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Your data matches 82 different statistics following compositions of up to 3 maps.
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Matching statistic: St001420
(load all 30 compositions to match this statistic)
(load all 30 compositions to match this statistic)
St001420: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 = 2 - 2
1 => 0 = 2 - 2
00 => 0 = 2 - 2
01 => 1 = 3 - 2
10 => 1 = 3 - 2
11 => 0 = 2 - 2
000 => 0 = 2 - 2
001 => 1 = 3 - 2
010 => 1 = 3 - 2
011 => 1 = 3 - 2
100 => 1 = 3 - 2
101 => 1 = 3 - 2
110 => 1 = 3 - 2
111 => 0 = 2 - 2
0000 => 0 = 2 - 2
0001 => 1 = 3 - 2
0010 => 1 = 3 - 2
0011 => 2 = 4 - 2
0100 => 1 = 3 - 2
0101 => 2 = 4 - 2
0110 => 1 = 3 - 2
0111 => 1 = 3 - 2
1000 => 1 = 3 - 2
1001 => 1 = 3 - 2
1010 => 2 = 4 - 2
1011 => 1 = 3 - 2
1100 => 2 = 4 - 2
1101 => 1 = 3 - 2
1110 => 1 = 3 - 2
1111 => 0 = 2 - 2
Description
Half the length of a longest factor which is its own reverse-complement of a binary word.
Matching statistic: St000291
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Mp00096: Binary words —Foata bijection⟶ Binary words
Mp00261: Binary words —Burrows-Wheeler⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00261: Binary words —Burrows-Wheeler⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 0 => 0 = 2 - 2
1 => 1 => 1 => 0 = 2 - 2
00 => 00 => 00 => 0 = 2 - 2
01 => 01 => 10 => 1 = 3 - 2
10 => 10 => 10 => 1 = 3 - 2
11 => 11 => 11 => 0 = 2 - 2
000 => 000 => 000 => 0 = 2 - 2
001 => 001 => 100 => 1 = 3 - 2
010 => 100 => 100 => 1 = 3 - 2
011 => 011 => 110 => 1 = 3 - 2
100 => 010 => 100 => 1 = 3 - 2
101 => 101 => 110 => 1 = 3 - 2
110 => 110 => 110 => 1 = 3 - 2
111 => 111 => 111 => 0 = 2 - 2
0000 => 0000 => 0000 => 0 = 2 - 2
0001 => 0001 => 1000 => 1 = 3 - 2
0010 => 1000 => 1000 => 1 = 3 - 2
0011 => 0011 => 1010 => 2 = 4 - 2
0100 => 0100 => 1000 => 1 = 3 - 2
0101 => 1001 => 1010 => 2 = 4 - 2
0110 => 1010 => 1100 => 1 = 3 - 2
0111 => 0111 => 1110 => 1 = 3 - 2
1000 => 0010 => 1000 => 1 = 3 - 2
1001 => 0101 => 1100 => 1 = 3 - 2
1010 => 1100 => 1010 => 2 = 4 - 2
1011 => 1011 => 1110 => 1 = 3 - 2
1100 => 0110 => 1010 => 2 = 4 - 2
1101 => 1101 => 1110 => 1 = 3 - 2
1110 => 1110 => 1110 => 1 = 3 - 2
1111 => 1111 => 1111 => 0 = 2 - 2
Description
The number of descents of a binary word.
Matching statistic: St001738
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00136: Binary words —rotate back-to-front⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001738: 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
St001738: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => [2] => ([],2)
=> 2
1 => 1 => [1,1] => ([(0,1)],2)
=> 2
00 => 00 => [3] => ([],3)
=> 2
01 => 10 => [1,2] => ([(1,2)],3)
=> 3
10 => 01 => [2,1] => ([(0,2),(1,2)],3)
=> 3
11 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
000 => 000 => [4] => ([],4)
=> 2
001 => 100 => [1,3] => ([(2,3)],4)
=> 3
010 => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
011 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
100 => 010 => [2,2] => ([(1,3),(2,3)],4)
=> 3
101 => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
110 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
111 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
0000 => 0000 => [5] => ([],5)
=> 2
0001 => 1000 => [1,4] => ([(3,4)],5)
=> 3
0010 => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
0011 => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
0100 => 0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
0101 => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4
0110 => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
0111 => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
1000 => 0100 => [2,3] => ([(2,4),(3,4)],5)
=> 3
1001 => 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
1010 => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
1011 => 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
1100 => 0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
1101 => 1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
1110 => 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
1111 => 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)
=> 2
Description
The minimal order of a graph which is not an induced subgraph of the given graph.
