Your data matches 4 different statistics following compositions of up to 3 maps.
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Matching statistic: St000792
Mapping
St000792: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0
1 => 1
00 => 0
01 => 2
10 => 1
11 => 3
000 => 0
001 => 1
010 => 2
011 => 3
100 => 1
101 => 0
110 => 3
111 => 2
0000 => 0
0001 => 4
0010 => 1
0011 => 5
0100 => 2
0101 => 6
0110 => 3
0111 => 7
1000 => 1
1001 => 5
1010 => 0
1011 => 4
1100 => 3
1101 => 7
1110 => 2
1111 => 6
00000 => 0
00001 => 1
00010 => 4
00011 => 5
00100 => 1
00101 => 0
00110 => 5
00111 => 4
01000 => 2
01001 => 3
01010 => 6
01011 => 7
01100 => 3
01101 => 2
01110 => 7
01111 => 6
10000 => 1
10001 => 0
10010 => 5
10011 => 4
Description
The Grundy value for the game of ruler on a binary word. Two players alternately may switch any consecutive sequence of numbers that ends with a 1. The player facing the word which has only 0's looses.
Matching statistic: St000454
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 44%
Values
0 => [2] => ([],2)
 => 0
1 => [1,1] => ([(0,1)],2)
 => 1
00 => [3] => ([],3)
 => 0
01 => [2,1] => ([(0,2),(1,2)],3)
 => ? = 3
10 => [1,2] => ([(1,2)],3)
 => 1
11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
000 => [4] => ([],4)
 => 0
001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {0,1,2,3}
010 => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,1,2,3}
011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,1,2,3}
100 => [1,3] => ([(2,3)],4)
 => 1
101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,1,2,3}
110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
0000 => [5] => ([],5)
 => 0
0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,1,4,5,5,6,6,7,7}
0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
0100 => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {0,1,4,5,5,6,6,7,7}
0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,1,4,5,5,6,6,7,7}
0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,1,4,5,5,6,6,7,7}
0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,1,4,5,5,6,6,7,7}
1000 => [1,4] => ([(3,4)],5)
 => 1
1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,1,4,5,5,6,6,7,7}
1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,1,4,5,5,6,6,7,7}
1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,1,4,5,5,6,6,7,7}
1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 2
1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,1,4,5,5,6,6,7,7}
1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
00000 => [6] => ([],6)
 => 0
00001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,1,1,1,2,2,3,3,4,4,4,5,5,5,6,6,6,6,7,7,7,7}
00010 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2
00011 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,1,1,1,2,2,3,3,4,4,4,5,5,5,6,6,6,6,7,7,7,7}
00100 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,1,1,1,2,2,3,3,4,4,4,5,5,5,6,6,6,6,7,7,7,7}
00101 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,1,1,1,2,2,3,3,4,4,4,5,5,5,6,6,6,6,7,7,7,7}
00110 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
00111 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,1,1,1,2,2,3,3,4,4,4,5,5,5,6,6,6,6,7,7,7,7}
01000 => [2,4] => ([(3,5),(4,5)],6)
 => ? ∊ {0,0,0,1,1,1,2,2,3,3,4,4,4,5,5,5,6,6,6,6,7,7,7,7}
01001 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,1,1,1,2,2,3,3,4,4,4,5,5,5,6,6,6,6,7,7,7,7}
01010 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,1,1,1,2,2,3,3,4,4,4,5,5,5,6,6,6,6,7,7,7,7}
01011 => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,1,1,1,2,2,3,3,4,4,4,5,5,5,6,6,6,6,7,7,7,7}
01100 => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,1,1,1,2,2,3,3,4,4,4,5,5,5,6,6,6,6,7,7,7,7}
01101 => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,1,1,1,2,2,3,3,4,4,4,5,5,5,6,6,6,6,7,7,7,7}
01110 => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,1,1,1,2,2,3,3,4,4,4,5,5,5,6,6,6,6,7,7,7,7}
01111 => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,1,1,1,2,2,3,3,4,4,4,5,5,5,6,6,6,6,7,7,7,7}
10000 => [1,5] => ([(4,5)],6)
 => 1
10001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,1,1,1,2,2,3,3,4,4,4,5,5,5,6,6,6,6,7,7,7,7}
10010 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,1,1,1,2,2,3,3,4,4,4,5,5,5,6,6,6,6,7,7,7,7}
10011 => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,1,1,1,2,2,3,3,4,4,4,5,5,5,6,6,6,6,7,7,7,7}
10100 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,1,1,1,2,2,3,3,4,4,4,5,5,5,6,6,6,6,7,7,7,7}
10101 => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,1,1,1,2,2,3,3,4,4,4,5,5,5,6,6,6,6,7,7,7,7}
10110 => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,1,1,1,2,2,3,3,4,4,4,5,5,5,6,6,6,6,7,7,7,7}
10111 => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,1,1,1,2,2,3,3,4,4,4,5,5,5,6,6,6,6,7,7,7,7}
11000 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => 2
11001 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,1,1,1,2,2,3,3,4,4,4,5,5,5,6,6,6,6,7,7,7,7}
11010 => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,1,1,1,2,2,3,3,4,4,4,5,5,5,6,6,6,6,7,7,7,7}
11011 => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,1,1,1,2,2,3,3,4,4,4,5,5,5,6,6,6,6,7,7,7,7}
11100 => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
11101 => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,1,1,1,2,2,3,3,4,4,4,5,5,5,6,6,6,6,7,7,7,7}
11110 => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
11111 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
000000 => [7] => ([],7)
 => 0
000001 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7}
000010 => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7}
000011 => [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7}
000100 => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 2
000101 => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7}
000110 => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7}
000111 => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7}
001000 => [3,4] => ([(3,6),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7}
001001 => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7}
001010 => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7}
001011 => [3,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7}
001100 => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3
001101 => [3,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7}
001110 => [3,1,1,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7}
100000 => [1,6] => ([(5,6)],7)
 => 1
110000 => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
 => 2
111000 => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3
