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Your data matches 61 different statistics following compositions of up to 3 maps.
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Matching statistic: St000259
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => ([],1)
=> ([],1)
=> 0
1 => [1] => ([],1)
=> ([],1)
=> 0
00 => [2] => ([],2)
=> ([],1)
=> 0
01 => [1,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
10 => [1,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
11 => [2] => ([],2)
=> ([],1)
=> 0
000 => [3] => ([],3)
=> ([],1)
=> 0
001 => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
110 => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
111 => [3] => ([],3)
=> ([],1)
=> 0
0000 => [4] => ([],4)
=> ([],1)
=> 0
0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
0101 => [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)
=> 1
0110 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
1010 => [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)
=> 1
1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
1110 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
1111 => [4] => ([],4)
=> ([],1)
=> 0
00000 => [5] => ([],5)
=> ([],1)
=> 0
00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
00010 => [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)
=> 1
00101 => [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)
=> 1
00110 => [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)
=> 2
01001 => [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)
=> 2
01010 => [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)
=> 1
01101 => [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)
=> 2
01110 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
10010 => [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)
=> 2
10101 => [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)
=> 1
10110 => [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)
=> 2
11001 => [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)
=> 2
11010 => [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)
=> 1
11101 => [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)
=> 1
11110 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
11111 => [5] => ([],5)
=> ([],1)
=> 0
000000 => [6] => ([],6)
=> ([],1)
=> 0
000001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
000010 => [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)
=> 1
000101 => [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)
=> 1
000110 => [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)
=> 2
001001 => [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)
=> 2
001010 => [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)
=> 1
001101 => [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)
=> 2
001110 => [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)
=> 2
010001 => [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)
=> 2
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St001093
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(load all 3 compositions to match this statistic)
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001093: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001093: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => ([],1)
=> ([],1)
=> 1 = 0 + 1
1 => [1] => ([],1)
=> ([],1)
=> 1 = 0 + 1
00 => [2] => ([],2)
=> ([],1)
=> 1 = 0 + 1
01 => [1,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
10 => [1,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
11 => [2] => ([],2)
=> ([],1)
=> 1 = 0 + 1
000 => [3] => ([],3)
=> ([],1)
=> 1 = 0 + 1
001 => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
110 => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
111 => [3] => ([],3)
=> ([],1)
=> 1 = 0 + 1
0000 => [4] => ([],4)
=> ([],1)
=> 1 = 0 + 1
0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
0101 => [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)
=> 2 = 1 + 1
0110 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
1010 => [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)
=> 2 = 1 + 1
1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
1110 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
1111 => [4] => ([],4)
=> ([],1)
=> 1 = 0 + 1
00000 => [5] => ([],5)
=> ([],1)
=> 1 = 0 + 1
00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
00010 => [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)
=> 2 = 1 + 1
00101 => [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)
=> 2 = 1 + 1
00110 => [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)
=> 3 = 2 + 1
01001 => [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)
=> 3 = 2 + 1
01010 => [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)
=> 2 = 1 + 1
01101 => [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)
=> 3 = 2 + 1
01110 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
10010 => [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)
=> 3 = 2 + 1
10101 => [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)
=> 2 = 1 + 1
10110 => [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)
=> 3 = 2 + 1
11001 => [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)
=> 3 = 2 + 1
11010 => [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)
=> 2 = 1 + 1
11101 => [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)
=> 2 = 1 + 1
11110 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
11111 => [5] => ([],5)
=> ([],1)
=> 1 = 0 + 1
000000 => [6] => ([],6)
=> ([],1)
=> 1 = 0 + 1
000001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 2 = 1 + 1
000010 => [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)
=> 2 = 1 + 1
000101 => [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)
=> 2 = 1 + 1
000110 => [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)
=> 3 = 2 + 1
001001 => [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)
=> 3 = 2 + 1
001010 => [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)
=> 2 = 1 + 1
001101 => [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)
=> 3 = 2 + 1
001110 => [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)
=> 3 = 2 + 1
010001 => [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)
=> 3 = 2 + 1
Description
The detour number of a graph.
This is the number of vertices in a longest induced path in a graph.
Note that [1] defines the detour number as the number of edges in a longest induced path, which is unsuitable for the empty graph.