For example, the graph with two isolated vertices is not an induced subgraph of the complete graph on three vertices.
By contrast, the minimal number of vertices of a graph which is not a subgraph of a graph is one plus the clique number [[St000097]].
Matching statistic: St001165
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(load all 2 compositions to match this statistic)
Mp00136: Binary words —rotate back-to-front⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001165: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001165: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => [1] => [1,0]
=> 1 = 2 - 1
1 => 1 => [1] => [1,0]
=> 1 = 2 - 1
00 => 00 => [2] => [1,1,0,0]
=> 1 = 2 - 1
01 => 10 => [1,1] => [1,0,1,0]
=> 2 = 3 - 1
10 => 01 => [1,1] => [1,0,1,0]
=> 2 = 3 - 1
11 => 11 => [2] => [1,1,0,0]
=> 1 = 2 - 1
000 => 000 => [3] => [1,1,1,0,0,0]
=> 1 = 2 - 1
001 => 100 => [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
010 => 001 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
011 => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 2 = 3 - 1
100 => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 2 = 3 - 1
101 => 110 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
110 => 011 => [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
111 => 111 => [3] => [1,1,1,0,0,0]
=> 1 = 2 - 1
0000 => 0000 => [4] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
0001 => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
0010 => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
0011 => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
0100 => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
0101 => 1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 3 = 4 - 1
0110 => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
0111 => 1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
1000 => 0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
1001 => 1100 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
1010 => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 3 = 4 - 1
1011 => 1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
1100 => 0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
1101 => 1110 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
1110 => 0111 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
1111 => 1111 => [4] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
Description
Number of simple modules with even projective dimension in the corresponding Nakayama algebra.
Matching statistic: St001471
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Mp00136: Binary words —rotate back-to-front⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001471: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001471: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => [1] => [1,0]
=> 1 = 2 - 1
1 => 1 => [1] => [1,0]
=> 1 = 2 - 1
00 => 00 => [2] => [1,1,0,0]
=> 1 = 2 - 1
01 => 10 => [1,1] => [1,0,1,0]
=> 2 = 3 - 1
10 => 01 => [1,1] => [1,0,1,0]
=> 2 = 3 - 1
11 => 11 => [2] => [1,1,0,0]
=> 1 = 2 - 1
000 => 000 => [3] => [1,1,1,0,0,0]
=> 1 = 2 - 1
001 => 100 => [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
010 => 001 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
011 => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 2 = 3 - 1
100 => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 2 = 3 - 1
101 => 110 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
110 => 011 => [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
111 => 111 => [3] => [1,1,1,0,0,0]
=> 1 = 2 - 1
0000 => 0000 => [4] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
0001 => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
0010 => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
0011 => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
0100 => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
0101 => 1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 3 = 4 - 1
0110 => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
0111 => 1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
1000 => 0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
1001 => 1100 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
1010 => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 3 = 4 - 1
1011 => 1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
1100 => 0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
1101 => 1110 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
1110 => 0111 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
1111 => 1111 => [4] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
Description
The magnitude of a Dyck path.
The magnitude of a finite dimensional algebra with invertible Cartan matrix C is defined as the sum of all entries of the inverse of C.
We define the magnitude of a Dyck path as the magnitude of the corresponding LNakayama algebra.
