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Your data matches 120 different statistics following compositions of up to 3 maps.
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Matching statistic: St000288
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(load all 10 compositions to match this statistic)
Mp00267: Signed permutations —signs⟶ 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] => 0 => 0 = 1 - 1
[-1] => 1 => 1 = 2 - 1
[1,2] => 00 => 0 = 1 - 1
[1,-2] => 01 => 1 = 2 - 1
[-1,2] => 10 => 1 = 2 - 1
[-1,-2] => 11 => 2 = 3 - 1
[2,1] => 00 => 0 = 1 - 1
[2,-1] => 01 => 1 = 2 - 1
[-2,1] => 10 => 1 = 2 - 1
[-2,-1] => 11 => 2 = 3 - 1
[1,2,3] => 000 => 0 = 1 - 1
[1,2,-3] => 001 => 1 = 2 - 1
[1,-2,3] => 010 => 1 = 2 - 1
[1,-2,-3] => 011 => 2 = 3 - 1
[-1,2,3] => 100 => 1 = 2 - 1
[-1,2,-3] => 101 => 2 = 3 - 1
[-1,-2,3] => 110 => 2 = 3 - 1
[-1,-2,-3] => 111 => 3 = 4 - 1
[1,3,2] => 000 => 0 = 1 - 1
[1,3,-2] => 001 => 1 = 2 - 1
[1,-3,2] => 010 => 1 = 2 - 1
[1,-3,-2] => 011 => 2 = 3 - 1
[-1,3,2] => 100 => 1 = 2 - 1
[-1,3,-2] => 101 => 2 = 3 - 1
[-1,-3,2] => 110 => 2 = 3 - 1
[-1,-3,-2] => 111 => 3 = 4 - 1
[2,1,3] => 000 => 0 = 1 - 1
[2,1,-3] => 001 => 1 = 2 - 1
[2,-1,3] => 010 => 1 = 2 - 1
[2,-1,-3] => 011 => 2 = 3 - 1
[-2,1,3] => 100 => 1 = 2 - 1
[-2,1,-3] => 101 => 2 = 3 - 1
[-2,-1,3] => 110 => 2 = 3 - 1
[-2,-1,-3] => 111 => 3 = 4 - 1
[2,3,1] => 000 => 0 = 1 - 1
[2,3,-1] => 001 => 1 = 2 - 1
[2,-3,1] => 010 => 1 = 2 - 1
[2,-3,-1] => 011 => 2 = 3 - 1
[-2,3,1] => 100 => 1 = 2 - 1
[-2,3,-1] => 101 => 2 = 3 - 1
[-2,-3,1] => 110 => 2 = 3 - 1
[-2,-3,-1] => 111 => 3 = 4 - 1
[3,1,2] => 000 => 0 = 1 - 1
[3,1,-2] => 001 => 1 = 2 - 1
[3,-1,2] => 010 => 1 = 2 - 1
[3,-1,-2] => 011 => 2 = 3 - 1
[-3,1,2] => 100 => 1 = 2 - 1
[-3,1,-2] => 101 => 2 = 3 - 1
[-3,-1,2] => 110 => 2 = 3 - 1
[-3,-1,-2] => 111 => 3 = 4 - 1
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000010
Mp00267: Signed permutations —signs⟶ 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] => 0 => [2] => [2]
=> 1