Matching statistic: St000264
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 33% ●values known / values provided: 52%●distinct values known / distinct values provided: 33%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 33% ●values known / values provided: 52%●distinct values known / distinct values provided: 33%
Values
0 => [1] => [1] => ([],1)
=> ? = 0 + 1
1 => [1] => [1] => ([],1)
=> ? = 0 + 1
00 => [2] => [1] => ([],1)
=> ? = 0 + 1
01 => [1,1] => [2] => ([],2)
=> ? = 1 + 1
10 => [1,1] => [2] => ([],2)
=> ? = 1 + 1
11 => [2] => [1] => ([],1)
=> ? = 0 + 1
000 => [3] => [1] => ([],1)
=> ? = 0 + 1
001 => [2,1] => [1,1] => ([(0,1)],2)
=> ? = 1 + 1
010 => [1,1,1] => [3] => ([],3)
=> ? = 1 + 1
101 => [1,1,1] => [3] => ([],3)
=> ? = 1 + 1
110 => [2,1] => [1,1] => ([(0,1)],2)
=> ? = 1 + 1
111 => [3] => [1] => ([],1)
=> ? = 0 + 1
0000 => [4] => [1] => ([],1)
=> ? = 0 + 1
0001 => [3,1] => [1,1] => ([(0,1)],2)
=> ? = 1 + 1
0010 => [2,1,1] => [1,2] => ([(1,2)],3)
=> ? = 1 + 1
0101 => [1,1,1,1] => [4] => ([],4)
=> ? = 1 + 1
0110 => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
1001 => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
1010 => [1,1,1,1] => [4] => ([],4)
=> ? = 1 + 1
1101 => [2,1,1] => [1,2] => ([(1,2)],3)
=> ? = 1 + 1
1110 => [3,1] => [1,1] => ([(0,1)],2)
=> ? = 1 + 1
1111 => [4] => [1] => ([],1)
=> ? = 0 + 1
00000 => [5] => [1] => ([],1)
=> ? = 0 + 1
00001 => [4,1] => [1,1] => ([(0,1)],2)
=> ? = 1 + 1
00010 => [3,1,1] => [1,2] => ([(1,2)],3)
=> ? = 1 + 1
00101 => [2,1,1,1] => [1,3] => ([(2,3)],4)
=> ? = 1 + 1
00110 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> ? = 2 + 1
01001 => [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
01010 => [1,1,1,1,1] => [5] => ([],5)
=> ? = 1 + 1
01101 => [1,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
01110 => [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
10001 => [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
10010 => [1,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
10101 => [1,1,1,1,1] => [5] => ([],5)
=> ? = 1 + 1
10110 => [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
11001 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> ? = 2 + 1
11010 => [2,1,1,1] => [1,3] => ([(2,3)],4)
=> ? = 1 + 1
11101 => [3,1,1] => [1,2] => ([(1,2)],3)
=> ? = 1 + 1
11110 => [4,1] => [1,1] => ([(0,1)],2)
=> ? = 1 + 1
11111 => [5] => [1] => ([],1)
=> ? = 0 + 1
000000 => [6] => [1] => ([],1)
=> ? = 0 + 1
000001 => [5,1] => [1,1] => ([(0,1)],2)
=> ? = 1 + 1
000010 => [4,1,1] => [1,2] => ([(1,2)],3)
=> ? = 1 + 1
000101 => [3,1,1,1] => [1,3] => ([(2,3)],4)
=> ? = 1 + 1
000110 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
001001 => [2,1,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
001010 => [2,1,1,1,1] => [1,4] => ([(3,4)],5)
=> ? = 1 + 1
001101 => [2,2,1,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 2 + 1
001110 => [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
010001 => [1,1,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
010010 => [1,1,2,1,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
010101 => [1,1,1,1,1,1] => [6] => ([],6)
=> ? = 1 + 1
010110 => [1,1,1,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
011001 => [1,2,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
011010 => [1,2,1,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
011101 => [1,3,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
011110 => [1,4,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
100001 => [1,4,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
100010 => [1,3,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
100101 => [1,2,1,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
100110 => [1,2,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
101001 => [1,1,1,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
101010 => [1,1,1,1,1,1] => [6] => ([],6)
=> ? = 1 + 1
101101 => [1,1,2,1,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
101110 => [1,1,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
110001 => [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
110010 => [2,2,1,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 2 + 1
110101 => [2,1,1,1,1] => [1,4] => ([(3,4)],5)
=> ? = 1 + 1
110110 => [2,1,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
111001 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
111010 => [3,1,1,1] => [1,3] => ([(2,3)],4)
=> ? = 1 + 1
111101 => [4,1,1] => [1,2] => ([(1,2)],3)
=> ? = 1 + 1
111110 => [5,1] => [1,1] => ([(0,1)],2)
=> ? = 1 + 1
111111 => [6] => [1] => ([],1)
=> ? = 0 + 1
0000000 => [7] => [1] => ([],1)
=> ? = 0 + 1
