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Your data matches 73 different statistics following compositions of up to 3 maps.
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Matching statistic: St001669
(load all 17 compositions to match this statistic)
(load all 17 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
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: 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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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
Description
The number of rises of length 1 of a Dyck path.
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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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
Description
The number of parts equal to 1 in a partition.
Matching statistic: St000674
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000674: Dyck paths ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000674: Dyck paths ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%
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]
=> ? ∊ {0,1,2,2}
[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]
=> ? ∊ {0,1,2,2}
[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]
=> ? ∊ {0,1,2,2}
[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]
=> ? ∊ {0,1,2,2}
Description
The number of hills of a Dyck path.
A hill is a peak with up step starting and down step ending at height zero.
Matching statistic: St001817
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001817: Signed permutations ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 83%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001817: Signed permutations ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 83%
Values
[1,0]
=> [1,1,0,0]
=> [1,2] => [1,2] => 4 = 2 + 2
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [3,1,2] => [3,1,2] => 2 = 0 + 2
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 6 = 4 + 2
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [3,4,1,2] => 4 = 2 + 2
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,2,4] => 4 = 2 + 2
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => 4 = 2 + 2
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4,1,2,3] => 2 = 0 + 2
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 8 = 6 + 2
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [3,4,5,1,2] => ? ∊ {0,0,1,1,2,2,4,4,4} + 2
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? ∊ {0,0,1,1,2,2,4,4,4} + 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [3,1,5,2,4] => ? ∊ {0,0,1,1,2,2,4,4,4} + 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [3,5,1,2,4] => ? ∊ {0,0,1,1,2,2,4,4,4} + 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,1,2,4,5] => ? ∊ {0,0,1,1,2,2,4,4,4} + 2
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => [1,4,5,2,3] => 6 = 4 + 2
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => [1,4,2,3,5] => 6 = 4 + 2
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => [4,1,5,2,3] => ? ∊ {0,0,1,1,2,2,4,4,4} + 2
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? ∊ {0,0,1,1,2,2,4,4,4} + 2
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => [4,1,2,3,5] => ? ∊ {0,0,1,1,2,2,4,4,4} + 2
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => [1,2,5,3,4] => 6 = 4 + 2
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => [1,5,2,3,4] => 4 = 2 + 2
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => [5,1,2,3,4] => ? ∊ {0,0,1,1,2,2,4,4,4} + 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 10 = 8 + 2
Description
The number of flag weak exceedances of a signed permutation.
This is the number of negative entries plus twice the number of weak exceedances of the signed permutation.
Matching statistic: St001885
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001885: Binary words ⟶ ℤResult quality: 50% ●values known / values provided: 55%●distinct values known / distinct values provided: 50%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001885: Binary words ⟶ ℤResult quality: 50% ●values known / values provided: 55%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> []
=> ?
=> ? => ? = 2
[1,0,1,0]
=> [1]
=> []
=> => ? ∊ {0,4}
[1,1,0,0]
=> []
=> ?
=> ? => ? ∊ {0,4}
[1,0,1,0,1,0]
=> [2,1]
=> [1]
=> 10 => 2
[1,0,1,1,0,0]
=> [1,1]
=> [1]
=> 10 => 2
[1,1,0,0,1,0]
=> [2]
=> []
=> => ? ∊ {0,2,6}
[1,1,0,1,0,0]
=> [1]
=> []
=> => ? ∊ {0,2,6}
[1,1,1,0,0,0]
=> []
=> ?
=> ? => ? ∊ {0,2,6}
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [2,1]
=> 1010 => 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [2,1]
=> 1010 => 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,1]
=> 110 => 4
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> 110 => 4
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 110 => 4
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [2]
=> 100 => 4
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> 100 => 4
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> 10 => 2
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 10 => 2
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 10 => 2
[1,1,1,0,0,0,1,0]
=> [3]
=> []
=> => ? ∊ {0,0,4,8}
[1,1,1,0,0,1,0,0]
=> [2]
=> []
=> => ? ∊ {0,0,4,8}
[1,1,1,0,1,0,0,0]
=> [1]
=> []
=> => ? ∊ {0,0,4,8}
[1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? => ? ∊ {0,0,4,8}
Description
The number of binary words with the same proper border set.
The proper border set of a binary word $w$ is the set of proper prefixes which are also suffixes of $w$.
For example, $0010000010$, $0010100010$ and $0010110010$ are the words with proper border set $\{0, 0010\}$, whereas $0010010010$ has proper border set $\{0, 0010, 0010010\}$.