[-1] => 1 => [1,1] => [1,1]
=> 2
[1,2] => 00 => [3] => [3]
=> 1
[1,-2] => 01 => [2,1] => [2,1]
=> 2
[-1,2] => 10 => [1,2] => [2,1]
=> 2
[-1,-2] => 11 => [1,1,1] => [1,1,1]
=> 3
[2,1] => 00 => [3] => [3]
=> 1
[2,-1] => 01 => [2,1] => [2,1]
=> 2
[-2,1] => 10 => [1,2] => [2,1]
=> 2
[-2,-1] => 11 => [1,1,1] => [1,1,1]
=> 3
[1,2,3] => 000 => [4] => [4]
=> 1
[1,2,-3] => 001 => [3,1] => [3,1]
=> 2
[1,-2,3] => 010 => [2,2] => [2,2]
=> 2
[1,-2,-3] => 011 => [2,1,1] => [2,1,1]
=> 3
[-1,2,3] => 100 => [1,3] => [3,1]
=> 2
[-1,2,-3] => 101 => [1,2,1] => [2,1,1]
=> 3
[-1,-2,3] => 110 => [1,1,2] => [2,1,1]
=> 3
[-1,-2,-3] => 111 => [1,1,1,1] => [1,1,1,1]
=> 4
[1,3,2] => 000 => [4] => [4]
=> 1
[1,3,-2] => 001 => [3,1] => [3,1]
=> 2
[1,-3,2] => 010 => [2,2] => [2,2]
=> 2
[1,-3,-2] => 011 => [2,1,1] => [2,1,1]
=> 3
[-1,3,2] => 100 => [1,3] => [3,1]
=> 2
[-1,3,-2] => 101 => [1,2,1] => [2,1,1]
=> 3
[-1,-3,2] => 110 => [1,1,2] => [2,1,1]
=> 3
[-1,-3,-2] => 111 => [1,1,1,1] => [1,1,1,1]
=> 4
[2,1,3] => 000 => [4] => [4]
=> 1
[2,1,-3] => 001 => [3,1] => [3,1]
=> 2
[2,-1,3] => 010 => [2,2] => [2,2]
=> 2
[2,-1,-3] => 011 => [2,1,1] => [2,1,1]
=> 3
[-2,1,3] => 100 => [1,3] => [3,1]
=> 2
[-2,1,-3] => 101 => [1,2,1] => [2,1,1]
=> 3
[-2,-1,3] => 110 => [1,1,2] => [2,1,1]
=> 3
[-2,-1,-3] => 111 => [1,1,1,1] => [1,1,1,1]
=> 4
[2,3,1] => 000 => [4] => [4]
=> 1
[2,3,-1] => 001 => [3,1] => [3,1]
=> 2
[2,-3,1] => 010 => [2,2] => [2,2]
=> 2
[2,-3,-1] => 011 => [2,1,1] => [2,1,1]
=> 3
[-2,3,1] => 100 => [1,3] => [3,1]
=> 2
[-2,3,-1] => 101 => [1,2,1] => [2,1,1]
=> 3
[-2,-3,1] => 110 => [1,1,2] => [2,1,1]
=> 3
[-2,-3,-1] => 111 => [1,1,1,1] => [1,1,1,1]
=> 4
[3,1,2] => 000 => [4] => [4]
=> 1
[3,1,-2] => 001 => [3,1] => [3,1]
=> 2
[3,-1,2] => 010 => [2,2] => [2,2]
=> 2
[3,-1,-2] => 011 => [2,1,1] => [2,1,1]
=> 3
[-3,1,2] => 100 => [1,3] => [3,1]
=> 2
[-3,1,-2] => 101 => [1,2,1] => [2,1,1]
=> 3
[-3,-1,2] => 110 => [1,1,2] => [2,1,1]
=> 3
[-3,-1,-2] => 111 => [1,1,1,1] => [1,1,1,1]
=> 4
Description
The length of the partition.