0000001 => [6,1] => [1,1] => ([(0,1)],2)
=> ? = 1 + 1
0000010 => [5,1,1] => [1,2] => ([(1,2)],3)
=> ? = 1 + 1
0000101 => [4,1,1,1] => [1,3] => ([(2,3)],4)
=> ? = 1 + 1
0000110 => [4,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
0001001 => [3,1,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
0001101 => [3,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
0010001 => [2,1,3,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
0010010 => [2,1,2,1,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
0010110 => [2,1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
0011101 => [2,3,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
0011110 => [2,4,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
0100001 => [1,1,4,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
0100010 => [1,1,3,1,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
0100101 => [1,1,2,1,1,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
0100110 => [1,1,2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
0101001 => [1,1,1,1,2,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
0101101 => [1,1,1,2,1,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
0101110 => [1,1,1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
0110001 => [1,2,3,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
0110010 => [1,2,2,1,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
0110101 => [1,2,1,1,1,1] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
0110110 => [1,2,1,2,1] => [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)
=> 3 = 2 + 1
0111001 => [1,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
0111010 => [1,3,1,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
0111101 => [1,4,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
Matching statistic: St000897
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000897: Integer partitions ⟶ ℤResult quality: 46% ●values known / values provided: 46%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000897: Integer partitions ⟶ ℤResult quality: 46% ●values known / values provided: 46%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [1,0]
=> []
=> 0
1 => [1] => [1,0]
=> []
=> 0
00 => [2] => [1,1,0,0]
=> []
=> 0
01 => [1,1] => [1,0,1,0]
=> [1]
=> 1
10 => [1,1] => [1,0,1,0]
=> [1]
=> 1
11 => [2] => [1,1,0,0]
=> []
=> 0
000 => [3] => [1,1,1,0,0,0]
=> []
=> 0
001 => [2,1] => [1,1,0,0,1,0]
=> [2]
=> 1
010 => [1,1,1] => [1,0,1,0,1,0]
=> [2,1]
=> 1
101 => [1,1,1] => [1,0,1,0,1,0]
=> [2,1]
=> 1
110 => [2,1] => [1,1,0,0,1,0]
=> [2]
=> 1
111 => [3] => [1,1,1,0,0,0]
=> []
=> 0
0000 => [4] => [1,1,1,1,0,0,0,0]
=> []
=> 0
0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [3]
=> 1
0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 1
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 1
0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 2
1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 2
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 1
1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 1
1110 => [3,1] => [1,1,1,0,0,0,1,0]
=> [3]
=> 1
1111 => [4] => [1,1,1,1,0,0,0,0]
=> []
=> 0
00000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
00001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1
00010 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 1
00101 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 1
00110 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 2
01001 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 2
01010 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 1
01101 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 2
01110 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 2
10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 2
10010 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 2
10101 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 1
10110 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 2
11001 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 2
11010 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 1
11101 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 1
11110 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1
11111 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
000000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> 0
000001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> 1
000010 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [5,4]
=> 1
000101 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> 1
000110 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> 2
001001 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> ? = 2
