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Your data matches 40 different statistics following compositions of up to 3 maps.
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Matching statistic: St000288
(load all 28 compositions to match this statistic)
(load all 28 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 - 1
[2]
=> 0 => 0 = 1 - 1
[1,1]
=> 11 => 2 = 3 - 1
[3]
=> 1 => 1 = 2 - 1
[2,1]
=> 01 => 1 = 2 - 1
[1,1,1]
=> 111 => 3 = 4 - 1
[4]
=> 0 => 0 = 1 - 1
[3,1]
=> 11 => 2 = 3 - 1
[2,2]
=> 00 => 0 = 1 - 1
[1,1,1,1]
=> 1111 => 4 = 5 - 1
[5]
=> 1 => 1 = 2 - 1
[4,1]
=> 01 => 1 = 2 - 1
[3,2]
=> 10 => 1 = 2 - 1
[3,1,1]
=> 111 => 3 = 4 - 1
[2,2,1]
=> 001 => 1 = 2 - 1
[1,1,1,1,1]
=> 11111 => 5 = 6 - 1
[6]
=> 0 => 0 = 1 - 1
[5,1]
=> 11 => 2 = 3 - 1
[4,2]
=> 00 => 0 = 1 - 1
[3,3]
=> 11 => 2 = 3 - 1
[3,1,1,1]
=> 1111 => 4 = 5 - 1
[2,2,2]
=> 000 => 0 = 1 - 1
[1,1,1,1,1,1]
=> 111111 => 6 = 7 - 1
[7]
=> 1 => 1 = 2 - 1
[6,1]
=> 01 => 1 = 2 - 1
[5,2]
=> 10 => 1 = 2 - 1
[5,1,1]
=> 111 => 3 = 4 - 1
[4,3]
=> 01 => 1 = 2 - 1
[4,2,1]
=> 001 => 1 = 2 - 1
[3,3,1]
=> 111 => 3 = 4 - 1
[3,2,2]
=> 100 => 1 = 2 - 1
[3,1,1,1,1]
=> 11111 => 5 = 6 - 1
[2,2,2,1]
=> 0001 => 1 = 2 - 1
[1,1,1,1,1,1,1]
=> 1111111 => 7 = 8 - 1
[8]
=> 0 => 0 = 1 - 1
[7,1]
=> 11 => 2 = 3 - 1
[6,2]
=> 00 => 0 = 1 - 1
[5,3]
=> 11 => 2 = 3 - 1
[5,1,1,1]
=> 1111 => 4 = 5 - 1
[4,4]
=> 00 => 0 = 1 - 1
[4,2,2]
=> 000 => 0 = 1 - 1
[3,3,2]
=> 110 => 2 = 3 - 1
[3,3,1,1]
=> 1111 => 4 = 5 - 1
[3,1,1,1,1,1]
=> 111111 => 6 = 7 - 1
[2,2,2,2]
=> 0000 => 0 = 1 - 1
[9]
=> 1 => 1 = 2 - 1
[8,1]
=> 01 => 1 = 2 - 1
[7,2]
=> 10 => 1 = 2 - 1
[7,1,1]
=> 111 => 3 = 4 - 1
[6,3]
=> 01 => 1 = 2 - 1
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000392
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00317: Integer partitions —odd parts⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1 => 1 = 2 - 1
[2]
=> 0 => 0 = 1 - 1
[1,1]
=> 11 => 2 = 3 - 1
[3]
=> 1 => 1 = 2 - 1
[2,1]
=> 01 => 1 = 2 - 1
[1,1,1]
=> 111 => 3 = 4 - 1
[4]
=> 0 => 0 = 1 - 1
[3,1]
=> 11 => 2 = 3 - 1
[2,2]
=> 00 => 0 = 1 - 1
[1,1,1,1]
=> 1111 => 4 = 5 - 1
[5]
=> 1 => 1 = 2 - 1
[4,1]
=> 01 => 1 = 2 - 1
[3,2]