Matching statistic: St000011
Mp00267: Signed permutations —signs⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0 => [2] => [1,1,0,0]
=> 1
[-1] => 1 => [1,1] => [1,0,1,0]
=> 2
[1,2] => 00 => [3] => [1,1,1,0,0,0]
=> 1
[1,-2] => 01 => [2,1] => [1,1,0,0,1,0]
=> 2
[-1,2] => 10 => [1,2] => [1,0,1,1,0,0]
=> 2
[-1,-2] => 11 => [1,1,1] => [1,0,1,0,1,0]
=> 3
[2,1] => 00 => [3] => [1,1,1,0,0,0]
=> 1
[2,-1] => 01 => [2,1] => [1,1,0,0,1,0]
=> 2
[-2,1] => 10 => [1,2] => [1,0,1,1,0,0]
=> 2
[-2,-1] => 11 => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,2,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,2,-3] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[1,-2,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,-2,-3] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3
[-1,2,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[-1,2,-3] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3
[-1,-2,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3
[-1,-2,-3] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[1,3,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,3,-2] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[1,-3,2] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,-3,-2] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3
[-1,3,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[-1,3,-2] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3
[-1,-3,2] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3
[-1,-3,-2] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[2,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 1
[2,1,-3] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[2,-1,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[2,-1,-3] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3
[-2,1,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[-2,1,-3] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3
[-2,-1,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3
[-2,-1,-3] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[2,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 1
[2,3,-1] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[2,-3,1] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[2,-3,-1] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3
[-2,3,1] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[-2,3,-1] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3
[-2,-3,1] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3
[-2,-3,-1] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[3,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 1
[3,1,-2] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[3,-1,2] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,-1,-2] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3
[-3,1,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[-3,1,-2] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3
[-3,-1,2] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3
[-3,-1,-2] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
Description
The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St000015
Mp00267: Signed permutations —signs⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000015: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000015: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0 => [2] => [1,1,0,0]
=> 1
[-1] => 1 => [1,1] => [1,0,1,0]
=> 2
[1,2] => 00 => [3] => [1,1,1,0,0,0]
=> 1
[1,-2] => 01 => [2,1] => [1,1,0,0,1,0]
=> 2
[-1,2] => 10 => [1,2] => [1,0,1,1,0,0]
=> 2
[-1,-2] => 11 => [1,1,1] => [1,0,1,0,1,0]
=> 3
[2,1] => 00 => [3] => [1,1,1,0,0,0]
=> 1
[2,-1] => 01 => [2,1] => [1,1,0,0,1,0]
=> 2
[-2,1] => 10 => [1,2] => [1,0,1,1,0,0]
=> 2
[-2,-1] => 11 => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,2,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,2,-3] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[1,-2,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,-2,-3] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3
[-1,2,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[-1,2,-3] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3
[-1,-2,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3
[-1,-2,-3] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[1,3,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,3,-2] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[1,-3,2] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,-3,-2] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3
[-1,3,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[-1,3,-2] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3
[-1,-3,2] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3
[-1,-3,-2] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[2,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 1
[2,1,-3] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[2,-1,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[2,-1,-3] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3
[-2,1,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[-2,1,-3] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3
[-2,-1,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3
[-2,-1,-3] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[2,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 1
[2,3,-1] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[2,-3,1] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[2,-3,-1] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3
[-2,3,1] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[-2,3,-1] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3
[-2,-3,1] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3
[-2,-3,-1] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[3,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 1
[3,1,-2] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[3,-1,2] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,-1,-2] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3
[-3,1,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[-3,1,-2] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3
[-3,-1,2] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3
[-3,-1,-2] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
Description
The number of peaks of a Dyck path.