001010 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> ? = 1
001101 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> ? = 2
001110 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> 2
010001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> 2
010010 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> ? = 2
010101 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> ? = 1
010110 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> ? = 2
011001 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> ? = 2
011010 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> ? = 2
011101 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> 2
011110 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> 2
100001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> 2
100101 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> ? = 2
100110 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> ? = 2
101001 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> ? = 2
101010 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> ? = 1
101101 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> ? = 2
110010 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> ? = 2
110101 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> ? = 1
110110 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> ? = 2
0000101 => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [6,5,4]
=> ? = 1
0000110 => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [6,4,4]
=> ? = 2
0001001 => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3]
=> ? = 2
0001010 => [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3]
=> ? = 1
0001101 => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3]
=> ? = 2
0001110 => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3]
=> ? = 2
0010001 => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3,2]
=> ? = 2
0010010 => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2]
=> ? = 2
0010101 => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2]
=> ? = 1
0010110 => [2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,2]
=> ? = 2
0011001 => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2]
=> ? = 2
0011010 => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2]
=> ? = 2
0011101 => [2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,2,2]
=> ? = 2
0011110 => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,2,2,2]
=> ? = 2
0100001 => [1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,2,2,2,1]
=> ? = 2
0100010 => [1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,2,2,1]
=> ? = 2
0100101 => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2,1]
=> ? = 2
0100110 => [1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2,1]
=> ? = 2
0101001 => [1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,2,1]
=> ? = 2
0101010 => [1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1]
=> ? = 1
0101101 => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2,1]
=> ? = 2
0101110 => [1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3,2,1]
=> ? = 2
0110001 => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3,1,1]
=> ? = 2
0110010 => [1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,1,1]
=> ? = 2
0110101 => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,1]
=> ? = 2
0110110 => [1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,1,1]
=> ? = 2
0111001 => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [6,4,4,1,1,1]
=> ? = 2
0111010 => [1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1,1,1]
=> ? = 2
0111101 => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [6,5,1,1,1,1]
=> ? = 2
1000010 => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [6,5,1,1,1,1]
=> ? = 2
1000101 => [1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1,1,1]
=> ? = 2
1000110 => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [6,4,4,1,1,1]
=> ? = 2
1001001 => [1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,1,1]
=> ? = 2
1001010 => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,1]
=> ? = 2
Description
The number of different multiplicities of parts of an integer partition.
Matching statistic: St000455
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 33%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 33%
Values
0 => [1] => ([],1)
=> ([],1)
=> ? = 0 - 2
1 => [1] => ([],1)
=> ([],1)
=> ? = 0 - 2
00 => [2] => ([],2)
=> ([],1)
=> ? = 0 - 2
01 => [1,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> -1 = 1 - 2
10 => [1,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> -1 = 1 - 2
11 => [2] => ([],2)
=> ([],1)
=> ? = 0 - 2