=> 10 => 1 = 2 - 1
[3,1,1]
=> 111 => 3 = 4 - 1
[2,2,1]
=> 001 => 1 = 2 - 1
[1,1,1,1,1]
=> 11111 => 5 = 6 - 1
[6]
=> 0 => 0 = 1 - 1
[5,1]
=> 11 => 2 = 3 - 1
[4,2]
=> 00 => 0 = 1 - 1
[3,3]
=> 11 => 2 = 3 - 1
[3,1,1,1]
=> 1111 => 4 = 5 - 1
[2,2,2]
=> 000 => 0 = 1 - 1
[1,1,1,1,1,1]
=> 111111 => 6 = 7 - 1
[7]
=> 1 => 1 = 2 - 1
[6,1]
=> 01 => 1 = 2 - 1
[5,2]
=> 10 => 1 = 2 - 1
[5,1,1]
=> 111 => 3 = 4 - 1
[4,3]
=> 01 => 1 = 2 - 1
[4,2,1]
=> 001 => 1 = 2 - 1
[3,3,1]
=> 111 => 3 = 4 - 1
[3,2,2]
=> 100 => 1 = 2 - 1
[3,1,1,1,1]
=> 11111 => 5 = 6 - 1
[2,2,2,1]
=> 0001 => 1 = 2 - 1
[1,1,1,1,1,1,1]
=> 1111111 => 7 = 8 - 1
[8]
=> 0 => 0 = 1 - 1
[7,1]
=> 11 => 2 = 3 - 1
[6,2]
=> 00 => 0 = 1 - 1
[5,3]
=> 11 => 2 = 3 - 1
[5,1,1,1]
=> 1111 => 4 = 5 - 1
[4,4]
=> 00 => 0 = 1 - 1
[4,2,2]
=> 000 => 0 = 1 - 1
[3,3,2]
=> 110 => 2 = 3 - 1
[3,3,1,1]
=> 1111 => 4 = 5 - 1
[3,1,1,1,1,1]
=> 111111 => 6 = 7 - 1
[2,2,2,2]
=> 0000 => 0 = 1 - 1
[9]
=> 1 => 1 = 2 - 1
[8,1]
=> 01 => 1 = 2 - 1
[7,2]
=> 10 => 1 = 2 - 1
[7,1,1]
=> 111 => 3 = 4 - 1
[6,3]
=> 01 => 1 = 2 - 1
Description
The length of the longest run of ones in a binary word.
Matching statistic: St001372
(load all 13 compositions to match this statistic)
(load all 13 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 - 1
[2]
=> 0 => 0 = 1 - 1
[1,1]
=> 11 => 2 = 3 - 1
[3]
=> 1 => 1 = 2 - 1
[2,1]
=> 01 => 1 = 2 - 1
[1,1,1]
=> 111 => 3 = 4 - 1
[4]
=> 0 => 0 = 1 - 1
[3,1]
=> 11 => 2 = 3 - 1
[2,2]
=> 00 => 0 = 1 - 1
[1,1,1,1]
=> 1111 => 4 = 5 - 1
[5]
=> 1 => 1 = 2 - 1
[4,1]
=> 01 => 1 = 2 - 1
[3,2]
=> 10 => 1 = 2 - 1
[3,1,1]
=> 111 => 3 = 4 - 1
[2,2,1]
=> 001 => 1 = 2 - 1
[1,1,1,1,1]
=> 11111 => 5 = 6 - 1
[6]
=> 0 => 0 = 1 - 1
[5,1]
=> 11 => 2 = 3 - 1
[4,2]
=> 00 => 0 = 1 - 1
[3,3]
=> 11 => 2 = 3 - 1
[3,1,1,1]
=> 1111 => 4 = 5 - 1
[2,2,2]
=> 000 => 0 = 1 - 1
[1,1,1,1,1,1]
=> 111111 => 6 = 7 - 1
[7]
=> 1 => 1 = 2 - 1
[6,1]
=> 01 => 1 = 2 - 1
[5,2]
=> 10 => 1 = 2 - 1
[5,1,1]
=> 111 => 3 = 4 - 1
[4,3]
=> 01 => 1 = 2 - 1
[4,2,1]
=> 001 => 1 = 2 - 1
[3,3,1]
=> 111 => 3 = 4 - 1
[3,2,2]
=> 100 => 1 = 2 - 1
[3,1,1,1,1]
=> 11111 => 5 = 6 - 1
[2,2,2,1]
=> 0001 => 1 = 2 - 1
[1,1,1,1,1,1,1]
=> 1111111 => 7 = 8 - 1