Matching statistic: St000093
Mp00267: Signed permutations —signs⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000093: 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
St000093: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0 => [2] => ([],2)
=> 2
[-1] => 1 => [1,1] => ([(0,1)],2)
=> 1
[1,2] => 00 => [3] => ([],3)
=> 3
[1,-2] => 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[-1,2] => 10 => [1,2] => ([(1,2)],3)
=> 2
[-1,-2] => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[2,1] => 00 => [3] => ([],3)
=> 3
[2,-1] => 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[-2,1] => 10 => [1,2] => ([(1,2)],3)
=> 2
[-2,-1] => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2,3] => 000 => [4] => ([],4)
=> 4
[1,2,-3] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,-2,3] => 010 => [2,2] => ([(1,3),(2,3)],4)
=> 3
[1,-2,-3] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[-1,2,3] => 100 => [1,3] => ([(2,3)],4)
=> 3
[-1,2,-3] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[-1,-2,3] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[-1,-2,-3] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,3,2] => 000 => [4] => ([],4)
=> 4
[1,3,-2] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,-3,2] => 010 => [2,2] => ([(1,3),(2,3)],4)
=> 3
[1,-3,-2] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[-1,3,2] => 100 => [1,3] => ([(2,3)],4)
=> 3
[-1,3,-2] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[-1,-3,2] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[-1,-3,-2] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,1,3] => 000 => [4] => ([],4)
=> 4
[2,1,-3] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[2,-1,3] => 010 => [2,2] => ([(1,3),(2,3)],4)
=> 3
[2,-1,-3] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[-2,1,3] => 100 => [1,3] => ([(2,3)],4)
=> 3
[-2,1,-3] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[-2,-1,3] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[-2,-1,-3] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,3,1] => 000 => [4] => ([],4)
=> 4
[2,3,-1] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[2,-3,1] => 010 => [2,2] => ([(1,3),(2,3)],4)
=> 3
[2,-3,-1] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[-2,3,1] => 100 => [1,3] => ([(2,3)],4)
=> 3
[-2,3,-1] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[-2,-3,1] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[-2,-3,-1] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,1,2] => 000 => [4] => ([],4)
=> 4
[3,1,-2] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[3,-1,2] => 010 => [2,2] => ([(1,3),(2,3)],4)
=> 3
[3,-1,-2] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[-3,1,2] => 100 => [1,3] => ([(2,3)],4)
=> 3
[-3,1,-2] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[-3,-1,2] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[-3,-1,-2] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
Description
The cardinality of a maximal independent set of vertices of a graph.
An independent set of a graph is a set of pairwise non-adjacent vertices. A maximum independent set is an independent set of maximum cardinality. This statistic is also called the independence number or stability number $\alpha(G)$ of $G$.
Matching statistic: St000097
Mp00267: Signed permutations —signs⟶ 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] => 0 => [2] => ([],2)
=> 1
[-1] => 1 => [1,1] => ([(0,1)],2)
=> 2
[1,2] => 00 => [3] => ([],3)
=> 1
[1,-2] => 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[-1,2] => 10 => [1,2] => ([(1,2)],3)
=> 2
[-1,-2] => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[2,1] => 00 => [3] => ([],3)
=> 1
[2,-1] => 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[-2,1] => 10 => [1,2] => ([(1,2)],3)
=> 2
[-2,-1] => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,2,3] => 000 => [4] => ([],4)
=> 1
[1,2,-3] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,-2,3] => 010 => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,-2,-3] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-1,2,3] => 100 => [1,3] => ([(2,3)],4)
=> 2
[-1,2,-3] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-1,-2,3] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[-1,-2,-3] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,3,2] => 000 => [4] => ([],4)
=> 1
[1,3,-2] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,-3,2] => 010 => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,-3,-2] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-1,3,2] => 100 => [1,3] => ([(2,3)],4)
=> 2
[-1,3,-2] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-1,-3,2] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[-1,-3,-2] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[2,1,3] => 000 => [4] => ([],4)
=> 1
[2,1,-3] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[2,-1,3] => 010 => [2,2] => ([(1,3),(2,3)],4)
=> 2
[2,-1,-3] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-2,1,3] => 100 => [1,3] => ([(2,3)],4)
=> 2
[-2,1,-3] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-2,-1,3] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[-2,-1,-3] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[2,3,1] => 000 => [4] => ([],4)
=> 1
[2,3,-1] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[2,-3,1] => 010 => [2,2] => ([(1,3),(2,3)],4)
=> 2
[2,-3,-1] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-2,3,1] => 100 => [1,3] => ([(2,3)],4)
=> 2
[-2,3,-1] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-2,-3,1] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[-2,-3,-1] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[3,1,2] => 000 => [4] => ([],4)
=> 1
[3,1,-2] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[3,-1,2] => 010 => [2,2] => ([(1,3),(2,3)],4)
=> 2
[3,-1,-2] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-3,1,2] => 100 => [1,3] => ([(2,3)],4)
=> 2
[-3,1,-2] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-3,-1,2] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[-3,-1,-2] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
