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Your data matches 12 different statistics following compositions of up to 3 maps.
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Matching statistic: St001721
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(load all 2 compositions to match this statistic)
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St001721: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00094: Integer compositions —to binary word⟶ Binary words
St001721: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => 10 => 1
1 => [1,1] => 11 => 0
01 => [2,1] => 101 => 2
10 => [1,2] => 110 => 1
11 => [1,1,1] => 111 => 0
010 => [2,2] => 1010 => 3
011 => [2,1,1] => 1011 => 3
101 => [1,2,1] => 1101 => 2
110 => [1,1,2] => 1110 => 1
111 => [1,1,1,1] => 1111 => 0
0101 => [2,2,1] => 10101 => 4
0110 => [2,1,2] => 10110 => 4
0111 => [2,1,1,1] => 10111 => 4
1010 => [1,2,2] => 11010 => 3
1011 => [1,2,1,1] => 11011 => 3
1101 => [1,1,2,1] => 11101 => 2
1110 => [1,1,1,2] => 11110 => 1
1111 => [1,1,1,1,1] => 11111 => 0
01010 => [2,2,2] => 101010 => 5
01011 => [2,2,1,1] => 101011 => 5
01101 => [2,1,2,1] => 101101 => 5
01110 => [2,1,1,2] => 101110 => 5
01111 => [2,1,1,1,1] => 101111 => 5
10101 => [1,2,2,1] => 110101 => 4
10110 => [1,2,1,2] => 110110 => 4
10111 => [1,2,1,1,1] => 110111 => 4
11010 => [1,1,2,2] => 111010 => 3
11011 => [1,1,2,1,1] => 111011 => 3
11101 => [1,1,1,2,1] => 111101 => 2
11110 => [1,1,1,1,2] => 111110 => 1
11111 => [1,1,1,1,1,1] => 111111 => 0
010101 => [2,2,2,1] => 1010101 => 6
010110 => [2,2,1,2] => 1010110 => 6
010111 => [2,2,1,1,1] => 1010111 => 6
011010 => [2,1,2,2] => 1011010 => 6
011011 => [2,1,2,1,1] => 1011011 => 6
011101 => [2,1,1,2,1] => 1011101 => 6
011110 => [2,1,1,1,2] => 1011110 => 6
011111 => [2,1,1,1,1,1] => 1011111 => 6
101010 => [1,2,2,2] => 1101010 => 5
101011 => [1,2,2,1,1] => 1101011 => 5
101101 => [1,2,1,2,1] => 1101101 => 5
101110 => [1,2,1,1,2] => 1101110 => 5
101111 => [1,2,1,1,1,1] => 1101111 => 5
110101 => [1,1,2,2,1] => 1110101 => 4
110110 => [1,1,2,1,2] => 1110110 => 4
110111 => [1,1,2,1,1,1] => 1110111 => 4
111010 => [1,1,1,2,2] => 1111010 => 3
111011 => [1,1,1,2,1,1] => 1111011 => 3
111101 => [1,1,1,1,2,1] => 1111101 => 2
Description
The degree of a binary word.
A valley in a binary word is a letter $0$ which is not immediately followed by a $1$. A peak is a letter $1$ which is not immediately followed by a $0$.
Let $f$ be the map that replaces every valley with a peak. The degree of a binary word $w$ is the number of times $f$ has to be applied to obtain a binary word without zeros.