[8]
=> 0 => 0 = 1 - 1
[7,1]
=> 11 => 2 = 3 - 1
[6,2]
=> 00 => 0 = 1 - 1
[5,3]
=> 11 => 2 = 3 - 1
[5,1,1,1]
=> 1111 => 4 = 5 - 1
[4,4]
=> 00 => 0 = 1 - 1
[4,2,2]
=> 000 => 0 = 1 - 1
[3,3,2]
=> 110 => 2 = 3 - 1
[3,3,1,1]
=> 1111 => 4 = 5 - 1
[3,1,1,1,1,1]
=> 111111 => 6 = 7 - 1
[2,2,2,2]
=> 0000 => 0 = 1 - 1
[9]
=> 1 => 1 = 2 - 1
[8,1]
=> 01 => 1 = 2 - 1
[7,2]
=> 10 => 1 = 2 - 1
[7,1,1]
=> 111 => 3 = 4 - 1
[6,3]
=> 01 => 1 = 2 - 1
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: St001419
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00317: Integer partitions —odd parts⟶ Binary words
St001419: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001419: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1 => 1 = 2 - 1
[2]
=> 0 => 0 = 1 - 1
[1,1]
=> 11 => 2 = 3 - 1
[3]
=> 1 => 1 = 2 - 1
[2,1]
=> 01 => 1 = 2 - 1
[1,1,1]
=> 111 => 3 = 4 - 1
[4]
=> 0 => 0 = 1 - 1
[3,1]
=> 11 => 2 = 3 - 1
[2,2]
=> 00 => 0 = 1 - 1
[1,1,1,1]
=> 1111 => 4 = 5 - 1
[5]
=> 1 => 1 = 2 - 1
[4,1]
=> 01 => 1 = 2 - 1
[3,2]
=> 10 => 1 = 2 - 1
[3,1,1]
=> 111 => 3 = 4 - 1
[2,2,1]
=> 001 => 1 = 2 - 1
[1,1,1,1,1]
=> 11111 => 5 = 6 - 1
[6]
=> 0 => 0 = 1 - 1
[5,1]
=> 11 => 2 = 3 - 1
[4,2]
=> 00 => 0 = 1 - 1
[3,3]
=> 11 => 2 = 3 - 1
[3,1,1,1]
=> 1111 => 4 = 5 - 1
[2,2,2]
=> 000 => 0 = 1 - 1
[1,1,1,1,1,1]
=> 111111 => 6 = 7 - 1
[7]
=> 1 => 1 = 2 - 1
[6,1]
=> 01 => 1 = 2 - 1
[5,2]
=> 10 => 1 = 2 - 1
[5,1,1]
=> 111 => 3 = 4 - 1
[4,3]
=> 01 => 1 = 2 - 1
[4,2,1]
=> 001 => 1 = 2 - 1
[3,3,1]
=> 111 => 3 = 4 - 1
[3,2,2]
=> 100 => 1 = 2 - 1
[3,1,1,1,1]
=> 11111 => 5 = 6 - 1
[2,2,2,1]
=> 0001 => 1 = 2 - 1
[1,1,1,1,1,1,1]
=> 1111111 => 7 = 8 - 1
[8]
=> 0 => 0 = 1 - 1
[7,1]
=> 11 => 2 = 3 - 1
[6,2]
=> 00 => 0 = 1 - 1
[5,3]
=> 11 => 2 = 3 - 1
[5,1,1,1]
=> 1111 => 4 = 5 - 1
[4,4]
=> 00 => 0 = 1 - 1
[4,2,2]
=> 000 => 0 = 1 - 1
[3,3,2]
=> 110 => 2 = 3 - 1
[3,3,1,1]
=> 1111 => 4 = 5 - 1
[3,1,1,1,1,1]
=> 111111 => 6 = 7 - 1
[2,2,2,2]
=> 0000 => 0 = 1 - 1
[9]
=> 1 => 1 = 2 - 1
[8,1]
=> 01 => 1 = 2 - 1
[7,2]
=> 10 => 1 = 2 - 1
[7,1,1]
=> 111 => 3 = 4 - 1
[6,3]
=> 01 => 1 = 2 - 1
Description
The length of the longest palindromic factor beginning with a one of a binary word.