000 => [3] => ([],3)
=> ([],1)
=> ? = 0 - 2
001 => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> -1 = 1 - 2
010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
110 => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> -1 = 1 - 2
111 => [3] => ([],3)
=> ([],1)
=> ? = 0 - 2
0000 => [4] => ([],4)
=> ([],1)
=> ? = 0 - 2
0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> -1 = 1 - 2
0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
0101 => [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)
=> -1 = 1 - 2
0110 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 2
1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 2
1010 => [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)
=> -1 = 1 - 2
1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
1110 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> -1 = 1 - 2
1111 => [4] => ([],4)
=> ([],1)
=> ? = 0 - 2
00000 => [5] => ([],5)
=> ([],1)
=> ? = 0 - 2
00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> -1 = 1 - 2
00010 => [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)
=> -1 = 1 - 2
00101 => [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)
=> -1 = 1 - 2
00110 => [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)
=> ? = 2 - 2
01001 => [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)
=> ? = 2 - 2
01010 => [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)
=> -1 = 1 - 2
01101 => [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)
=> ? = 2 - 2
01110 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 2
10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 2
10010 => [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)
=> ? = 2 - 2
10101 => [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)
=> -1 = 1 - 2
10110 => [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)
=> ? = 2 - 2
11001 => [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)
=> ? = 2 - 2
11010 => [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)
=> -1 = 1 - 2
11101 => [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)
=> -1 = 1 - 2
11110 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> -1 = 1 - 2
11111 => [5] => ([],5)
=> ([],1)
=> ? = 0 - 2
000000 => [6] => ([],6)
=> ([],1)
=> ? = 0 - 2
000001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> -1 = 1 - 2
000010 => [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)
=> -1 = 1 - 2
000101 => [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)
=> -1 = 1 - 2
000110 => [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)
=> ? = 2 - 2
001001 => [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)
=> ? = 2 - 2
001010 => [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)
=> -1 = 1 - 2
001101 => [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)
=> ? = 2 - 2
001110 => [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)
=> ? = 2 - 2
010001 => [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)
=> ? = 2 - 2
010010 => [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)
=> ? = 2 - 2
010101 => [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)
=> -1 = 1 - 2
010110 => [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)
=> ? = 2 - 2
011001 => [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)
=> ? = 2 - 2
011010 => [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)
=> ? = 2 - 2
011101 => [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)
=> ? = 2 - 2
011110 => [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)
=> ? = 2 - 2
100001 => [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)
=> ? = 2 - 2
100010 => [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)
=> ? = 2 - 2
100101 => [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)
=> ? = 2 - 2
100110 => [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)
=> ? = 2 - 2
101001 => [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)
=> ? = 2 - 2
101010 => [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)
=> -1 = 1 - 2
101101 => [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)
=> ? = 2 - 2
101110 => [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)
=> ? = 2 - 2
110001 => [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)
=> ? = 2 - 2
110010 => [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)
=> ? = 2 - 2
110101 => [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)
=> -1 = 1 - 2
110110 => [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)
=> ? = 2 - 2
111001 => [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)
=> ? = 2 - 2
111010 => [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)
=> -1 = 1 - 2
