3,2,1,2] => [3,2,2,1]
=> 1
[1,1,1,0,1,0,0,0]
=> 11101000 => [3,1,1,3] => [3,3,1,1]
=> 2
[1,1,1,1,0,0,0,0]
=> 11110000 => [4,4] => [4,4]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => [1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1]
=> 10
[1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => [1,1,1,1,1,1,2,2] => [2,2,1,1,1,1,1,1]
=> 6
[1,0,1,0,1,1,0,0,1,0]
=> 1010110010 => [1,1,1,1,2,2,1,1] => [2,2,1,1,1,1,1,1]
=> 6
[1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => [1,1,1,1,2,1,1,2] => [2,2,1,1,1,1,1,1]
=> 6
[1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => [1,1,1,1,3,3] => [3,3,1,1,1,1]
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> 1011001010 => [1,1,2,2,1,1,1,1] => [2,2,1,1,1,1,1,1]
=> 6
[1,0,1,1,0,0,1,1,0,0]
=> 1011001100 => [1,1,2,2,2,2] => [2,2,2,2,1,1]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> 1011010010 => [1,1,2,1,1,2,1,1] => [2,2,1,1,1,1,1,1]
=> 6
[1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => [1,1,2,1,1,1,1,2] => [2,2,1,1,1,1,1,1]
=> 6
[1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => [1,1,2,1,2,3] => [3,2,2,1,1,1]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> 1011100010 => [1,1,3,3,1,1] => [3,3,1,1,1,1]
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => [1,1,3,2,1,2] => [3,2,2,1,1,1]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => [1,1,3,1,1,3] => [3,3,1,1,1,1]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => [1,1,4,4] => [4,4,1,1]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> 1100101010 => [2,2,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1]
=> 6
[1,1,0,0,1,0,1,1,0,0]
=> 1100101100 => [2,2,1,1,2,2] => [2,2,2,2,1,1]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> 1100110010 => [2,2,2,2,1,1] => [2,2,2,2,1,1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> 1100110100 => [2,2,2,1,1,2] => [2,2,2,2,1,1]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => [2,2,3,3] => [3,3,2,2]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> 1101001010 => [2,1,1,2,1,1,1,1] => [2,2,1,1,1,1,1,1]
=> 6
[1,1,0,1,0,0,1,1,0,0]
=> 1101001100 => [2,1,1,2,2,2] => [2,2,2,2,1,1]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> 1101010010 => [2,1,1,1,1,2,1,1] => [2,2,1,1,1,1,1,1]
=> 6
[1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => [2,1,1,1,1,1,1,2] => [2,2,1,1,1,1,1,1]
=> 6
[1,1,0,1,0,1,1,0,0,0]
=> 1101011000 => [2,1,1,1,2,3] => [3,2,2,1,1,1]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => [2,1,2,3,1,1] => [3,2,2,1,1,1]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> 1101100100 => [2,1,2,2,1,2] => [2,2,2,2,1,1]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> 1101101000 => [2,1,2,1,1,3] => [3,2,2,1,1,1]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> 1101110000 => [2,1,3,4] => [4,3,2,1]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 10101010101010 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1] => ?
=> ? = 14
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 10101010101100 => [1,1,1,1,1,1,1,1,1,1,2,2] => ?
=> ? = 10
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> 10101010110010 => [1,1,1,1,1,1,1,1,2,2,1,1] => ?
=> ? = 10
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 10101010110100 => [1,1,1,1,1,1,1,1,2,1,1,2] => ?
=> ? = 10
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 10101010111000 => [1,1,1,1,1,1,1,1,3,3] => ?
=> ? = 8
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> 10101011001010 => [1,1,1,1,1,1,2,2,1,1,1,1] => ?
=> ? = 10
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> 10101011001100 => [1,1,1,1,1,1,2,2,2,2] => ?
=> ? = 6
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> 10101011010010 => [1,1,1,1,1,1,2,1,1,2,1,1] => ?