Matching statistic: St000162
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(load all 2 compositions to match this statistic)
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000162: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000162: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [1,0]
=> [1] => 0 = 2 - 2
1 => [1] => [1,0]
=> [1] => 0 = 2 - 2
00 => [2] => [1,1,0,0]
=> [1,2] => 0 = 2 - 2
01 => [1,1] => [1,0,1,0]
=> [2,1] => 1 = 3 - 2
10 => [1,1] => [1,0,1,0]
=> [2,1] => 1 = 3 - 2
11 => [2] => [1,1,0,0]
=> [1,2] => 0 = 2 - 2
000 => [3] => [1,1,1,0,0,0]
=> [1,2,3] => 0 = 2 - 2
001 => [2,1] => [1,1,0,0,1,0]
=> [3,1,2] => 1 = 3 - 2
010 => [1,1,1] => [1,0,1,0,1,0]
=> [2,1,3] => 1 = 3 - 2
011 => [1,2] => [1,0,1,1,0,0]
=> [2,3,1] => 1 = 3 - 2
100 => [1,2] => [1,0,1,1,0,0]
=> [2,3,1] => 1 = 3 - 2
101 => [1,1,1] => [1,0,1,0,1,0]
=> [2,1,3] => 1 = 3 - 2
110 => [2,1] => [1,1,0,0,1,0]
=> [3,1,2] => 1 = 3 - 2
111 => [3] => [1,1,1,0,0,0]
=> [1,2,3] => 0 = 2 - 2
0000 => [4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0 = 2 - 2
0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1 = 3 - 2
0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => 1 = 3 - 2
0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2 = 4 - 2
0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => 1 = 3 - 2
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 2 = 4 - 2
0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => 1 = 3 - 2
0111 => [1,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1 = 3 - 2
1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1 = 3 - 2
1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => 1 = 3 - 2
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 2 = 4 - 2
1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => 1 = 3 - 2
1100 => [2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2 = 4 - 2
1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => 1 = 3 - 2
1110 => [3,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1 = 3 - 2
1111 => [4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0 = 2 - 2
Description
The number of nontrivial cycles in the cycle decomposition of a permutation.
This statistic is equal to the difference of the number of cycles of $\pi$ (see [[St000031]]) and the number of fixed points of $\pi$ (see [[St000022]]).
Matching statistic: St000292
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(load all 2 compositions to match this statistic)
Mp00096: Binary words —Foata bijection⟶ Binary words
Mp00261: Binary words —Burrows-Wheeler⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00261: Binary words —Burrows-Wheeler⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 0 => 0 => 0 = 2 - 2
1 => 1 => 1 => 1 => 0 = 2 - 2
00 => 00 => 00 => 00 => 0 = 2 - 2
01 => 01 => 10 => 01 => 1 = 3 - 2
10 => 10 => 10 => 01 => 1 = 3 - 2
11 => 11 => 11 => 11 => 0 = 2 - 2
000 => 000 => 000 => 000 => 0 = 2 - 2
001 => 001 => 100 => 001 => 1 = 3 - 2
010 => 100 => 100 => 001 => 1 = 3 - 2
011 => 011 => 110 => 011 => 1 = 3 - 2
100 => 010 => 100 => 001 => 1 = 3 - 2
101 => 101 => 110 => 011 => 1 = 3 - 2
110 => 110 => 110 => 011 => 1 = 3 - 2
111 => 111 => 111 => 111 => 0 = 2 - 2
0000 => 0000 => 0000 => 0000 => 0 = 2 - 2
0001 => 0001 => 1000 => 0001 => 1 = 3 - 2
0010 => 1000 => 1000 => 0001 => 1 = 3 - 2
0011 => 0011 => 1010 => 0101 => 2 = 4 - 2
0100 => 0100 => 1000 => 0001 => 1 = 3 - 2
0101 => 1001 => 1010 => 0101 => 2 = 4 - 2
0110 => 1010 => 1100 => 0011 => 1 = 3 - 2
0111 => 0111 => 1110 => 0111 => 1 = 3 - 2
1000 => 0010 => 1000 => 0001 => 1 = 3 - 2
1001 => 0101 => 1100 => 0011 => 1 = 3 - 2
1010 => 1100 => 1010 => 0101 => 2 = 4 - 2
1011 => 1011 => 1110 => 0111 => 1 = 3 - 2
1100 => 0110 => 1010 => 0101 => 2 = 4 - 2
1101 => 1101 => 1110 => 0111 => 1 = 3 - 2
1110 => 1110 => 1110 => 0111 => 1 = 3 - 2
1111 => 1111 => 1111 => 1111 => 0 = 2 - 2
Description
The number of ascents of a binary word.