111101 => [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)
=> -1 = 1 - 2
111110 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> -1 = 1 - 2
111111 => [6] => ([],6)
=> ([],1)
=> ? = 0 - 2
0000000 => [7] => ([],7)
=> ([],1)
=> ? = 0 - 2
0000001 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1)],2)
=> -1 = 1 - 2
0000010 => [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)
=> -1 = 1 - 2
0000101 => [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)
=> -1 = 1 - 2
0000110 => [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)
=> ? = 2 - 2
0001001 => [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)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
0001010 => [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)
=> -1 = 1 - 2
0001101 => [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)
=> ? = 2 - 2
0001110 => [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)
=> ? = 2 - 2
0010001 => [2,1,3,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
0010101 => [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)
=> -1 = 1 - 2
0101010 => [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)
=> -1 = 1 - 2
1010101 => [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)
=> -1 = 1 - 2
1101010 => [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)
=> -1 = 1 - 2
1110101 => [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)
=> -1 = 1 - 2
1111010 => [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)
=> -1 = 1 - 2
1111101 => [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)
=> -1 = 1 - 2
1111110 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1)],2)
=> -1 = 1 - 2
00001010 => [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)
=> -1 = 1 - 2
11110101 => [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)
=> -1 = 1 - 2
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Matching statistic: St000527
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00136: Binary words —rotate back-to-front⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000527: Posets ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 100%
Mp00262: Binary words —poset of factors⟶ Posets
St000527: Posets ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 100%
Values
0 => 0 => ([(0,1)],2)
=> 1 = 0 + 1
1 => 1 => ([(0,1)],2)
=> 1 = 0 + 1
00 => 00 => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
01 => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
10 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
11 => 11 => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
000 => 000 => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
001 => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
010 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
101 => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
110 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
111 => 111 => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
0000 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
0001 => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 2 = 1 + 1
0010 => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 2 = 1 + 1
0101 => 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)
=> ? = 1 + 1
0110 => 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)
=> 3 = 2 + 1
1001 => 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)
=> 3 = 2 + 1
1010 => 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)
=> ? = 1 + 1
1101 => 1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 2 = 1 + 1
1110 => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 2 = 1 + 1
1111 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
00000 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
00001 => 10000 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> 2 = 1 + 1
00010 => 00001 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> 2 = 1 + 1
00101 => 10010 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ? = 1 + 1
00110 => 00011 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> 3 = 2 + 1
01001 => 10100 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ? = 2 + 1
01010 => 00101 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ? = 1 + 1
01101 => 10110 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ? = 2 + 1
01110 => 00111 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> 3 = 2 + 1
10001 => 11000 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> 3 = 2 + 1
10010 => 01001 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ? = 2 + 1
10101 => 11010 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ? = 1 + 1