Matching statistic: St001459
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001459: Graphs ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 67%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001459: Graphs ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [[1],[2]]
=> [2,1] => ([(0,1)],2)
=> 2
[1,0,1,0]
=> [[1,3],[2,4]]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 4
[1,1,0,0]
=> [[1,2],[3,4]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 0
[1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 6
[1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> [2,5,6,1,3,4] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 2
[1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> [3,4,6,1,2,5] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 2
[1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> [3,5,6,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 0
[1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> [2,4,6,8,1,3,5,7] => ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,1,1,2,2,2,4,4,4,4,4,4,8}
[1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> [2,4,7,8,1,3,5,6] => ([(0,7),(1,5),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,1,1,2,2,2,4,4,4,4,4,4,8}
[1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> [2,5,6,8,1,3,4,7] => ([(0,7),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,0,1,1,2,2,2,4,4,4,4,4,4,8}
[1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> [2,5,7,8,1,3,4,6] => ([(0,7),(1,5),(1,6),(2,3),(2,4),(2,7),(3,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,1,1,2,2,2,4,4,4,4,4,4,8}
[1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> [2,6,7,8,1,3,4,5] => ([(0,7),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,0,1,1,2,2,2,4,4,4,4,4,4,8}
[1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> [3,4,6,8,1,2,5,7] => ([(0,7),(1,5),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,1,1,2,2,2,4,4,4,4,4,4,8}
[1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> [3,4,7,8,1,2,5,6] => ([(0,6),(0,7),(1,4),(1,5),(2,4),(2,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? ∊ {0,0,1,1,2,2,2,4,4,4,4,4,4,8}
[1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> [3,5,6,8,1,2,4,7] => ([(0,7),(1,5),(1,6),(2,3),(2,4),(2,7),(3,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,1,1,2,2,2,4,4,4,4,4,4,8}
[1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> [3,5,7,8,1,2,4,6] => ([(0,6),(0,7),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? ∊ {0,0,1,1,2,2,2,4,4,4,4,4,4,8}
[1,1,0,1,1,0,0,0]
=> [[1,2,4,5],[3,6,7,8]]
=> [3,6,7,8,1,2,4,5] => ([(0,6),(0,7),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? ∊ {0,0,1,1,2,2,2,4,4,4,4,4,4,8}
[1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> [4,5,6,8,1,2,3,7] => ([(0,7),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,0,1,1,2,2,2,4,4,4,4,4,4,8}
[1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> [4,5,7,8,1,2,3,6] => ([(0,6),(0,7),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? ∊ {0,0,1,1,2,2,2,4,4,4,4,4,4,8}
[1,1,1,0,1,0,0,0]
=> [[1,2,3,5],[4,6,7,8]]
=> [4,6,7,8,1,2,3,5] => ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? ∊ {0,0,1,1,2,2,2,4,4,4,4,4,4,8}
[1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ? ∊ {0,0,1,1,2,2,2,4,4,4,4,4,4,8}
Description
The number of zero columns in the nullspace of a graph.
Matching statistic: St001605
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001605: Integer partitions ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 50%
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001605: Integer partitions ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [1,0]
=> [1,0]
=> []
=> ? = 2
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> []
=> ? ∊ {0,4}
[1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1]
=> ? ∊ {0,4}
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,2,2,2,6}
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2]
=> ? ∊ {0,2,2,2,6}
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> ? ∊ {0,2,2,2,6}
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2]
=> ? ∊ {0,2,2,2,6}
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [1]
=> ? ∊ {0,2,2,2,6}
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,4,4,4,8}
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> ? ∊ {0,0,4,4,4,8}
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 2
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 4
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 4
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> ? ∊ {0,0,4,4,4,8}
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> ? ∊ {0,0,4,4,4,8}
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 4
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> ? ∊ {0,0,4,4,4,8}
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {0,0,4,4,4,8}
Description
The number of colourings of a cycle such that the multiplicities of colours are given by a partition.