=> ? = 10
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> 10101011010100 => [1,1,1,1,1,1,2,1,1,1,1,2] => ?
=> ? = 10
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> 10101011011000 => [1,1,1,1,1,1,2,1,2,3] => ?
=> ? = 7
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> 10101011100010 => [1,1,1,1,1,1,3,3,1,1] => ?
=> ? = 8
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> 10101011100100 => [1,1,1,1,1,1,3,2,1,2] => ?
=> ? = 7
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> 10101011101000 => [1,1,1,1,1,1,3,1,1,3] => ?
=> ? = 8
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> 10101011110000 => [1,1,1,1,1,1,4,4] => ?
=> ? = 6
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> 10101100101010 => [1,1,1,1,2,2,1,1,1,1,1,1] => ?
=> ? = 10
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> 10101100101100 => [1,1,1,1,2,2,1,1,2,2] => ?
=> ? = 6
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> 10101100110010 => [1,1,1,1,2,2,2,2,1,1] => ?
=> ? = 6
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> 10101100110100 => [1,1,1,1,2,2,2,1,1,2] => ?
=> ? = 6
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> 10101100111000 => [1,1,1,1,2,2,3,3] => ?
=> ? = 4
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> 10101101001010 => [1,1,1,1,2,1,1,2,1,1,1,1] => ?
=> ? = 10
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> 10101101001100 => [1,1,1,1,2,1,1,2,2,2] => ?
=> ? = 6
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> 10101101010010 => [1,1,1,1,2,1,1,1,1,2,1,1] => ?
=> ? = 10
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> 10101101010100 => [1,1,1,1,2,1,1,1,1,1,1,2] => ?
=> ? = 10
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> 10101101011000 => [1,1,1,1,2,1,1,1,2,3] => ?
=> ? = 7
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> 10101101100010 => [1,1,1,1,2,1,2,3,1,1] => ?
=> ? = 7
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> 10101101100100 => [1,1,1,1,2,1,2,2,1,2] => ?
=> ? = 6
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> 10101101101000 => [1,1,1,1,2,1,2,1,1,3] => ?
=> ? = 7
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> 10101101110000 => [1,1,1,1,2,1,3,4] => ?
=> ? = 5
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> 10101110001010 => [1,1,1,1,3,3,1,1,1,1] => ?
=> ? = 8
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> 10101110001100 => [1,1,1,1,3,3,2,2] => ?
=> ? = 4
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> 10101110010010 => [1,1,1,1,3,2,1,2,1,1] => ?
=> ? = 7
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> 10101110010100 => [1,1,1,1,3,2,1,1,1,2] => ?
=> ? = 7
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> 10101110011000 => [1,1,1,1,3,2,2,3] => ?
=> ? = 4
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> 10101110100010 => [1,1,1,1,3,1,1,3,1,1] => ?
=> ? = 8
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> 10101110100100 => [1,1,1,1,3,1,1,2,1,2] => ?
=> ? = 7
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> 10101110101000 => [1,1,1,1,3,1,1,1,1,3] => ?
=> ? = 8
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> 10101110110000 => [1,1,1,1,3,1,2,4] => ?
=> ? = 5
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> 10101111000010 => [1,1,1,1,4,4,1,1] => ?
=> ? = 6
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> 10101111000100 => [1,1,1,1,4,3,1,2] => ?
=> ? = 5
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> 10101111001000 => [1,1,1,1,4,2,1,3] => ?
=> ? = 5
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> 10101111010000 => [1,1,1,1,4,1,1,4] => ?
=> ? = 6
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> 10101111100000 => [1,1,1,1,5,5] => ?
=> ? = 4
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> 10110010101010 => [1,1,2,2,1,1,1,1,1,1,1,1] => ?
=> ? = 10
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> 10110010101100 => [1,1,2,2,1,1,1,1,2,2] => ?