Matching statistic: St000171
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000171: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000171: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1] => ([(0,1)],2)
=> 1
1 => [1,1] => [2] => ([],2)
=> 0
01 => [2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2
10 => [1,2] => [1,2] => ([(1,2)],3)
=> 1
11 => [1,1,1] => [3] => ([],3)
=> 0
010 => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
011 => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
101 => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 2
110 => [1,1,2] => [1,3] => ([(2,3)],4)
=> 1
111 => [1,1,1,1] => [4] => ([],4)
=> 0
0101 => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
0110 => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
0111 => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
1010 => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
1011 => [1,2,1,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
1101 => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2
1110 => [1,1,1,2] => [1,4] => ([(3,4)],5)
=> 1
1111 => [1,1,1,1,1] => [5] => ([],5)
=> 0
01010 => [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
01011 => [2,2,1,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
01101 => [2,1,2,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
01110 => [2,1,1,2] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
01111 => [2,1,1,1,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 5
10101 => [1,2,2,1] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
10110 => [1,2,1,2] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
10111 => [1,2,1,1,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
11010 => [1,1,2,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3
11011 => [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 3
11101 => [1,1,1,2,1] => [2,4] => ([(3,5),(4,5)],6)
=> 2
11110 => [1,1,1,1,2] => [1,5] => ([(4,5)],6)
=> 1
11111 => [1,1,1,1,1,1] => [6] => ([],6)
=> 0
010101 => [2,2,2,1] => [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
010110 => [2,2,1,2] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
010111 => [2,2,1,1,1] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
011010 => [2,1,2,2] => [1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
011011 => [2,1,2,1,1] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
011101 => [2,1,1,2,1] => [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
011110 => [2,1,1,1,2] => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
011111 => [2,1,1,1,1,1] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 6
101010 => [1,2,2,2] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
101011 => [1,2,2,1,1] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
101101 => [1,2,1,2,1] => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
101110 => [1,2,1,1,2] => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
101111 => [1,2,1,1,1,1] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 5
110101 => [1,1,2,2,1] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
110110 => [1,1,2,1,2] => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
110111 => [1,1,2,1,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 4
111010 => [1,1,1,2,2] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> 3
111011 => [1,1,1,2,1,1] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 3
111101 => [1,1,1,1,2,1] => [2,5] => ([(4,6),(5,6)],7)
=> 2
Description
The degree of the graph.
This is the maximal vertex degree of a graph.
Matching statistic: St000645
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000645: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000645: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 1
1 => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 0
01 => [2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
10 => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
11 => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 3
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 4
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 3
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> 5
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> 5
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> 5
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 5
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 5
10101 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> 4
10110 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> 4
10111 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 4
11010 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 3
11011 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3
11101 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 2
11110 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1
11111 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
010101 => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> 6
010110 => [2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> 6
010111 => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> 6
011010 => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> 6
011011 => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> 6
011101 => [2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> 6
011110 => [2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> 6
011111 => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 6
101010 => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> 5
101011 => [1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> 5
101101 => [1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> 5
101110 => [1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> 5
101111 => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> 5
110101 => [1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> 4
110110 => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> 4
110111 => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 4
111010 => [1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> 3
111011 => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> 3
111101 => [1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 2
Description
The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between.
For a Dyck path $D = D_1 \cdots D_{2n}$ with peaks in positions $i_1 < \ldots < i_k$ and valleys in positions $j_1 < \ldots < j_{k-1}$, this statistic is given by
$$
\sum_{a=1}^{k-1} (j_a-i_a)(i_{a+1}-j_a)
$$
Matching statistic: St000987
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000987: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000987: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1] => ([(0,1)],2)
=> 1
1 => [1,1] => [2] => ([],2)
=> 0
01 => [2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2
10 => [1,2] => [1,2] => ([(1,2)],3)
=> 1
11 => [1,1,1] => [3] => ([],3)
=> 0
010 => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
011 => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
101 => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 2
110 => [1,1,2] => [1,3] => ([(2,3)],4)
=> 1
111 => [1,1,1,1] => [4] => ([],4)
=> 0
0101 => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
0110 => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
