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Your data matches 70 different statistics following compositions of up to 3 maps.
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Matching statistic: St000148
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
St000148: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1
[2]
=> 0
[1,1]
=> 2
[3]
=> 1
[2,1]
=> 1
[1,1,1]
=> 3
[4]
=> 0
[3,1]
=> 2
[2,2]
=> 0
[2,1,1]
=> 2
[1,1,1,1]
=> 4
[5]
=> 1
[4,1]
=> 1
[3,2]
=> 1
[3,1,1]
=> 3
[2,2,1]
=> 1
[2,1,1,1]
=> 3
[1,1,1,1,1]
=> 5
[6]
=> 0
[5,1]
=> 2
[4,2]
=> 0
[4,1,1]
=> 2
[3,3]
=> 2
[3,2,1]
=> 2
[3,1,1,1]
=> 4
[2,2,2]
=> 0
[2,2,1,1]
=> 2
[2,1,1,1,1]
=> 4
[1,1,1,1,1,1]
=> 6
Description
The number of odd parts of a partition.
Matching statistic: St000992
St000992: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1
[2]
=> 2
[1,1]
=> 0
[3]
=> 3
[2,1]
=> 1
[1,1,1]
=> 1
[4]
=> 4
[3,1]
=> 2
[2,2]
=> 0
[2,1,1]
=> 2
[1,1,1,1]
=> 0
[5]
=> 5
[4,1]
=> 3
[3,2]
=> 1
[3,1,1]
=> 3
[2,2,1]
=> 1
[2,1,1,1]
=> 1
[1,1,1,1,1]
=> 1
[6]
=> 6
[5,1]
=> 4
[4,2]
=> 2
[4,1,1]
=> 4
[3,3]
=> 0
[3,2,1]
=> 2
[3,1,1,1]
=> 2
[2,2,2]
=> 2
[2,2,1,1]
=> 0
[2,1,1,1,1]
=> 2
[1,1,1,1,1,1]
=> 0
Description
The alternating sum of the parts of an integer partition.
For a partition $\lambda = (\lambda_1,\ldots,\lambda_k)$, this is $\lambda_1 - \lambda_2 + \cdots \pm \lambda_k$.
Matching statistic: St000288
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00317: Integer partitions —odd parts⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1 => 1
[2]
=> 0 => 0
[1,1]
=> 11 => 2
[3]
=> 1 => 1
[2,1]
=> 01 => 1
[1,1,1]
=> 111 => 3
[4]
=> 0 => 0
[3,1]
=> 11 => 2
[2,2]
=> 00 => 0
[2,1,1]
=> 011 => 2
[1,1,1,1]
=> 1111 => 4
[5]
=> 1 => 1
[4,1]
=> 01 => 1
[3,2]
=> 10 => 1
[3,1,1]
=> 111 => 3
[2,2,1]
=> 001 => 1
[2,1,1,1]
=> 0111 => 3
[1,1,1,1,1]
=> 11111 => 5
[6]
=> 0 => 0
[5,1]
=> 11 => 2
[4,2]
=> 00 => 0
[4,1,1]
=> 011 => 2
[3,3]
=> 11 => 2
[3,2,1]
=> 101 => 2
[3,1,1,1]
=> 1111 => 4
[2,2,2]
=> 000 => 0
[2,2,1,1]
=> 0011 => 2
[2,1,1,1,1]
=> 01111 => 4
[1,1,1,1,1,1]
=> 111111 => 6
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St001372
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00317: Integer partitions —odd parts⟶ Binary words
St001372: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001372: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1 => 1
[2]
=> 0 => 0
[1,1]
=> 11 => 2
[3]
=> 1 => 1
[2,1]
=> 01 => 1
[1,1,1]
=> 111 => 3
[4]
=> 0 => 0
[3,1]
=> 11 => 2
[2,2]
=> 00 => 0
[2,1,1]
=> 011 => 2
[1,1,1,1]
=> 1111 => 4
[5]
=> 1 => 1
[4,1]
=> 01 => 1
[3,2]
=> 10 => 1
[3,1,1]
=> 111 => 3
[2,2,1]
=> 001 => 1
[2,1,1,1]
=> 0111 => 3
[1,1,1,1,1]
=> 11111 => 5
[6]
=> 0 => 0
[5,1]
=> 11 => 2
[4,2]
=> 00 => 0
[4,1,1]
=> 011 => 2
[3,3]
=> 11 => 2
[3,2,1]
=> 101 => 2
[3,1,1,1]
=> 1111 => 4
[2,2,2]
=> 000 => 0
[2,2,1,1]
=> 0011 => 2
[2,1,1,1,1]
=> 01111 => 4
[1,1,1,1,1,1]
=> 111111 => 6
Description
The length of a longest cyclic run of ones of a binary word.
Consider the binary word as a cyclic arrangement of ones and zeros. Then this statistic is the length of the longest continuous sequence of ones in this arrangement.