111100 => [1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 4
111110 => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
111111 => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 6
Description
The largest eigenvalue of a graph if it is integral. If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St001645
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 44%
Values
0 => [2] => ([],2)
 => ([],1)
 => 1 = 0 + 1
1 => [1,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 2 = 1 + 1
00 => [3] => ([],3)
 => ([],1)
 => 1 = 0 + 1
01 => [2,1] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 2 = 1 + 1
10 => [1,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? = 3 + 1
11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
000 => [4] => ([],4)
 => ([],1)
 => 1 = 0 + 1
001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 2 = 1 + 1
010 => [2,2] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {0,1,2,3} + 1
011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
100 => [1,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {0,1,2,3} + 1
101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,1,2,3} + 1
110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,1,2,3} + 1
111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
0000 => [5] => ([],5)
 => ([],1)
 => 1 = 0 + 1
0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2 = 1 + 1
0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? ∊ {0,1,2,3,4,5,5,6,6,7,7} + 1
0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
0100 => [2,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? ∊ {0,1,2,3,4,5,5,6,6,7,7} + 1
0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,1,2,3,4,5,5,6,6,7,7} + 1
0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,1,2,3,4,5,5,6,6,7,7} + 1
0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
1000 => [1,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? ∊ {0,1,2,3,4,5,5,6,6,7,7} + 1
1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,1,2,3,4,5,5,6,6,7,7} + 1
1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,1,2,3,4,5,5,6,6,7,7} + 1
1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,1,2,3,4,5,5,6,6,7,7} + 1
1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,1,2,3,4,5,5,6,6,7,7} + 1
1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,1,2,3,4,5,5,6,6,7,7} + 1
1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,1,2,3,4,5,5,6,6,7,7} + 1
1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
00000 => [6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
00001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 1 + 1
00010 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,6,6,6,6,7,7,7,7} + 1
00011 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
00100 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,6,6,6,6,7,7,7,7} + 1
00101 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,6,6,6,6,7,7,7,7} + 1
00110 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,6,6,6,6,7,7,7,7} + 1
00111 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
01000 => [2,4] => ([(3,5),(4,5)],6)
 => ([(1,2)],3)
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,6,6,6,6,7,7,7,7} + 1
01001 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,6,6,6,6,7,7,7,7} + 1
01010 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,6,6,6,6,7,7,7,7} + 1
01011 => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,6,6,6,6,7,7,7,7} + 1
01100 => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,6,6,6,6,7,7,7,7} + 1
01101 => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,6,6,6,6,7,7,7,7} + 1
01110 => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,6,6,6,6,7,7,7,7} + 1
01111 => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
10000 => [1,5] => ([(4,5)],6)
 => ([(1,2)],3)
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,6,6,6,6,7,7,7,7} + 1
10001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,6,6,6,6,7,7,7,7} + 1
10010 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,6,6,6,6,7,7,7,7} + 1
10011 => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,6,6,6,6,7,7,7,7} + 1
10100 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,6,6,6,6,7,7,7,7} + 1
10101 => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,6,6,6,6,7,7,7,7} + 1
10110 => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,6,6,6,6,7,7,7,7} + 1
10111 => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
11000 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,6,6,6,6,7,7,7,7} + 1
11001 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,6,6,6,6,7,7,7,7} + 1
11010 => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,6,6,6,6,7,7,7,7} + 1
11011 => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,6,6,6,6,7,7,7,7} + 1
11100 => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,6,6,6,6,7,7,7,7} + 1
11101 => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,6,6,6,6,7,7,7,7} + 1
11110 => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,6,6,6,6,7,7,7,7} + 1
11111 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
000000 => [7] => ([],7)
 => ([],1)
 => 1 = 0 + 1
000001 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,1)],2)
 => 2 = 1 + 1
000010 => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7} + 1
000011 => [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
000100 => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7} + 1
000101 => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7} + 1
000110 => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7} + 1
000111 => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
001000 => [3,4] => ([(3,6),(4,6),(5,6)],7)
 => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7} + 1
001001 => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7} + 1
001010 => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7} + 1
001011 => [3,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7} + 1
001100 => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7} + 1
001111 => [3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
010111 => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
011111 => [2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
100111 => [1,3,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
101111 => [1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
110111 => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
111111 => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
0001111 => [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
Description
The pebbling number of a connected graph.