0111 => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
1010 => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
1011 => [1,2,1,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
1101 => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2
1110 => [1,1,1,2] => [1,4] => ([(3,4)],5)
=> 1
1111 => [1,1,1,1,1] => [5] => ([],5)
=> 0
01010 => [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
01011 => [2,2,1,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
01101 => [2,1,2,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
01110 => [2,1,1,2] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
01111 => [2,1,1,1,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 5
10101 => [1,2,2,1] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
10110 => [1,2,1,2] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
10111 => [1,2,1,1,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
11010 => [1,1,2,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3
11011 => [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 3
11101 => [1,1,1,2,1] => [2,4] => ([(3,5),(4,5)],6)
=> 2
11110 => [1,1,1,1,2] => [1,5] => ([(4,5)],6)
=> 1
11111 => [1,1,1,1,1,1] => [6] => ([],6)
=> 0
010101 => [2,2,2,1] => [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
010110 => [2,2,1,2] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
010111 => [2,2,1,1,1] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
011010 => [2,1,2,2] => [1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
011011 => [2,1,2,1,1] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
011101 => [2,1,1,2,1] => [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
011110 => [2,1,1,1,2] => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
011111 => [2,1,1,1,1,1] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 6
101010 => [1,2,2,2] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
101011 => [1,2,2,1,1] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
101101 => [1,2,1,2,1] => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
101110 => [1,2,1,1,2] => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
101111 => [1,2,1,1,1,1] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 5
110101 => [1,1,2,2,1] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
110110 => [1,1,2,1,2] => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
110111 => [1,1,2,1,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 4
111010 => [1,1,1,2,2] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> 3
111011 => [1,1,1,2,1,1] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 3
111101 => [1,1,1,1,2,1] => [2,5] => ([(4,6),(5,6)],7)
=> 2
Description
The number of positive eigenvalues of the Laplacian matrix of the graph.
This is the number of vertices minus the number of connected components of the graph.
Matching statistic: St001232
Mp00105: Binary words —complement⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001232: 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
St001232: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 1 => [1,1] => [1,0,1,0]
=> 1
1 => 0 => [2] => [1,1,0,0]
=> 0
01 => 10 => [1,2] => [1,0,1,1,0,0]
=> 2
10 => 01 => [2,1] => [1,1,0,0,1,0]
=> 1
11 => 00 => [3] => [1,1,1,0,0,0]
=> 0
010 => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3
011 => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
101 => 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
110 => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
111 => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 0
0101 => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4
0110 => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 4
0111 => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4
1010 => 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
1011 => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
1101 => 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
1110 => 0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
1111 => 0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
01010 => 10101 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5
01011 => 10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 5
01101 => 10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 5
01110 => 10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5
01111 => 10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5
10101 => 01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 4
10110 => 01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 4
10111 => 01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 4
11010 => 00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3
11011 => 00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
11101 => 00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2
11110 => 00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
11111 => 00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
010101 => 101010 => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> 6
010110 => 101001 => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> 6
010111 => 101000 => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> 6
011010 => 100101 => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> 6
011011 => 100100 => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> 6
011101 => 100010 => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> 6
011110 => 100001 => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
011111 => 100000 => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
101010 => 010101 => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> 5
101011 => 010100 => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> 5
101101 => 010010 => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> 5
101110 => 010001 => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> 5
101111 => 010000 => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> 5
110101 => 001010 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> 4
110110 => 001001 => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> 4
110111 => 001000 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> 4
111010 => 000101 => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> 3
111011 => 000100 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> 3
111101 => 000010 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> 2
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000476
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000476: Dyck paths ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000476: Dyck paths ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1] => [1,0,1,0]
=> 1
1 => [1,1] => [2] => [1,1,0,0]
=> 0
01 => [2,1] => [2,1] => [1,1,0,0,1,0]
=> 2
10 => [1,2] => [1,2] => [1,0,1,1,0,0]
=> 1
11 => [1,1,1] => [3] => [1,1,1,0,0,0]
=> 0
010 => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3
011 => [2,1,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
101 => [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
110 => [1,1,2] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
111 => [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 0
0101 => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 4