=> ? = 6
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> 10110010110010 => [1,1,2,2,1,1,2,2,1,1] => ?
=> ? = 6
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> 10110010110100 => [1,1,2,2,1,1,2,1,1,2] => ?
=> ? = 6
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> 10110010111000 => [1,1,2,2,1,1,3,3] => ?
=> ? = 4
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> 10110011001010 => [1,1,2,2,2,2,1,1,1,1] => ?
=> ? = 6
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> 10110011001100 => [1,1,2,2,2,2,2,2] => ?
=> ? = 2
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> 10110011010010 => [1,1,2,2,2,1,1,2,1,1] => ?
=> ? = 6
Description
The number of parts equal to 1 in a partition.
Matching statistic: St000445
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 57%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 57%
Values
[1,0]
=> 10 => [1,1] => [1,0,1,0]
=> 2
[1,0,1,0]
=> 1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[1,1,0,0]
=> 1100 => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[1,0,1,0,1,0]
=> 101010 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[1,0,1,1,0,0]
=> 101100 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,0,1,0]
=> 110010 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> 2
[1,1,0,1,0,0]
=> 110100 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 2
[1,1,1,0,0,0]
=> 111000 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> 10101010 => [1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 8
[1,0,1,0,1,1,0,0]
=> 10101100 => [1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> 4
[1,0,1,1,0,0,1,0]
=> 10110010 => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> 4
[1,0,1,1,0,1,0,0]
=> 10110100 => [1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> 4
[1,0,1,1,1,0,0,0]
=> 10111000 => [1,1,3,3] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> 2
[1,1,0,0,1,0,1,0]
=> 11001010 => [2,2,1,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> 4
[1,1,0,0,1,1,0,0]
=> 11001100 => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> 0
[1,1,0,1,0,0,1,0]
=> 11010010 => [2,1,1,2,1,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> 4
[1,1,0,1,0,1,0,0]
=> 11010100 => [2,1,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 4
[1,1,0,1,1,0,0,0]
=> 11011000 => [2,1,2,3] => [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> 11100010 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> 2
[1,1,1,0,0,1,0,0]
=> 11100100 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> 11101000 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> 2
[1,1,1,1,0,0,0,0]
=> 11110000 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => [1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 10
[1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => [1,1,1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> 6
[1,0,1,0,1,1,0,0,1,0]
=> 1010110010 => [1,1,1,1,2,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 6
[1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => [1,1,1,1,2,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 6
[1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => [1,1,1,1,3,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 4
[1,0,1,1,0,0,1,0,1,0]
=> 1011001010 => [1,1,2,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 6
[1,0,1,1,0,0,1,1,0,0]
=> 1011001100 => [1,1,2,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> 1011010010 => [1,1,2,1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 6
[1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => [1,1,2,1,1,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 6
[1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => [1,1,2,1,2,3] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3
[1,0,1,1,1,0,0,0,1,0]
=> 1011100010 => [1,1,3,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 4
[1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => [1,1,3,2,1,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3
[1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => [1,1,3,1,1,3] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 4
[1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => [1,1,4,4] => [1,0,1,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 2
[1,1,0,0,1,0,1,0,1,0]
=> 1100101010 => [2,2,1,1,1,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[1,1,0,0,1,0,1,1,0,0]
=> 1100101100 => [2,2,1,1,2,2] => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 2
[1,1,0,0,1,1,0,0,1,0]
=> 1100110010 => [2,2,2,2,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 2
[1,1,0,0,1,1,0,1,0,0]
=> 1100110100 => [2,2,2,1,1,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 2