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
Mp00267: Signed permutations —signs⟶ 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] => 0 => [2] => ([],2)
=> 1
[-1] => 1 => [1,1] => ([(0,1)],2)
=> 2
[1,2] => 00 => [3] => ([],3)
=> 1
[1,-2] => 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[-1,2] => 10 => [1,2] => ([(1,2)],3)
=> 2
[-1,-2] => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[2,1] => 00 => [3] => ([],3)
=> 1
[2,-1] => 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[-2,1] => 10 => [1,2] => ([(1,2)],3)
=> 2
[-2,-1] => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,2,3] => 000 => [4] => ([],4)
=> 1
[1,2,-3] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,-2,3] => 010 => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,-2,-3] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-1,2,3] => 100 => [1,3] => ([(2,3)],4)
=> 2
[-1,2,-3] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-1,-2,3] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[-1,-2,-3] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,3,2] => 000 => [4] => ([],4)
=> 1
[1,3,-2] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,-3,2] => 010 => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,-3,-2] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-1,3,2] => 100 => [1,3] => ([(2,3)],4)
=> 2
[-1,3,-2] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-1,-3,2] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[-1,-3,-2] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[2,1,3] => 000 => [4] => ([],4)
=> 1
[2,1,-3] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[2,-1,3] => 010 => [2,2] => ([(1,3),(2,3)],4)
=> 2
[2,-1,-3] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-2,1,3] => 100 => [1,3] => ([(2,3)],4)
=> 2
[-2,1,-3] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-2,-1,3] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[-2,-1,-3] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[2,3,1] => 000 => [4] => ([],4)
=> 1
[2,3,-1] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[2,-3,1] => 010 => [2,2] => ([(1,3),(2,3)],4)
=> 2
[2,-3,-1] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-2,3,1] => 100 => [1,3] => ([(2,3)],4)
=> 2
[-2,3,-1] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-2,-3,1] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[-2,-3,-1] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[3,1,2] => 000 => [4] => ([],4)
=> 1
[3,1,-2] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[3,-1,2] => 010 => [2,2] => ([(1,3),(2,3)],4)
=> 2
[3,-1,-2] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-3,1,2] => 100 => [1,3] => ([(2,3)],4)
=> 2
[-3,1,-2] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-3,-1,2] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[-3,-1,-2] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
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: St000172
Mp00267: Signed permutations —signs⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000172: 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
St000172: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0 => [2] => ([],2)
=> 1
[-1] => 1 => [1,1] => ([(0,1)],2)
=> 2
[1,2] => 00 => [3] => ([],3)
=> 1
[1,-2] => 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[-1,2] => 10 => [1,2] => ([(1,2)],3)
=> 2
[-1,-2] => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[2,1] => 00 => [3] => ([],3)
=> 1
[2,-1] => 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[-2,1] => 10 => [1,2] => ([(1,2)],3)
=> 2
[-2,-1] => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,2,3] => 000 => [4] => ([],4)
=> 1
[1,2,-3] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,-2,3] => 010 => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,-2,-3] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-1,2,3] => 100 => [1,3] => ([(2,3)],4)
=> 2
[-1,2,-3] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-1,-2,3] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[-1,-2,-3] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,3,2] => 000 => [4] => ([],4)
=> 1
[1,3,-2] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,-3,2] => 010 => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,-3,-2] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-1,3,2] => 100 => [1,3] => ([(2,3)],4)
=> 2
[-1,3,-2] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-1,-3,2] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[-1,-3,-2] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[2,1,3] => 000 => [4] => ([],4)
=> 1
[2,1,-3] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[2,-1,3] => 010 => [2,2] => ([(1,3),(2,3)],4)
=> 2
[2,-1,-3] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-2,1,3] => 100 => [1,3] => ([(2,3)],4)
=> 2
[-2,1,-3] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-2,-1,3] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[-2,-1,-3] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[2,3,1] => 000 => [4] => ([],4)
=> 1
[2,3,-1] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[2,-3,1] => 010 => [2,2] => ([(1,3),(2,3)],4)
=> 2
[2,-3,-1] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-2,3,1] => 100 => [1,3] => ([(2,3)],4)
=> 2
[-2,3,-1] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-2,-3,1] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[-2,-3,-1] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[3,1,2] => 000 => [4] => ([],4)
=> 1
[3,1,-2] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[3,-1,2] => 010 => [2,2] => ([(1,3),(2,3)],4)
=> 2
[3,-1,-2] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-3,1,2] => 100 => [1,3] => ([(2,3)],4)
=> 2
[-3,1,-2] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-3,-1,2] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[-3,-1,-2] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
Description
The Grundy number of a graph.