0110 => [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 4
0111 => [2,1,1,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
1010 => [1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 3
1011 => [1,2,1,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
1101 => [1,1,2,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
1110 => [1,1,1,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
1111 => [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
01010 => [2,2,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5
01011 => [2,2,1,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 5
01101 => [2,1,2,1] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 5
01110 => [2,1,1,2] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5
01111 => [2,1,1,1,1] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 5
10101 => [1,2,2,1] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 4
10110 => [1,2,1,2] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 4
10111 => [1,2,1,1,1] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4
11010 => [1,1,2,2] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 3
11011 => [1,1,2,1,1] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
11101 => [1,1,1,2,1] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
11110 => [1,1,1,1,2] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
11111 => [1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
010101 => [2,2,2,1] => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> 6
010110 => [2,2,1,2] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> 6
010111 => [2,2,1,1,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> 6
011010 => [2,1,2,2] => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> 6
011011 => [2,1,2,1,1] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> 6
011101 => [2,1,1,2,1] => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> 6
011110 => [2,1,1,1,2] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
011111 => [2,1,1,1,1,1] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 6
101010 => [1,2,2,2] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> 5
101011 => [1,2,2,1,1] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> 5
101101 => [1,2,1,2,1] => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> 5
101110 => [1,2,1,1,2] => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> 5
101111 => [1,2,1,1,1,1] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> 5
110101 => [1,1,2,2,1] => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> 4
110110 => [1,1,2,1,2] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> 4
110111 => [1,1,2,1,1,1] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> 4
111010 => [1,1,1,2,2] => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> 3
111011 => [1,1,1,2,1,1] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> 3
111101 => [1,1,1,1,2,1] => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> 2
=> [1] => [1] => [1,0]
=> ? = 0
Description
The sum of the semi-lengths of tunnels before a valley of a Dyck path.
For each valley $v$ in a Dyck path $D$ there is a corresponding tunnel, which
is the factor $T_v = s_i\dots s_j$ of $D$ where $s_i$ is the step after the first intersection of $D$ with the line $y = ht(v)$ to the left of $s_j$. This statistic is
$$
\sum_v (j_v-i_v)/2.
$$
Matching statistic: St001725
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001725: Graphs ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001725: Graphs ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
1 => [1,1] => [2] => ([],2)
=> 1 = 0 + 1
01 => [2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
10 => [1,2] => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
11 => [1,1,1] => [3] => ([],3)
=> 1 = 0 + 1
010 => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
011 => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 4 = 3 + 1
101 => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
110 => [1,1,2] => [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
111 => [1,1,1,1] => [4] => ([],4)
=> 1 = 0 + 1
0101 => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
0110 => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
0111 => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 5 = 4 + 1
1010 => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
1011 => [1,2,1,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 4 = 3 + 1
1101 => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 3 = 2 + 1
1110 => [1,1,1,2] => [1,4] => ([(3,4)],5)
=> 2 = 1 + 1
1111 => [1,1,1,1,1] => [5] => ([],5)
=> 1 = 0 + 1
01010 => [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
01011 => [2,2,1,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
01101 => [2,1,2,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
01110 => [2,1,1,2] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
01111 => [2,1,1,1,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 6 = 5 + 1
10101 => [1,2,2,1] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
10110 => [1,2,1,2] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
10111 => [1,2,1,1,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 5 = 4 + 1
11010 => [1,1,2,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
11011 => [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 4 = 3 + 1
11101 => [1,1,1,2,1] => [2,4] => ([(3,5),(4,5)],6)
=> 3 = 2 + 1
11110 => [1,1,1,1,2] => [1,5] => ([(4,5)],6)
=> 2 = 1 + 1
11111 => [1,1,1,1,1,1] => [6] => ([],6)
=> 1 = 0 + 1
010101 => [2,2,2,1] => [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6 + 1
010110 => [2,2,1,2] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6 + 1
010111 => [2,2,1,1,1] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
011010 => [2,1,2,2] => [1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
011011 => [2,1,2,1,1] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
011101 => [2,1,1,2,1] => [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
011110 => [2,1,1,1,2] => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
011111 => [2,1,1,1,1,1] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 7 = 6 + 1
101010 => [1,2,2,2] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6 = 5 + 1
101011 => [1,2,2,1,1] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6 = 5 + 1
101101 => [1,2,1,2,1] => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6 = 5 + 1
101110 => [1,2,1,1,2] => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 6 = 5 + 1
101111 => [1,2,1,1,1,1] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 6 = 5 + 1
110101 => [1,1,2,2,1] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5 = 4 + 1
110110 => [1,1,2,1,2] => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 5 = 4 + 1
110111 => [1,1,2,1,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 5 = 4 + 1
111010 => [1,1,1,2,2] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 3 + 1
111011 => [1,1,1,2,1,1] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 4 = 3 + 1
111101 => [1,1,1,1,2,1] => [2,5] => ([(4,6),(5,6)],7)
=> 3 = 2 + 1
111110 => [1,1,1,1,1,2] => [1,6] => ([(5,6)],7)
=> 2 = 1 + 1
111111 => [1,1,1,1,1,1,1] => [7] => ([],7)
=> 1 = 0 + 1
Description
The harmonious chromatic number of a graph.