[1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => [2,2,3,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 0
[1,1,0,1,0,0,1,0,1,0]
=> 1101001010 => [2,1,1,2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 6
[1,1,0,1,0,0,1,1,0,0]
=> 1101001100 => [2,1,1,2,2,2] => [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 2
[1,1,0,1,0,1,0,0,1,0]
=> 1101010010 => [2,1,1,1,1,2,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 6
[1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => [2,1,1,1,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 6
[1,1,0,1,0,1,1,0,0,0]
=> 1101011000 => [2,1,1,1,2,3] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3
[1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => [2,1,2,3,1,1] => [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3
[1,1,0,1,1,0,0,1,0,0]
=> 1101100100 => [2,1,2,2,1,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 2
[1,1,0,1,1,0,1,0,0,0]
=> 1101101000 => [2,1,2,1,1,3] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3
[1,1,0,1,1,1,0,0,0,0]
=> 1101110000 => [2,1,3,4] => [1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> 1110001010 => [3,3,1,1,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 4
[1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => [3,3,2,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0
[1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => [3,2,1,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 3
[1,1,1,0,0,1,0,1,0,0]
=> 1110010100 => [3,2,1,1,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 3
[1,1,1,0,0,1,1,0,0,0]
=> 1110011000 => [3,2,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0
[1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => [3,1,1,3,1,1] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 4
[1,1,1,0,1,0,0,1,0,0]
=> 1110100100 => [3,1,1,2,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3
[1,1,1,0,1,0,1,0,0,0]
=> 1110101000 => [3,1,1,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 4
[1,1,1,0,1,1,0,0,0,0]
=> 1110110000 => [3,1,2,4] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => [4,4,1,1] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 2
[1,1,1,1,0,0,0,1,0,0]
=> 1111000100 => [4,3,1,2] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> 1111001000 => [4,2,1,3] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> 1111010000 => [4,1,1,4] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2
[1,1,1,1,1,0,0,0,0,0]
=> 1111100000 => [5,5] => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> 101010101010 => [1,1,1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 12
[1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101100 => [1,1,1,1,1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 8
[1,0,1,0,1,0,1,1,0,0,1,0]
=> 101010110010 => [1,1,1,1,1,1,2,2,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 8
[1,0,1,0,1,0,1,1,0,1,0,0]
=> 101010110100 => [1,1,1,1,1,1,2,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 8
[1,0,1,0,1,0,1,1,1,0,0,0]
=> 101010111000 => [1,1,1,1,1,1,3,3] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 6
[1,0,1,0,1,1,0,0,1,0,1,0]
=> 101011001010 => [1,1,1,1,2,2,1,1,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 8
[1,0,1,0,1,1,0,0,1,1,0,0]
=> 101011001100 => [1,1,1,1,2,2,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 4
[1,0,1,0,1,1,0,1,0,0,1,0]
=> 101011010010 => [1,1,1,1,2,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 8
[1,0,1,0,1,1,0,1,0,1,0,0]
=> 101011010100 => [1,1,1,1,2,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 8
[1,0,1,0,1,1,0,1,1,0,0,0]
=> 101011011000 => [1,1,1,1,2,1,2,3] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 5
[1,0,1,0,1,1,1,0,0,0,1,0]
=> 101011100010 => [1,1,1,1,3,3,1,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 6
[1,0,1,0,1,1,1,0,0,1,0,0]
=> 101011100100 => [1,1,1,1,3,2,1,2] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 5
[1,0,1,0,1,1,1,0,1,0,0,0]
=> 101011101000 => [1,1,1,1,3,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 6
[1,0,1,0,1,1,1,1,0,0,0,0]
=> 101011110000 => [1,1,1,1,4,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[1,0,1,1,0,0,1,0,1,0,1,0]
=> 101100101010 => [1,1,2,2,1,1,1,1,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 8
[1,0,1,1,0,0,1,0,1,1,0,0]
=> 101100101100 => [1,1,2,2,1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 4
[1,0,1,1,0,0,1,1,0,0,1,0]
=> 101100110010 => [1,1,2,2,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 4
[1,0,1,1,0,0,1,1,0,1,0,0]
=> 101100110100 => [1,1,2,2,2,1,1,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 4
Description
The number of rises of length 1 of a Dyck path.
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