Matching statistic: St000994
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000994: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000994: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [1,0]
=> [1] => 0 = 2 - 2
1 => [1] => [1,0]
=> [1] => 0 = 2 - 2
00 => [2] => [1,1,0,0]
=> [1,2] => 0 = 2 - 2
01 => [1,1] => [1,0,1,0]
=> [2,1] => 1 = 3 - 2
10 => [1,1] => [1,0,1,0]
=> [2,1] => 1 = 3 - 2
11 => [2] => [1,1,0,0]
=> [1,2] => 0 = 2 - 2
000 => [3] => [1,1,1,0,0,0]
=> [1,2,3] => 0 = 2 - 2
001 => [2,1] => [1,1,0,0,1,0]
=> [3,1,2] => 1 = 3 - 2
010 => [1,1,1] => [1,0,1,0,1,0]
=> [2,1,3] => 1 = 3 - 2
011 => [1,2] => [1,0,1,1,0,0]
=> [2,3,1] => 1 = 3 - 2
100 => [1,2] => [1,0,1,1,0,0]
=> [2,3,1] => 1 = 3 - 2
101 => [1,1,1] => [1,0,1,0,1,0]
=> [2,1,3] => 1 = 3 - 2
110 => [2,1] => [1,1,0,0,1,0]
=> [3,1,2] => 1 = 3 - 2
111 => [3] => [1,1,1,0,0,0]
=> [1,2,3] => 0 = 2 - 2
0000 => [4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0 = 2 - 2
0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1 = 3 - 2
0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => 1 = 3 - 2
0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2 = 4 - 2
0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => 1 = 3 - 2
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 2 = 4 - 2
0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => 1 = 3 - 2
0111 => [1,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1 = 3 - 2
1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1 = 3 - 2
1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => 1 = 3 - 2
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 2 = 4 - 2
1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => 1 = 3 - 2
1100 => [2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2 = 4 - 2
1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => 1 = 3 - 2
1110 => [3,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1 = 3 - 2
1111 => [4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0 = 2 - 2
Description
The number of cycle peaks and the number of cycle valleys of a permutation.
A '''cycle peak''' of a permutation $\pi$ is an index $i$ such that $\pi^{-1}(i) < i > \pi(i)$. Analogously, a '''cycle valley''' is an index $i$ such that $\pi^{-1}(i) > i < \pi(i)$.
Clearly, every cycle of $\pi$ contains as many peaks as valleys.
Matching statistic: St000098
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
0 => ([(0,1)],2)
=> ([],2)
=> 1 = 2 - 1
1 => ([(0,1)],2)
=> ([],2)
=> 1 = 2 - 1
00 => ([(0,2),(2,1)],3)
=> ([],3)
=> 1 = 2 - 1
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 2 = 3 - 1
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 2 = 3 - 1
11 => ([(0,2),(2,1)],3)
=> ([],3)
=> 1 = 2 - 1
000 => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 2 - 1
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2 = 3 - 1
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> 2 = 3 - 1
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2 = 3 - 1
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2 = 3 - 1
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> 2 = 3 - 1
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2 = 3 - 1
111 => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 2 - 1
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1 = 2 - 1
0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> 2 = 3 - 1
0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
=> ? = 3 - 1
0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(2,5),(2,8),(3,4),(3,8),(4,7),(5,7),(6,7),(6,8),(7,8)],9)
=> ? = 4 - 1
0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
=> ? = 3 - 1
0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(2,7),(3,6),(4,5)],8)
=> ? = 4 - 1
0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ([(2,5),(3,4),(3,7),(4,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3 - 1
0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> 2 = 3 - 1
1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> 2 = 3 - 1
1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ([(2,5),(3,4),(3,7),(4,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3 - 1
1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(2,7),(3,6),(4,5)],8)
=> ? = 4 - 1
1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
=> ? = 3 - 1
1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(2,5),(2,8),(3,4),(3,8),(4,7),(5,7),(6,7),(6,8),(7,8)],9)
=> ? = 4 - 1
1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
=> ? = 3 - 1
1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> 2 = 3 - 1
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1 = 2 - 1
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: St001335
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
0 => ([(0,1)],2)
=> ([(0,1)],2)
=> 0 = 2 - 2
1 => ([(0,1)],2)
=> ([(0,1)],2)
=> 0 = 2 - 2
00 => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 2 - 2
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 1 = 3 - 2
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 1 = 3 - 2
11 => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 2 - 2
000 => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 2 - 2
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> 1 = 3 - 2
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 1 = 3 - 2