The Grundy number $\Gamma(G)$ is defined to be the largest $k$ such that $G$ admits a greedy $k$-coloring. Any order of the vertices of $G$ induces a greedy coloring by assigning to the $i$-th vertex in this order the smallest positive integer such that the partial coloring remains a proper coloring.
In particular, we have that $\chi(G) \leq \Gamma(G) \leq \Delta(G) + 1$, where $\chi(G)$ is the chromatic number of $G$ ([[St000098]]), and where $\Delta(G)$ is the maximal degree of a vertex of $G$ ([[St000171]]).
Matching statistic: St000299
Mp00267: Signed permutations —signs⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000299: 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
St000299: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0 => [2] => ([],2)
=> 1
[-1] => 1 => [1,1] => ([(0,1)],2)
=> 2
[1,2] => 00 => [3] => ([],3)
=> 1
[1,-2] => 01 => [2,1] => ([(0,2),(1,2)],3)
=> 3
[-1,2] => 10 => [1,2] => ([(1,2)],3)
=> 2
[-1,-2] => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[2,1] => 00 => [3] => ([],3)
=> 1
[2,-1] => 01 => [2,1] => ([(0,2),(1,2)],3)
=> 3
[-2,1] => 10 => [1,2] => ([(1,2)],3)
=> 2
[-2,-1] => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,2,3] => 000 => [4] => ([],4)
=> 1
[1,2,-3] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 4
[1,-2,3] => 010 => [2,2] => ([(1,3),(2,3)],4)
=> 3
[1,-2,-3] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-1,2,3] => 100 => [1,3] => ([(2,3)],4)
=> 2
[-1,2,-3] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-1,-2,3] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[-1,-2,-3] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,3,2] => 000 => [4] => ([],4)
=> 1
[1,3,-2] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 4
[1,-3,2] => 010 => [2,2] => ([(1,3),(2,3)],4)
=> 3
[1,-3,-2] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-1,3,2] => 100 => [1,3] => ([(2,3)],4)
=> 2
[-1,3,-2] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-1,-3,2] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[-1,-3,-2] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,1,3] => 000 => [4] => ([],4)
=> 1
[2,1,-3] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 4
[2,-1,3] => 010 => [2,2] => ([(1,3),(2,3)],4)
=> 3
[2,-1,-3] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-2,1,3] => 100 => [1,3] => ([(2,3)],4)
=> 2
[-2,1,-3] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-2,-1,3] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[-2,-1,-3] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,3,1] => 000 => [4] => ([],4)
=> 1
[2,3,-1] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 4
[2,-3,1] => 010 => [2,2] => ([(1,3),(2,3)],4)
=> 3
[2,-3,-1] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-2,3,1] => 100 => [1,3] => ([(2,3)],4)
=> 2
[-2,3,-1] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-2,-3,1] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[-2,-3,-1] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,1,2] => 000 => [4] => ([],4)
=> 1
[3,1,-2] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 4
[3,-1,2] => 010 => [2,2] => ([(1,3),(2,3)],4)
=> 3
[3,-1,-2] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-3,1,2] => 100 => [1,3] => ([(2,3)],4)
=> 2
[-3,1,-2] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-3,-1,2] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[-3,-1,-2] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
Description
The number of nonisomorphic vertex-induced subtrees.