Matching statistic: St000010
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1 => [1,1] => [1,1]
=> 2
[2]
=> 0 => [2] => [2]
=> 1
[1,1]
=> 11 => [1,1,1] => [1,1,1]
=> 3
[3]
=> 1 => [1,1] => [1,1]
=> 2
[2,1]
=> 01 => [2,1] => [2,1]
=> 2
[1,1,1]
=> 111 => [1,1,1,1] => [1,1,1,1]
=> 4
[4]
=> 0 => [2] => [2]
=> 1
[3,1]
=> 11 => [1,1,1] => [1,1,1]
=> 3
[2,2]
=> 00 => [3] => [3]
=> 1
[1,1,1,1]
=> 1111 => [1,1,1,1,1] => [1,1,1,1,1]
=> 5
[5]
=> 1 => [1,1] => [1,1]
=> 2
[4,1]
=> 01 => [2,1] => [2,1]
=> 2
[3,2]
=> 10 => [1,2] => [2,1]
=> 2
[3,1,1]
=> 111 => [1,1,1,1] => [1,1,1,1]
=> 4
[2,2,1]
=> 001 => [3,1] => [3,1]
=> 2
[1,1,1,1,1]
=> 11111 => [1,1,1,1,1,1] => [1,1,1,1,1,1]
=> 6
[6]
=> 0 => [2] => [2]
=> 1
[5,1]
=> 11 => [1,1,1] => [1,1,1]
=> 3
[4,2]
=> 00 => [3] => [3]
=> 1
[3,3]
=> 11 => [1,1,1] => [1,1,1]
=> 3
[3,1,1,1]
=> 1111 => [1,1,1,1,1] => [1,1,1,1,1]
=> 5
[2,2,2]
=> 000 => [4] => [4]
=> 1
[1,1,1,1,1,1]
=> 111111 => [1,1,1,1,1,1,1] => [1,1,1,1,1,1,1]
=> 7
[7]
=> 1 => [1,1] => [1,1]
=> 2
[6,1]
=> 01 => [2,1] => [2,1]
=> 2
[5,2]
=> 10 => [1,2] => [2,1]
=> 2
[5,1,1]
=> 111 => [1,1,1,1] => [1,1,1,1]
=> 4
[4,3]
=> 01 => [2,1] => [2,1]
=> 2
[4,2,1]
=> 001 => [3,1] => [3,1]
=> 2
[3,3,1]
=> 111 => [1,1,1,1] => [1,1,1,1]
=> 4
[3,2,2]
=> 100 => [1,3] => [3,1]
=> 2
[3,1,1,1,1]
=> 11111 => [1,1,1,1,1,1] => [1,1,1,1,1,1]
=> 6
[2,2,2,1]
=> 0001 => [4,1] => [4,1]
=> 2
[1,1,1,1,1,1,1]
=> 1111111 => [1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1]
=> 8
[8]
=> 0 => [2] => [2]
=> 1
[7,1]
=> 11 => [1,1,1] => [1,1,1]
=> 3
[6,2]
=> 00 => [3] => [3]
=> 1
[5,3]
=> 11 => [1,1,1] => [1,1,1]
=> 3
[5,1,1,1]
=> 1111 => [1,1,1,1,1] => [1,1,1,1,1]
=> 5
[4,4]
=> 00 => [3] => [3]
=> 1
[4,2,2]
=> 000 => [4] => [4]
=> 1
[3,3,2]
=> 110 => [1,1,2] => [2,1,1]
=> 3
[3,3,1,1]
=> 1111 => [1,1,1,1,1] => [1,1,1,1,1]
=> 5
[3,1,1,1,1,1]
=> 111111 => [1,1,1,1,1,1,1] => [1,1,1,1,1,1,1]
=> 7
[2,2,2,2]
=> 0000 => [5] => [5]
=> 1
[9]
=> 1 => [1,1] => [1,1]
=> 2
[8,1]
=> 01 => [2,1] => [2,1]
=> 2
[7,2]
=> 10 => [1,2] => [2,1]
=> 2
[7,1,1]
=> 111 => [1,1,1,1] => [1,1,1,1]
=> 4
[6,3]
=> 01 => [2,1] => [2,1]
=> 2
Description
The length of the partition.