A harmonious colouring is a proper vertex colouring such that any pair of colours appears at most once on adjacent vertices.
Matching statistic: St001118
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001118: Graphs ⟶ ℤResult quality: 86% ●values known / values provided: 87%●distinct values known / distinct values provided: 86%
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001118: Graphs ⟶ ℤResult quality: 86% ●values known / values provided: 87%●distinct values known / distinct values provided: 86%
Values
0 => [2] => [1,1] => ([(0,1)],2)
=> 1
1 => [1,1] => [2] => ([],2)
=> ? = 0
01 => [2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2
10 => [1,2] => [1,2] => ([(1,2)],3)
=> 1
11 => [1,1,1] => [3] => ([],3)
=> ? = 0
010 => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
011 => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
101 => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 2
110 => [1,1,2] => [1,3] => ([(2,3)],4)
=> 1
111 => [1,1,1,1] => [4] => ([],4)
=> ? = 0
0101 => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
0110 => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
0111 => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
1010 => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
1011 => [1,2,1,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
1101 => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2
1110 => [1,1,1,2] => [1,4] => ([(3,4)],5)
=> 1
1111 => [1,1,1,1,1] => [5] => ([],5)
=> ? = 0
01010 => [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
01011 => [2,2,1,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
01101 => [2,1,2,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
01110 => [2,1,1,2] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
01111 => [2,1,1,1,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 5
10101 => [1,2,2,1] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
10110 => [1,2,1,2] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
10111 => [1,2,1,1,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
11010 => [1,1,2,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3
11011 => [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 3
11101 => [1,1,1,2,1] => [2,4] => ([(3,5),(4,5)],6)
=> 2
11110 => [1,1,1,1,2] => [1,5] => ([(4,5)],6)
=> 1
11111 => [1,1,1,1,1,1] => [6] => ([],6)
=> ? = 0
010101 => [2,2,2,1] => [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
010110 => [2,2,1,2] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
010111 => [2,2,1,1,1] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
011010 => [2,1,2,2] => [1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
011011 => [2,1,2,1,1] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
011101 => [2,1,1,2,1] => [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
011110 => [2,1,1,1,2] => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
011111 => [2,1,1,1,1,1] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 6
101010 => [1,2,2,2] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
101011 => [1,2,2,1,1] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
101101 => [1,2,1,2,1] => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
101110 => [1,2,1,1,2] => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
101111 => [1,2,1,1,1,1] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 5
110101 => [1,1,2,2,1] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
110110 => [1,1,2,1,2] => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
110111 => [1,1,2,1,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 4
111010 => [1,1,1,2,2] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> 3
111011 => [1,1,1,2,1,1] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 3
111101 => [1,1,1,1,2,1] => [2,5] => ([(4,6),(5,6)],7)
=> 2
111110 => [1,1,1,1,1,2] => [1,6] => ([(5,6)],7)
=> 1
111111 => [1,1,1,1,1,1,1] => [7] => ([],7)
=> ? = 0
=> [1] => [1] => ([],1)
=> ? = 0
Description
The acyclic chromatic index of a graph.
An acyclic edge coloring of a graph is a proper colouring of the edges of a graph such that the union of the edges colored with any two given colours is a forest.
The smallest number of colours such that such a colouring exists is the acyclic chromatic index.