Matching statistic: St000022
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [[1]]
=> [1] => 1
[2]
=> [[1,2]]
=> [1,2] => 2
[1,1]
=> [[1],[2]]
=> [2,1] => 0
[3]
=> [[1,2,3]]
=> [1,2,3] => 3
[2,1]
=> [[1,3],[2]]
=> [2,1,3] => 1
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 4
[3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 2
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 0
[2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 5
[4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 3
[3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 1
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 3
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 1
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 1
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => 6
[5,1]
=> [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => 4
[4,2]
=> [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => 2
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => 4
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => 0
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => 2
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => 2
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => 2
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => 0
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => 2
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => 0
Description
The number of fixed points of a permutation.
Matching statistic: St000392
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00224: Binary words —runsort⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1 => 1 => 1
[2]
=> 0 => 0 => 0
[1,1]
=> 11 => 11 => 2
[3]
=> 1 => 1 => 1
[2,1]
=> 01 => 01 => 1
[1,1,1]
=> 111 => 111 => 3
[4]
=> 0 => 0 => 0
[3,1]
=> 11 => 11 => 2
[2,2]
=> 00 => 00 => 0
[2,1,1]
=> 011 => 011 => 2
[1,1,1,1]
=> 1111 => 1111 => 4
[5]
=> 1 => 1 => 1
[4,1]
=> 01 => 01 => 1
[3,2]
=> 10 => 01 => 1
[3,1,1]
=> 111 => 111 => 3
[2,2,1]
=> 001 => 001 => 1
[2,1,1,1]
=> 0111 => 0111 => 3
[1,1,1,1,1]
=> 11111 => 11111 => 5
[6]
=> 0 => 0 => 0
[5,1]
=> 11 => 11 => 2
[4,2]
=> 00 => 00 => 0
[4,1,1]
=> 011 => 011 => 2
[3,3]
=> 11 => 11 => 2
[3,2,1]
=> 101 => 011 => 2
[3,1,1,1]
=> 1111 => 1111 => 4
[2,2,2]
=> 000 => 000 => 0
[2,2,1,1]
=> 0011 => 0011 => 2
[2,1,1,1,1]
=> 01111 => 01111 => 4
[1,1,1,1,1,1]
=> 111111 => 111111 => 6
Description
The length of the longest run of ones in a binary word.
Matching statistic: St001419
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St001419: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00224: Binary words —runsort⟶ Binary words
St001419: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1 => 1 => 1
[2]
=> 0 => 0 => 0
[1,1]
=> 11 => 11 => 2
[3]
=> 1 => 1 => 1
[2,1]
=> 01 => 01 => 1
[1,1,1]
=> 111 => 111 => 3
[4]
=> 0 => 0 => 0
[3,1]
=> 11 => 11 => 2
[2,2]
=> 00 => 00 => 0
[2,1,1]
=> 011 => 011 => 2
[1,1,1,1]
=> 1111 => 1111 => 4
[5]
=> 1 => 1 => 1
[4,1]
=> 01 => 01 => 1
[3,2]
=> 10 => 01 => 1
[3,1,1]
=> 111 => 111 => 3
[2,2,1]
=> 001 => 001 => 1
[2,1,1,1]
=> 0111 => 0111 => 3
[1,1,1,1,1]
=> 11111 => 11111 => 5
[6]
=> 0 => 0 => 0
[5,1]
=> 11 => 11 => 2
[4,2]
=> 00 => 00 => 0
[4,1,1]
=> 011 => 011 => 2
[3,3]
=> 11 => 11 => 2
[3,2,1]
=> 101 => 011 => 2
[3,1,1,1]
=> 1111 => 1111 => 4
[2,2,2]
=> 000 => 000 => 0
[2,2,1,1]
=> 0011 => 0011 => 2
[2,1,1,1,1]
=> 01111 => 01111 => 4
[1,1,1,1,1,1]
=> 111111 => 111111 => 6
Description
The length of the longest palindromic factor beginning with a one of a binary word.
Matching statistic: St000241
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000241: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000241: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [[1]]
=> [1] => [1] => 1
[2]
=> [[1,2]]
=> [1,2] => [2,1] => 2
[1,1]
=> [[1],[2]]
=> [2,1] => [1,2] => 0
[3]
=> [[1,2,3]]
=> [1,2,3] => [2,3,1] => 3
[2,1]
=> [[1,3],[2]]
=> [2,1,3] => [1,3,2] => 1
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => [2,1,3] => 1
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => [2,3,4,1] => 4
[3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => [1,3,4,2] => 2
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [4,1,2,3] => 0
[2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => [2,1,4,3] => 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [3,2,1,4] => 0
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => [2,3,4,5,1] => 5
[4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => [1,3,4,5,2] => 3
[3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [4,1,2,5,3] => 1
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [2,1,4,5,3] => 3
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [2,5,1,3,4] => 1
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [3,2,1,5,4] => 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [4,3,2,1,5] => 1
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [2,3,4,5,6,1] => 6
[5,1]
=> [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [1,3,4,5,6,2] => 4
[4,2]
=> [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [4,1,2,5,6,3] => 2
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [2,1,4,5,6,3] => 4
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [5,6,1,2,3,4] => 0
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => [2,5,1,3,6,4] => 2
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [3,2,1,5,6,4] => 2
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [6,3,4,1,2,5] => 2
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [3,2,6,1,4,5] => 0
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => [4,3,2,1,6,5] => 2
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [5,4,3,2,1,6] => 0
Description
The number of cyclical small excedances.