Matching statistic: St000097
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000097: 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
St000097: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1 => [1,1] => ([(0,1)],2)
=> 2
[2]
=> 0 => [2] => ([],2)
=> 1
[1,1]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[3]
=> 1 => [1,1] => ([(0,1)],2)
=> 2
[2,1]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,1,1]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[4]
=> 0 => [2] => ([],2)
=> 1
[3,1]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[2,2]
=> 00 => [3] => ([],3)
=> 1
[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)
=> 5
[5]
=> 1 => [1,1] => ([(0,1)],2)
=> 2
[4,1]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[3,2]
=> 10 => [1,2] => ([(1,2)],3)
=> 2
[3,1,1]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[2,2,1]
=> 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[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)
=> 6
[6]
=> 0 => [2] => ([],2)
=> 1
[5,1]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[4,2]
=> 00 => [3] => ([],3)
=> 1
[3,3]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[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)
=> 5
[2,2,2]
=> 000 => [4] => ([],4)
=> 1
[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)
=> 7
[7]
=> 1 => [1,1] => ([(0,1)],2)
=> 2
[6,1]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[5,2]
=> 10 => [1,2] => ([(1,2)],3)
=> 2
[5,1,1]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[4,3]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[4,2,1]
=> 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[3,3,1]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[3,2,2]
=> 100 => [1,3] => ([(2,3)],4)
=> 2
[3,1,1,1,1]
=> 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)
=> 6
[2,2,2,1]
=> 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,1,1,1,1,1]
=> 1111111 => [1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 8
[8]
=> 0 => [2] => ([],2)
=> 1
[7,1]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[6,2]
=> 00 => [3] => ([],3)
=> 1
[5,3]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[5,1,1,1]
=> 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)
=> 5
[4,4]
=> 00 => [3] => ([],3)
=> 1
[4,2,2]
=> 000 => [4] => ([],4)
=> 1
[3,3,2]
=> 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[3,3,1,1]
=> 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)
=> 5
[3,1,1,1,1,1]
=> 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)
=> 7
[2,2,2,2]
=> 0000 => [5] => ([],5)
=> 1
[9]
=> 1 => [1,1] => ([(0,1)],2)
=> 2
[8,1]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[7,2]
=> 10 => [1,2] => ([(1,2)],3)
=> 2
[7,1,1]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[6,3]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2
Description
The order of the largest clique of the graph.
A clique in a graph $G$ is a subset $U \subseteq V(G)$ such that any pair of vertices in $U$ are adjacent. I.e. the subgraph induced by $U$ is a complete graph.