Matching statistic: St000443
Mp00267: Signed permutations —signs⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000443: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000443: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0 => [2] => [1,1,0,0]
=> 2
[-1] => 1 => [1,1] => [1,0,1,0]
=> 1
[1,2] => 00 => [3] => [1,1,1,0,0,0]
=> 3
[1,-2] => 01 => [2,1] => [1,1,0,0,1,0]
=> 2
[-1,2] => 10 => [1,2] => [1,0,1,1,0,0]
=> 2
[-1,-2] => 11 => [1,1,1] => [1,0,1,0,1,0]
=> 1
[2,1] => 00 => [3] => [1,1,1,0,0,0]
=> 3
[2,-1] => 01 => [2,1] => [1,1,0,0,1,0]
=> 2
[-2,1] => 10 => [1,2] => [1,0,1,1,0,0]
=> 2
[-2,-1] => 11 => [1,1,1] => [1,0,1,0,1,0]
=> 1
[1,2,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4
[1,2,-3] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,-2,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 3
[1,-2,-3] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[-1,2,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[-1,2,-3] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[-1,-2,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
[-1,-2,-3] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1
[1,3,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4
[1,3,-2] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,-3,2] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 3
[1,-3,-2] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[-1,3,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[-1,3,-2] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[-1,-3,2] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
[-1,-3,-2] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1
[2,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4
[2,1,-3] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[2,-1,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 3
[2,-1,-3] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[-2,1,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[-2,1,-3] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[-2,-1,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
[-2,-1,-3] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1
[2,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4
[2,3,-1] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[2,-3,1] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 3
[2,-3,-1] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[-2,3,1] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[-2,3,-1] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[-2,-3,1] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
[-2,-3,-1] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1
[3,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4
[3,1,-2] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[3,-1,2] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 3
[3,-1,-2] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[-3,1,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[-3,1,-2] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[-3,-1,2] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
[-3,-1,-2] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1
Description
The number of long tunnels of a Dyck path.
A long tunnel of a Dyck path is a longest sequence of consecutive usual tunnels, i.e., a longest sequence of tunnels where the end point of one is the starting point of the next. See [1] for the definition of tunnels.
The following 110 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000453The number of distinct Laplacian eigenvalues of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000822The Hadwiger number of the graph. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001337The upper domination number of a graph. St001338The upper irredundance 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. St001674The number of vertices of the largest induced star graph in the graph. St001963The tree-depth of a graph. St000024The number of double up and double down steps of a Dyck path. St000053The number of valleys of the Dyck path. St000272The treewidth of a graph. St000306The bounce count of a Dyck path. St000340The number of non-final maximal constant sub-paths of length greater than one. St000362The size of a minimal vertex cover of a graph. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000536The pathwidth of a graph. St000691The number of changes of a binary word. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001277The degeneracy of a graph. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001358The largest degree of a regular subgraph of a graph. St001372The length of a longest cyclic run of ones of a binary word. St001480The number of simple summands of the module J^2/J^3. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001777The number of weak descents in an integer composition. St001812The biclique partition number of a graph. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001330The hat guessing number of a graph. St001875The number of simple modules with projective dimension at most 1. St000438The position of the last up step in a Dyck path. St000806The semiperimeter of the associated bargraph. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001838The number of nonempty primitive factors of a binary word. St001870The number of positive entries followed by a negative entry in a signed permutation. St001430The number of positive entries in a signed permutation. St000454The largest eigenvalue of a graph if it is integral. St000264The girth of a graph, which is not a tree. St001429The number of negative entries in a signed permutation. 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. 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. 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. St000981The length of the longest zigzag subpath. 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 $eA$. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. 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. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. 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. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition.
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