A cyclical small excedance is an index $i$ such that $\pi_i = i+1$ considered cyclically.
Matching statistic: St000272
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000272: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000272: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1 => [1,1] => ([(0,1)],2)
=> 1
[2]
=> 0 => [2] => ([],2)
=> 0
[1,1]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[3]
=> 1 => [1,1] => ([(0,1)],2)
=> 1
[2,1]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,1,1]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4]
=> 0 => [2] => ([],2)
=> 0
[3,1]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[2,2]
=> 00 => [3] => ([],3)
=> 0
[2,1,1]
=> 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,1,1]
=> 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[5]
=> 1 => [1,1] => ([(0,1)],2)
=> 1
[4,1]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 1
[3,2]
=> 10 => [1,2] => ([(1,2)],3)
=> 1
[3,1,1]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[2,2,1]
=> 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[2,1,1,1]
=> 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,1,1,1]
=> 11111 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[6]
=> 0 => [2] => ([],2)
=> 0
[5,1]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[4,2]
=> 00 => [3] => ([],3)
=> 0
[4,1,1]
=> 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,3]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[3,2,1]
=> 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,1,1,1]
=> 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[2,2,2]
=> 000 => [4] => ([],4)
=> 0
[2,2,1,1]
=> 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,1,1,1,1]
=> 01111 => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,1,1,1,1]
=> 111111 => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
Description
The treewidth of a graph.
A graph has treewidth zero if and only if it has no edges. A connected graph has treewidth at most one if and only if it is a tree. A connected graph has treewidth at most two if and only if it is a series-parallel graph.
Matching statistic: St000297
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00224: Binary words —runsort⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1 => 1 => 1 => 1
[2]
=> 0 => 0 => 0 => 0
[1,1]
=> 11 => 11 => 11 => 2
[3]
=> 1 => 1 => 1 => 1
[2,1]
=> 01 => 01 => 10 => 1
[1,1,1]
=> 111 => 111 => 111 => 3
[4]
=> 0 => 0 => 0 => 0
[3,1]
=> 11 => 11 => 11 => 2
[2,2]
=> 00 => 00 => 00 => 0
[2,1,1]
=> 011 => 011 => 110 => 2
[1,1,1,1]
=> 1111 => 1111 => 1111 => 4
[5]
=> 1 => 1 => 1 => 1
[4,1]
=> 01 => 01 => 10 => 1
[3,2]
=> 10 => 01 => 10 => 1
[3,1,1]
=> 111 => 111 => 111 => 3
[2,2,1]
=> 001 => 001 => 100 => 1
[2,1,1,1]
=> 0111 => 0111 => 1110 => 3
[1,1,1,1,1]
=> 11111 => 11111 => 11111 => 5
[6]
=> 0 => 0 => 0 => 0
[5,1]
=> 11 => 11 => 11 => 2
[4,2]
=> 00 => 00 => 00 => 0
[4,1,1]
=> 011 => 011 => 110 => 2
[3,3]
=> 11 => 11 => 11 => 2
[3,2,1]
=> 101 => 011 => 110 => 2
[3,1,1,1]
=> 1111 => 1111 => 1111 => 4
[2,2,2]
=> 000 => 000 => 000 => 0
[2,2,1,1]
=> 0011 => 0011 => 1100 => 2
[2,1,1,1,1]
=> 01111 => 01111 => 11110 => 4
[1,1,1,1,1,1]
=> 111111 => 111111 => 111111 => 6
Description
The number of leading ones in a binary word.
The following 60 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000362The size of a minimal vertex cover of a graph. St000475The number of parts equal to 1 in a partition. St000536The pathwidth of a graph. St000877The depth of the binary word interpreted as a path. St000885The number of critical steps in the Catalan decomposition of a binary word. St000895The number of ones on the main diagonal of an alternating sign matrix. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001176The size of a partition minus its first part. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St000010The length of the partition. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000172The Grundy number of a graph. St000326The position of the first one in a binary word after appending a 1 at the end. St000696The number of cycles in the breakpoint graph of a permutation. St001029The size of the core of a graph. St001116The game chromatic number of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St000247The number of singleton blocks of a set partition. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St000822The Hadwiger number of the graph. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001963The tree-depth of a graph. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001812The biclique partition number of a graph. St000894The trace of an alternating sign matrix. St001330The hat guessing number of a graph. St000806The semiperimeter of the associated bargraph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001903The number of fixed points of a parking function. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000884The number of isolated descents of a permutation. St000259The diameter of a connected graph. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001645The pebbling number of a connected graph. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001060The distinguishing index of a graph. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000090The variation of a composition. St000264The girth of a graph, which is not a tree. 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. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000850The number of 1/2-balanced pairs in a poset. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. 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. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
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