Matching statistic: St001232
(load all 2 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 44%
Values
0 => [2] => [1,1,0,0]
 => [1,0,1,0]
 => 1
1 => [1,1] => [1,0,1,0]
 => [1,1,0,0]
 => 0
00 => [3] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => ? = 3
01 => [2,1] => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 1
10 => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2
11 => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
000 => [4] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? ∊ {0,1,2,3}
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => ? ∊ {0,1,2,3}
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ? ∊ {0,1,2,3}
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => ? ∊ {0,1,2,3}
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]
 => 3
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,1,2,3,4,5,5,6,6,7,7}
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? ∊ {0,1,2,3,4,5,5,6,6,7,7}
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => ? ∊ {0,1,2,3,4,5,5,6,6,7,7}
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? ∊ {0,1,2,3,4,5,5,6,6,7,7}
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => ? ∊ {0,1,2,3,4,5,5,6,6,7,7}
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => ? ∊ {0,1,2,3,4,5,5,6,6,7,7}
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => ? ∊ {0,1,2,3,4,5,5,6,6,7,7}
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]
 => 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,1,2,3,4,5,5,6,6,7,7}
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => ? ∊ {0,1,2,3,4,5,5,6,6,7,7}
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ? ∊ {0,1,2,3,4,5,5,6,6,7,7}
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]
 => 2
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => ? ∊ {0,1,2,3,4,5,5,6,6,7,7}
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]
 => 3
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]
 => 4
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
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,5,6,6,6,6,7,7,7,7}
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,5,6,6,6,6,7,7,7,7}
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,5,6,6,6,6,7,7,7,7}
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,5,6,6,6,6,7,7,7,7}
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,5,6,6,6,6,7,7,7,7}
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,5,6,6,6,6,7,7,7,7}
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,5,6,6,6,6,7,7,7,7}
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,5,6,6,6,6,7,7,7,7}
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,5,6,6,6,6,7,7,7,7}
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,5,6,6,6,6,7,7,7,7}
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]
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,5,6,6,6,6,7,7,7,7}
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]
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,5,6,6,6,6,7,7,7,7}
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,5,6,6,6,6,7,7,7,7}
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]
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,5,6,6,6,6,7,7,7,7}
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]
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,5,6,6,6,6,7,7,7,7}
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]
 => 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,5,6,6,6,6,7,7,7,7}
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,5,6,6,6,6,7,7,7,7}
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,5,6,6,6,6,7,7,7,7}
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,5,6,6,6,6,7,7,7,7}
10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,5,6,6,6,6,7,7,7,7}
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]
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,5,6,6,6,6,7,7,7,7}
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]
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,5,6,6,6,6,7,7,7,7}
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]
 => 2
11000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,5,6,6,6,6,7,7,7,7}
11001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,5,6,6,6,6,7,7,7,7}
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]
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,5,6,6,6,6,7,7,7,7}
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
11100 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,5,6,6,6,6,7,7,7,7}
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]
 => 4
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]
 => 5
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
000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7}
000001 => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7}
000010 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7}
000011 => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7}
000100 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7}
000101 => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7}
000110 => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7}
000111 => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7}
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]
 => 1
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]
 => 2
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]
 => 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]
 => 4
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]
 => 5
111110 => [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => 6
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
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.