Matching statistic: St001117
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001117: Graphs ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 86%
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001117: Graphs ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 86%
Values
0 => [2] => [1,1] => ([(0,1)],2)
=> 1
1 => [1,1] => [2] => ([],2)
=> 0
01 => [2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2
10 => [1,2] => [1,2] => ([(1,2)],3)
=> 1
11 => [1,1,1] => [3] => ([],3)
=> 0
010 => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
011 => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
101 => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 2
110 => [1,1,2] => [1,3] => ([(2,3)],4)
=> 1
111 => [1,1,1,1] => [4] => ([],4)
=> 0
0101 => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
0110 => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
0111 => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
1010 => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
1011 => [1,2,1,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
1101 => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2
1110 => [1,1,1,2] => [1,4] => ([(3,4)],5)
=> 1
1111 => [1,1,1,1,1] => [5] => ([],5)
=> 0
01010 => [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
01011 => [2,2,1,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
01101 => [2,1,2,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
01110 => [2,1,1,2] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
01111 => [2,1,1,1,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 5
10101 => [1,2,2,1] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
10110 => [1,2,1,2] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
10111 => [1,2,1,1,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
11010 => [1,1,2,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3
11011 => [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 3
11101 => [1,1,1,2,1] => [2,4] => ([(3,5),(4,5)],6)
=> 2
11110 => [1,1,1,1,2] => [1,5] => ([(4,5)],6)
=> 1
11111 => [1,1,1,1,1,1] => [6] => ([],6)
=> 0
010101 => [2,2,2,1] => [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6
010110 => [2,2,1,2] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6
010111 => [2,2,1,1,1] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6
011010 => [2,1,2,2] => [1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6
011011 => [2,1,2,1,1] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6
011101 => [2,1,1,2,1] => [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6
011110 => [2,1,1,1,2] => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6
011111 => [2,1,1,1,1,1] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 6
101010 => [1,2,2,2] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
101011 => [1,2,2,1,1] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
101101 => [1,2,1,2,1] => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
101110 => [1,2,1,1,2] => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
101111 => [1,2,1,1,1,1] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 5
110101 => [1,1,2,2,1] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
110110 => [1,1,2,1,2] => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
110111 => [1,1,2,1,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 4
111010 => [1,1,1,2,2] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
111011 => [1,1,1,2,1,1] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? = 3
111101 => [1,1,1,1,2,1] => [2,5] => ([(4,6),(5,6)],7)
=> ? = 2
111110 => [1,1,1,1,1,2] => [1,6] => ([(5,6)],7)
=> ? = 1
111111 => [1,1,1,1,1,1,1] => [7] => ([],7)
=> ? = 0
=> [1] => [1] => ([],1)
=> 0
Description
The game chromatic index of a graph.
Two players, Alice and Bob, take turns colouring properly any uncolored edge of the graph. Alice begins. If it is not possible for either player to colour a edge, then Bob wins. If the graph is completely colored, Alice wins.
The game chromatic index is the smallest number of colours such that Alice has a winning strategy.
Matching statistic: St001225
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001225: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 86%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001225: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 86%
Values
0 => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 1
1 => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 0
01 => [2,1] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 2
10 => [1,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
11 => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 3
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 4
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 4
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 5
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 5
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 5
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 5
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
10101 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 4
10110 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> 4
10111 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 4
11010 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 3
11011 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 3
11101 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 2
11110 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
11111 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
010101 => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 6
010110 => [2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> ? = 6
010111 => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 6
011010 => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> ? = 6
011011 => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 6
011101 => [2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> ? = 6
011110 => [2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 6
011111 => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 6
101010 => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> ? = 5
101011 => [1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 5
101101 => [1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 5
101110 => [1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 5
101111 => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 5
110101 => [1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 4
110110 => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 4
110111 => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 4
111010 => [1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 3
111011 => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 3
111101 => [1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2
111110 => [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
111111 => [1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
=> [1] => [1,0]
=> [1,0]
=> 0
Description
The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra.
The following 2 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path.
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