Matching statistic: St000098
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000098: 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
St000098: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1 => [1,1] => ([(0,1)],2)
=> 2
[2]
=> 0 => [2] => ([],2)
=> 1
[1,1]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[3]
=> 1 => [1,1] => ([(0,1)],2)
=> 2
[2,1]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,1,1]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[4]
=> 0 => [2] => ([],2)
=> 1
[3,1]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[2,2]
=> 00 => [3] => ([],3)
=> 1
[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)
=> 5
[5]
=> 1 => [1,1] => ([(0,1)],2)
=> 2
[4,1]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[3,2]
=> 10 => [1,2] => ([(1,2)],3)
=> 2
[3,1,1]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[2,2,1]
=> 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[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)
=> 6
[6]
=> 0 => [2] => ([],2)
=> 1
[5,1]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[4,2]
=> 00 => [3] => ([],3)
=> 1
[3,3]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[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)
=> 5
[2,2,2]
=> 000 => [4] => ([],4)
=> 1
[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)
=> 7
[7]
=> 1 => [1,1] => ([(0,1)],2)
=> 2
[6,1]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[5,2]
=> 10 => [1,2] => ([(1,2)],3)
=> 2
[5,1,1]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[4,3]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[4,2,1]
=> 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[3,3,1]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[3,2,2]
=> 100 => [1,3] => ([(2,3)],4)
=> 2
[3,1,1,1,1]
=> 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)
=> 6
[2,2,2,1]
=> 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,1,1,1,1,1]
=> 1111111 => [1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 8
[8]
=> 0 => [2] => ([],2)
=> 1
[7,1]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[6,2]
=> 00 => [3] => ([],3)
=> 1
[5,3]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[5,1,1,1]
=> 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)
=> 5
[4,4]
=> 00 => [3] => ([],3)
=> 1
[4,2,2]
=> 000 => [4] => ([],4)
=> 1
[3,3,2]
=> 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[3,3,1,1]
=> 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)
=> 5
[3,1,1,1,1,1]
=> 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)
=> 7
[2,2,2,2]
=> 0000 => [5] => ([],5)
=> 1
[9]
=> 1 => [1,1] => ([(0,1)],2)
=> 2
[8,1]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[7,2]
=> 10 => [1,2] => ([(1,2)],3)
=> 2
[7,1,1]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[6,3]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2
Description
The chromatic number of a graph.
The minimal number of colors needed to color the vertices of the graph such that no two vertices which share an edge have the same color.
Matching statistic: St000326
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00105: Binary words —complement⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1 => 0 => 0 => 2
[2]
=> 0 => 1 => 1 => 1
[1,1]
=> 11 => 00 => 00 => 3
[3]
=> 1 => 0 => 0 => 2
[2,1]
=> 01 => 10 => 01 => 2
[1,1,1]
=> 111 => 000 => 000 => 4
[4]
=> 0 => 1 => 1 => 1
[3,1]
=> 11 => 00 => 00 => 3
[2,2]
=> 00 => 11 => 11 => 1
[1,1,1,1]
=> 1111 => 0000 => 0000 => 5
[5]
=> 1 => 0 => 0 => 2
[4,1]
=> 01 => 10 => 01 => 2
[3,2]
=> 10 => 01 => 01 => 2
[3,1,1]
=> 111 => 000 => 000 => 4
[2,2,1]
=> 001 => 110 => 011 => 2
[1,1,1,1,1]
=> 11111 => 00000 => 00000 => 6
[6]
=> 0 => 1 => 1 => 1
[5,1]
=> 11 => 00 => 00 => 3
[4,2]
=> 00 => 11 => 11 => 1
[3,3]
=> 11 => 00 => 00 => 3
[3,1,1,1]
=> 1111 => 0000 => 0000 => 5
[2,2,2]
=> 000 => 111 => 111 => 1
[1,1,1,1,1,1]
=> 111111 => 000000 => 000000 => 7
[7]
=> 1 => 0 => 0 => 2
[6,1]
=> 01 => 10 => 01 => 2
[5,2]
=> 10 => 01 => 01 => 2
[5,1,1]
=> 111 => 000 => 000 => 4
[4,3]
=> 01 => 10 => 01 => 2
[4,2,1]
=> 001 => 110 => 011 => 2
[3,3,1]
=> 111 => 000 => 000 => 4
[3,2,2]
=> 100 => 011 => 011 => 2
[3,1,1,1,1]
=> 11111 => 00000 => 00000 => 6
[2,2,2,1]
=> 0001 => 1110 => 0111 => 2
[1,1,1,1,1,1,1]
=> 1111111 => 0000000 => 0000000 => 8
[8]
=> 0 => 1 => 1 => 1
[7,1]
=> 11 => 00 => 00 => 3
[6,2]
=> 00 => 11 => 11 => 1
[5,3]
=> 11 => 00 => 00 => 3
[5,1,1,1]
=> 1111 => 0000 => 0000 => 5
[4,4]
=> 00 => 11 => 11 => 1
[4,2,2]
=> 000 => 111 => 111 => 1
[3,3,2]
=> 110 => 001 => 001 => 3
[3,3,1,1]
=> 1111 => 0000 => 0000 => 5
[3,1,1,1,1,1]
=> 111111 => 000000 => 000000 => 7
[2,2,2,2]
=> 0000 => 1111 => 1111 => 1
[9]
=> 1 => 0 => 0 => 2
[8,1]
=> 01 => 10 => 01 => 2
[7,2]
=> 10 => 01 => 01 => 2
[7,1,1]
=> 111 => 000 => 000 => 4
[6,3]
=> 01 => 10 => 01 => 2
Description
The position of the first one in a binary word after appending a 1 at the end.
Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Matching statistic: St000381
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00105: Binary words —complement⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1 => 0 => [2] => 2
[2]
=> 0 => 1 => [1,1] => 1
[1,1]
=> 11 => 00 => [3] => 3
[3]
=> 1 => 0 => [2] => 2
[2,1]
=> 01 => 10 => [1,2] => 2
[1,1,1]
=> 111 => 000 => [4] => 4
[4]
=> 0 => 1 => [1,1] => 1
[3,1]
=> 11 => 00 => [3] => 3
[2,2]
=> 00 => 11 => [1,1,1] => 1
[1,1,1,1]
=> 1111 => 0000 => [5] => 5
[5]
=> 1 => 0 => [2] => 2
[4,1]
=> 01 => 10 => [1,2] => 2
[3,2]
=> 10 => 01 => [2,1] => 2
[3,1,1]
=> 111 => 000 => [4] => 4
[2,2,1]
=> 001 => 110 => [1,1,2] => 2
[1,1,1,1,1]
=> 11111 => 00000 => [6] => 6
[6]
=> 0 => 1 => [1,1] => 1
[5,1]
=> 11 => 00 => [3] => 3
[4,2]
=> 00 => 11 => [1,1,1] => 1
[3,3]
=> 11 => 00 => [3] => 3
[3,1,1,1]
=> 1111 => 0000 => [5] => 5
[2,2,2]
=> 000 => 111 => [1,1,1,1] => 1
[1,1,1,1,1,1]
=> 111111 => 000000 => [7] => 7
[7]
=> 1 => 0 => [2] => 2
[6,1]
=> 01 => 10 => [1,2] => 2
[5,2]
=> 10 => 01 => [2,1] => 2
[5,1,1]
=> 111 => 000 => [4] => 4
[4,3]
=> 01 => 10 => [1,2] => 2
[4,2,1]
=> 001 => 110 => [1,1,2] => 2
[3,3,1]
=> 111 => 000 => [4] => 4
[3,2,2]
=> 100 => 011 => [2,1,1] => 2
[3,1,1,1,1]
=> 11111 => 00000 => [6] => 6
[2,2,2,1]
=> 0001 => 1110 => [1,1,1,2] => 2
[1,1,1,1,1,1,1]
=> 1111111 => 0000000 => [8] => 8
[8]
=> 0 => 1 => [1,1] => 1
[7,1]
=> 11 => 00 => [3] => 3
[6,2]
=> 00 => 11 => [1,1,1] => 1
[5,3]
=> 11 => 00 => [3] => 3
[5,1,1,1]
=> 1111 => 0000 => [5] => 5
[4,4]
=> 00 => 11 => [1,1,1] => 1
[4,2,2]
=> 000 => 111 => [1,1,1,1] => 1
[3,3,2]
=> 110 => 001 => [3,1] => 3
[3,3,1,1]
=> 1111 => 0000 => [5] => 5
[3,1,1,1,1,1]
=> 111111 => 000000 => [7] => 7
[2,2,2,2]
=> 0000 => 1111 => [1,1,1,1,1] => 1
[9]
=> 1 => 0 => [2] => 2
[8,1]
=> 01 => 10 => [1,2] => 2
[7,2]
=> 10 => 01 => [2,1] => 2
[7,1,1]
=> 111 => 000 => [4] => 4
[6,3]
=> 01 => 10 => [1,2] => 2
Description
The largest part of an integer composition.