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> 1 = 3 - 2
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> 1 = 3 - 2
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 1 = 3 - 2
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> 1 = 3 - 2
111 => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 2 - 2
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 2 - 2
0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> 1 = 3 - 2
0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(0,5),(0,8),(1,4),(1,6),(2,6),(2,8),(3,4),(3,5),(3,8),(4,7),(5,7),(6,7),(7,8)],9)
=> ? = 3 - 2
0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,6),(0,7),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 4 - 2
0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(0,5),(0,8),(1,4),(1,6),(2,6),(2,8),(3,4),(3,5),(3,8),(4,7),(5,7),(6,7),(7,8)],9)
=> ? = 3 - 2
0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(0,4),(0,5),(1,2),(1,3),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 4 - 2
0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ([(0,5),(0,6),(1,7),(1,8),(2,5),(2,6),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8)],9)
=> ? = 3 - 2
0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> 1 = 3 - 2
1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> 1 = 3 - 2
1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ([(0,5),(0,6),(1,7),(1,8),(2,5),(2,6),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8)],9)
=> ? = 3 - 2
1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(0,4),(0,5),(1,2),(1,3),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 4 - 2
1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(0,5),(0,8),(1,4),(1,6),(2,6),(2,8),(3,4),(3,5),(3,8),(4,7),(5,7),(6,7),(7,8)],9)
=> ? = 3 - 2
1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,6),(0,7),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 4 - 2
1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(0,5),(0,8),(1,4),(1,6),(2,6),(2,8),(3,4),(3,5),(3,8),(4,7),(5,7),(6,7),(7,8)],9)
=> ? = 3 - 2
1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> 1 = 3 - 2
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 2 - 2
Description
The cardinality of a minimal cycle-isolating set of a graph.
Let $\mathcal F$ be a set of graphs. A set of vertices $S$ is $\mathcal F$-isolating, if the subgraph induced by the vertices in the complement of the closed neighbourhood of $S$ does not contain any graph in $\mathcal F$.
This statistic returns the cardinality of the smallest isolating set when $\mathcal F$ contains all cycles.
The following 72 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000260The radius of a connected graph. St000455The second largest eigenvalue of a graph if it is integral. St000298The order dimension or Dushnik-Miller dimension of a poset. St000640The rank of the largest boolean interval in a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001642The Prague dimension of a graph. St001746The coalition number of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000307The number of rowmotion orbits of a poset. St000310The minimal degree of a vertex of a graph. St000544The cop number of a graph. St000785The number of distinct colouring schemes of a graph. St000822The Hadwiger number of the graph. St001029The size of the core of a graph. St001277The degeneracy of a graph. St001330The hat guessing number of a graph. St001358The largest degree of a regular subgraph of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001792The arboricity of a graph. St001883The mutual visibility number of a graph. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000272The treewidth of a graph. St000535The rank-width of a graph. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St000632The jump number of the poset. St001270The bandwidth of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St001644The dimension of a graph. St001743The discrepancy of a graph. St001826The maximal number of leaves on a vertex of a graph. St001962The proper pathwidth of a graph. St000258The burning number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St000259The diameter of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001057The Grundy value of the game of creating an independent set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001323The independence gap of a 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 $eA$. St001569The maximal modular displacement of a permutation. St000454The largest eigenvalue of a graph if it is integral. St000937The number of positive values of the symmetric group character corresponding to the partition. St001624The breadth of a lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001651The Frankl number of a lattice. St000264The girth of a graph, which is not a tree. St001118The acyclic chromatic index of a graph. 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. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. 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. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. 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. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition.
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