10110 => 01011 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ? = 2 + 1
11001 => 11100 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> 3 = 2 + 1
11010 => 01101 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ? = 1 + 1
11101 => 11110 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> 2 = 1 + 1
11110 => 01111 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> 2 = 1 + 1
11111 => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
000000 => 000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
000001 => 100000 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> 2 = 1 + 1
000010 => 000001 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> 2 = 1 + 1
000101 => 100010 => ([(0,3),(0,4),(1,2),(1,11),(1,15),(2,7),(2,12),(3,13),(3,14),(4,1),(4,13),(4,14),(6,9),(7,10),(8,6),(9,5),(10,5),(11,7),(12,9),(12,10),(13,8),(13,15),(14,8),(14,11),(15,6),(15,12)],16)
=> ? = 1 + 1
000110 => 000011 => ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> 3 = 2 + 1
001001 => 100100 => ([(0,2),(0,3),(1,11),(1,12),(2,13),(2,14),(3,1),(3,13),(3,14),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,9),(11,6),(11,9),(12,5),(12,6),(13,10),(13,11),(14,10),(14,12)],15)
=> ? = 2 + 1
001010 => 000101 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> ? = 1 + 1
001101 => 100110 => ([(0,3),(0,4),(1,11),(1,16),(2,10),(2,15),(3,2),(3,13),(3,14),(4,1),(4,13),(4,14),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,8),(12,9),(13,15),(13,16),(14,10),(14,11),(15,6),(15,12),(16,7),(16,12)],17)
=> ? = 2 + 1
001110 => 000111 => ([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
=> ? = 2 + 1
010001 => 101000 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> ? = 2 + 1
010010 => 001001 => ([(0,2),(0,3),(1,11),(1,12),(2,13),(2,14),(3,1),(3,13),(3,14),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,9),(11,6),(11,9),(12,5),(12,6),(13,10),(13,11),(14,10),(14,12)],15)
=> ? = 2 + 1
010101 => 101010 => ([(0,1),(0,2),(1,10),(1,11),(2,10),(2,11),(4,3),(5,3),(6,8),(6,9),(7,8),(7,9),(8,4),(8,5),(9,4),(9,5),(10,6),(10,7),(11,6),(11,7)],12)
=> ? = 1 + 1
010110 => 001011 => ([(0,3),(0,4),(1,11),(2,10),(3,2),(3,15),(3,16),(4,1),(4,15),(4,16),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,6),(12,14),(13,7),(13,14),(14,8),(14,9),(15,12),(15,13),(16,10),(16,11),(16,12),(16,13)],17)
=> ? = 2 + 1
011001 => 101100 => ([(0,3),(0,4),(1,11),(2,12),(2,13),(3,2),(3,15),(3,16),(4,1),(4,15),(4,16),(6,7),(7,9),(8,10),(9,5),(10,5),(11,8),(12,7),(12,14),(13,8),(13,14),(14,9),(14,10),(15,6),(15,12),(16,6),(16,11),(16,13)],17)
=> ? = 2 + 1
011010 => 001101 => ([(0,3),(0,4),(1,11),(2,12),(2,13),(3,2),(3,15),(3,16),(4,1),(4,15),(4,16),(6,7),(7,9),(8,10),(9,5),(10,5),(11,8),(12,7),(12,14),(13,8),(13,14),(14,9),(14,10),(15,6),(15,12),(16,6),(16,11),(16,13)],17)
=> ? = 2 + 1
011101 => 101110 => ([(0,3),(0,4),(1,2),(1,11),(1,15),(2,7),(2,12),(3,13),(3,14),(4,1),(4,13),(4,14),(6,9),(7,10),(8,6),(9,5),(10,5),(11,7),(12,9),(12,10),(13,8),(13,15),(14,8),(14,11),(15,6),(15,12)],16)
=> ? = 2 + 1
011110 => 001111 => ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> 3 = 2 + 1
100001 => 110000 => ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> 3 = 2 + 1
100010 => 010001 => ([(0,3),(0,4),(1,2),(1,11),(1,15),(2,7),(2,12),(3,13),(3,14),(4,1),(4,13),(4,14),(6,9),(7,10),(8,6),(9,5),(10,5),(11,7),(12,9),(12,10),(13,8),(13,15),(14,8),(14,11),(15,6),(15,12)],16)
=> ? = 2 + 1
100101 => 110010 => ([(0,3),(0,4),(1,11),(2,12),(2,13),(3,2),(3,15),(3,16),(4,1),(4,15),(4,16),(6,7),(7,9),(8,10),(9,5),(10,5),(11,8),(12,7),(12,14),(13,8),(13,14),(14,9),(14,10),(15,6),(15,12),(16,6),(16,11),(16,13)],17)
=> ? = 2 + 1
100110 => 010011 => ([(0,3),(0,4),(1,11),(2,12),(2,13),(3,2),(3,15),(3,16),(4,1),(4,15),(4,16),(6,7),(7,9),(8,10),(9,5),(10,5),(11,8),(12,7),(12,14),(13,8),(13,14),(14,9),(14,10),(15,6),(15,12),(16,6),(16,11),(16,13)],17)
=> ? = 2 + 1
101001 => 110100 => ([(0,3),(0,4),(1,11),(2,10),(3,2),(3,15),(3,16),(4,1),(4,15),(4,16),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,6),(12,14),(13,7),(13,14),(14,8),(14,9),(15,12),(15,13),(16,10),(16,11),(16,12),(16,13)],17)
=> ? = 2 + 1
101010 => 010101 => ([(0,1),(0,2),(1,10),(1,11),(2,10),(2,11),(4,3),(5,3),(6,8),(6,9),(7,8),(7,9),(8,4),(8,5),(9,4),(9,5),(10,6),(10,7),(11,6),(11,7)],12)
=> ? = 1 + 1
101101 => 110110 => ([(0,2),(0,3),(1,11),(1,12),(2,13),(2,14),(3,1),(3,13),(3,14),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,9),(11,6),(11,9),(12,5),(12,6),(13,10),(13,11),(14,10),(14,12)],15)
=> ? = 2 + 1
101110 => 010111 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> ? = 2 + 1
110001 => 111000 => ([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