Matching statistic: St000808
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St000808: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00105: Binary words —complement⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St000808: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1 => 0 => [2] => 2
[2]
=> 0 => 1 => [1,1] => 1
[1,1]
=> 11 => 00 => [3] => 3
[3]
=> 1 => 0 => [2] => 2
[2,1]
=> 01 => 10 => [1,2] => 2
[1,1,1]
=> 111 => 000 => [4] => 4
[4]
=> 0 => 1 => [1,1] => 1
[3,1]
=> 11 => 00 => [3] => 3
[2,2]
=> 00 => 11 => [1,1,1] => 1
[1,1,1,1]
=> 1111 => 0000 => [5] => 5
[5]
=> 1 => 0 => [2] => 2
[4,1]
=> 01 => 10 => [1,2] => 2
[3,2]
=> 10 => 01 => [2,1] => 2
[3,1,1]
=> 111 => 000 => [4] => 4
[2,2,1]
=> 001 => 110 => [1,1,2] => 2
[1,1,1,1,1]
=> 11111 => 00000 => [6] => 6
[6]
=> 0 => 1 => [1,1] => 1
[5,1]
=> 11 => 00 => [3] => 3
[4,2]
=> 00 => 11 => [1,1,1] => 1
[3,3]
=> 11 => 00 => [3] => 3
[3,1,1,1]
=> 1111 => 0000 => [5] => 5
[2,2,2]
=> 000 => 111 => [1,1,1,1] => 1
[1,1,1,1,1,1]
=> 111111 => 000000 => [7] => 7
[7]
=> 1 => 0 => [2] => 2
[6,1]
=> 01 => 10 => [1,2] => 2
[5,2]
=> 10 => 01 => [2,1] => 2
[5,1,1]
=> 111 => 000 => [4] => 4
[4,3]
=> 01 => 10 => [1,2] => 2
[4,2,1]
=> 001 => 110 => [1,1,2] => 2
[3,3,1]
=> 111 => 000 => [4] => 4
[3,2,2]
=> 100 => 011 => [2,1,1] => 2
[3,1,1,1,1]
=> 11111 => 00000 => [6] => 6
[2,2,2,1]
=> 0001 => 1110 => [1,1,1,2] => 2
[1,1,1,1,1,1,1]
=> 1111111 => 0000000 => [8] => 8
[8]
=> 0 => 1 => [1,1] => 1
[7,1]
=> 11 => 00 => [3] => 3
[6,2]
=> 00 => 11 => [1,1,1] => 1
[5,3]
=> 11 => 00 => [3] => 3
[5,1,1,1]
=> 1111 => 0000 => [5] => 5
[4,4]
=> 00 => 11 => [1,1,1] => 1
[4,2,2]
=> 000 => 111 => [1,1,1,1] => 1
[3,3,2]
=> 110 => 001 => [3,1] => 3
[3,3,1,1]
=> 1111 => 0000 => [5] => 5
[3,1,1,1,1,1]
=> 111111 => 000000 => [7] => 7
[2,2,2,2]
=> 0000 => 1111 => [1,1,1,1,1] => 1
[9]
=> 1 => 0 => [2] => 2
[8,1]
=> 01 => 10 => [1,2] => 2
[7,2]
=> 10 => 01 => [2,1] => 2
[7,1,1]
=> 111 => 000 => [4] => 4
[6,3]
=> 01 => 10 => [1,2] => 2
Description
The number of up steps of the associated bargraph.
Interpret the composition as the sequence of heights of the bars of a bargraph. This statistic is the number of up steps.
The following 30 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001330The hat guessing number of a graph. St000297The number of leading ones in a binary word. St000877The depth of the binary word interpreted as a path. St000148The number of odd parts of a partition. St001581The achromatic number of a graph. St000172The Grundy number of a graph. 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. St001670The connected partition number of a graph. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St001971The number of negative eigenvalues of the adjacency matrix of the graph. 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. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001235The global dimension of the corresponding Comp-Nakayama algebra. St000822The Hadwiger number of the graph. St000806The semiperimeter of the associated bargraph. St001812The biclique partition number of a graph. St000992The alternating sum of the parts of an integer partition. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St000022The number of fixed points of a permutation. St000895The number of ones on the main diagonal of an alternating sign matrix. St000696The number of cycles in the breakpoint graph of a permutation. St000264The girth of a graph, which is not a tree.
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