Two colourings are considered equal, if they are obtained by an action of the cyclic group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St000691
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St000691: Binary words ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 67%
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St000691: Binary words ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1010 => 3 = 2 + 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,0,0]
=> 111000 => 1 = 0 + 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 101010 => 5 = 4 + 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => 3 = 2 + 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 3 = 2 + 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 11101000 => 3 = 2 + 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 7 = 6 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1111010000 => ? ∊ {0,0,1,1,2,2,2,4,4,4,4,4,4,8} + 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? ∊ {0,0,1,1,2,2,2,4,4,4,4,4,4,8} + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1110011000 => ? ∊ {0,0,1,1,2,2,2,4,4,4,4,4,4,8} + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1111001000 => ? ∊ {0,0,1,1,2,2,2,4,4,4,4,4,4,8} + 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1110001010 => ? ∊ {0,0,1,1,2,2,2,4,4,4,4,4,4,8} + 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? ∊ {0,0,1,1,2,2,2,4,4,4,4,4,4,8} + 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1011100010 => ? ∊ {0,0,1,1,2,2,2,4,4,4,4,4,4,8} + 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1110110000 => ? ∊ {0,0,1,1,2,2,2,4,4,4,4,4,4,8} + 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1111100000 => ? ∊ {0,0,1,1,2,2,2,4,4,4,4,4,4,8} + 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? ∊ {0,0,1,1,2,2,2,4,4,4,4,4,4,8} + 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? ∊ {0,0,1,1,2,2,2,4,4,4,4,4,4,8} + 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? ∊ {0,0,1,1,2,2,2,4,4,4,4,4,4,8} + 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1110101000 => ? ∊ {0,0,1,1,2,2,2,4,4,4,4,4,4,8} + 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? ∊ {0,0,1,1,2,2,2,4,4,4,4,4,4,8} + 1
Description
The number of changes of a binary word.
This is the number of indices $i$ such that $w_i \neq w_{i+1}$.
Matching statistic: St001097
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001097: Integer partitions ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 50%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001097: Integer partitions ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> []
=> ?
=> ? = 2
[1,0,1,0]
=> [1]
=> []
=> ? ∊ {0,4}
[1,1,0,0]
=> []
=> ?
=> ? ∊ {0,4}
[1,0,1,0,1,0]
=> [2,1]
=> [1]
=> ? ∊ {0,2,2,2,6}
[1,0,1,1,0,0]
=> [1,1]
=> [1]
=> ? ∊ {0,2,2,2,6}
[1,1,0,0,1,0]
=> [2]
=> []
=> ? ∊ {0,2,2,2,6}
[1,1,0,1,0,0]
=> [1]
=> []
=> ? ∊ {0,2,2,2,6}
[1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {0,2,2,2,6}
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [2,1]
=> 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [2,1]
=> 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,1]
=> 2
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 2
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [2]
=> 0
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> 0
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> ? ∊ {4,4,4,4,4,4,8}
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> ? ∊ {4,4,4,4,4,4,8}
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> ? ∊ {4,4,4,4,4,4,8}
[1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ? ∊ {4,4,4,4,4,4,8}
[1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {4,4,4,4,4,4,8}
[1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {4,4,4,4,4,4,8}
[1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {4,4,4,4,4,4,8}
Description
The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders.
For a generating function $f$ the associated formal group law is the symmetric function $f(f^{(-1)}(x_1) + f^{(-1)}(x_2), \dots)$, see [1].
This statistic records the coefficient of the monomial symmetric function $m_\lambda$ in the formal group law for linear orders, with generating function $f(x) = x/(1-x)$, see [1, sec. 3.4].
This statistic gives the number of Smirnov arrangements of a set of letters with $\lambda_i$ of the $i$th letter, where a Smirnov word is a word with no repeated adjacent letters. e.g., [3,2,1] = > 10 since there are 10 Smirnov rearrangements of the word 'aaabbc': 'ababac', 'ababca', 'abacab', 'abacba', 'abcaba', 'acabab', 'acbaba', 'babaca', 'bacaba', 'cababa'.
The following 63 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000770The major index of an integer partition when read from bottom to top. St001545The second Elser number of a connected graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000567The sum of the products of all pairs of parts. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000668The least common multiple of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000706The product of the factorials of the multiplicities of an integer partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001568The smallest positive integer that does not appear twice in the partition. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000070The number of antichains in a poset. St000422The energy of a graph, if it is integral. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000264The girth of a graph, which is not a tree. St000454The largest eigenvalue of a graph if it is integral. St000478Another weight of a partition according to Alladi. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000815The number of semistandard Young tableaux of partition weight of given shape. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000934The 2-degree of an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001060The distinguishing index of a graph. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001405The number of bonds in a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001516The number of cyclic bonds of a permutation. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001637The number of (upper) dissectors of a poset. St001645The pebbling number of a connected graph. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000528The height of a poset. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000907The number of maximal antichains of minimal length in a poset. St000911The number of maximal antichains of maximal size in a poset. St000912The number of maximal antichains in a poset. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001343The dimension of the reduced incidence algebra of a poset. St000260The radius of a connected graph. St000487The length of the shortest cycle of a permutation. St001807The lower middle entry of a permutation. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001964The interval resolution global dimension of a poset.
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