=> ? = 2 + 1
110010 => 011001 => ([(0,3),(0,4),(1,11),(1,16),(2,10),(2,15),(3,2),(3,13),(3,14),(4,1),(4,13),(4,14),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,8),(12,9),(13,15),(13,16),(14,10),(14,11),(15,6),(15,12),(16,7),(16,12)],17)
=> ? = 2 + 1
110101 => 111010 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> ? = 1 + 1
110110 => 011011 => ([(0,2),(0,3),(1,11),(1,12),(2,13),(2,14),(3,1),(3,13),(3,14),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,9),(11,6),(11,9),(12,5),(12,6),(13,10),(13,11),(14,10),(14,12)],15)
=> ? = 2 + 1
111001 => 111100 => ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> 3 = 2 + 1
111010 => 011101 => ([(0,3),(0,4),(1,2),(1,11),(1,15),(2,7),(2,12),(3,13),(3,14),(4,1),(4,13),(4,14),(6,9),(7,10),(8,6),(9,5),(10,5),(11,7),(12,9),(12,10),(13,8),(13,15),(14,8),(14,11),(15,6),(15,12)],16)
=> ? = 1 + 1
111101 => 111110 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> 2 = 1 + 1
111110 => 011111 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> 2 = 1 + 1
111111 => 111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
0000000 => 0000000 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> 1 = 0 + 1
0000001 => 1000000 => ([(0,2),(0,7),(1,9),(2,8),(3,4),(3,11),(4,6),(4,10),(5,3),(5,13),(6,1),(6,12),(7,5),(7,8),(8,13),(10,12),(11,10),(12,9),(13,11)],14)
=> ? = 1 + 1
0000010 => 0000001 => ([(0,2),(0,7),(1,9),(2,8),(3,4),(3,11),(4,6),(4,10),(5,3),(5,13),(6,1),(6,12),(7,5),(7,8),(8,13),(10,12),(11,10),(12,9),(13,11)],14)
=> ? = 1 + 1
0000101 => 1000010 => ([(0,4),(0,5),(1,3),(1,9),(1,17),(2,14),(2,19),(3,2),(3,13),(3,18),(4,15),(4,16),(5,1),(5,15),(5,16),(7,11),(8,7),(9,13),(10,8),(11,6),(12,6),(13,14),(14,12),(15,9),(15,10),(16,10),(16,17),(17,8),(17,18),(18,7),(18,19),(19,11),(19,12)],20)
=> ? = 1 + 1
0000110 => 0000011 => ([(0,6),(0,7),(1,11),(2,5),(2,15),(3,13),(4,3),(4,17),(5,4),(5,16),(6,2),(6,14),(7,1),(7,14),(9,12),(10,9),(11,10),(12,8),(13,8),(14,11),(14,15),(15,10),(15,16),(16,9),(16,17),(17,12),(17,13)],18)
=> ? = 2 + 1
0001001 => 1000100 => ([(0,3),(0,4),(1,2),(1,18),(1,19),(2,7),(2,14),(3,15),(3,16),(4,1),(4,15),(4,16),(6,8),(7,9),(8,10),(9,11),(10,5),(11,5),(12,10),(12,11),(13,8),(13,12),(14,9),(14,12),(15,17),(15,19),(16,17),(16,18),(17,6),(17,13),(18,13),(18,14),(19,6),(19,7)],20)
=> ? = 2 + 1
0001010 => 0000101 => ([(0,4),(0,5),(1,3),(1,14),(2,12),(3,2),(3,18),(4,17),(4,19),(5,1),(5,17),(5,19),(7,8),(8,9),(9,10),(10,6),(11,6),(12,11),(13,8),(13,16),(14,16),(14,18),(15,10),(15,11),(16,9),(16,15),(17,7),(17,13),(18,12),(18,15),(19,7),(19,13),(19,14)],20)
=> ? = 1 + 1
0001101 => 1000110 => ([(0,4),(0,5),(1,13),(1,20),(2,3),(2,14),(2,21),(3,8),(3,16),(4,1),(4,17),(4,18),(5,2),(5,17),(5,18),(7,9),(8,10),(9,11),(10,12),(11,6),(12,6),(13,7),(14,8),(15,9),(15,19),(16,10),(16,19),(17,20),(17,21),(18,13),(18,14),(19,11),(19,12),(20,7),(20,15),(21,15),(21,16)],22)
=> ? = 2 + 1
0001110 => 0000111 => ([(0,6),(0,7),(1,4),(1,16),(2,5),(2,15),(3,13),(4,12),(5,3),(5,19),(6,1),(6,17),(7,2),(7,17),(9,11),(10,8),(11,8),(12,9),(13,10),(14,9),(14,18),(15,14),(15,19),(16,12),(16,14),(17,15),(17,16),(18,10),(18,11),(19,13),(19,18)],20)
=> ? = 2 + 1
0010001 => 1001000 => ([(0,3),(0,4),(1,11),(2,1),(2,15),(2,19),(3,17),(3,18),(4,2),(4,17),(4,18),(6,10),(7,8),(8,9),(9,5),(10,5),(11,6),(12,8),(12,13),(13,9),(13,10),(14,12),(14,16),(15,7),(15,12),(16,6),(16,13),(17,14),(17,15),(18,14),(18,19),(19,7),(19,11),(19,16)],20)
=> ? = 2 + 1
0010010 => 0001001 => ([(0,3),(0,4),(1,11),(2,1),(2,15),(2,19),(3,17),(3,18),(4,2),(4,17),(4,18),(6,10),(7,8),(8,9),(9,5),(10,5),(11,6),(12,8),(12,13),(13,9),(13,10),(14,12),(14,16),(15,7),(15,12),(16,6),(16,13),(17,14),(17,15),(18,14),(18,19),(19,7),(19,11),(19,16)],20)
=> ? = 2 + 1
0010101 => 1001010 => ([(0,2),(0,3),(1,5),(1,12),(2,18),(2,19),(3,1),(3,18),(3,19),(5,6),(6,7),(7,10),(8,11),(9,8),(10,4),(11,4),(12,6),(12,14),(13,9),(13,15),(14,7),(14,16),(15,8),(15,16),(16,10),(16,11),(17,9),(17,14),(17,15),(18,5),(18,13),(18,17),(19,12),(19,13),(19,17)],20)
=> ? = 1 + 1
0010110 => 0001011 => ([(0,4),(0,5),(1,13),(2,3),(2,20),(3,7),(4,1),(4,19),(4,21),(5,2),(5,19),(5,21),(7,9),(8,11),(9,12),(10,8),(11,6),(12,6),(13,10),(14,10),(14,17),(15,16),(15,17),(16,9),(16,18),(17,8),(17,18),(18,11),(18,12),(19,14),(19,15),(20,7),(20,16),(21,13),(21,14),(21,15),(21,20)],22)
=> ? = 2 + 1
0011001 => 1001100 => ([(0,3),(0,4),(1,18),(1,20),(2,17),(2,19),(3,1),(3,15),(3,16),(4,2),(4,15),(4,16),(6,8),(7,9),(8,10),(9,11),(10,5),(11,5),(12,10),(12,11),(13,8),(13,12),(14,9),(14,12),(15,19),(15,20),(16,17),(16,18),(17,13),(17,14),(18,6),(18,13),(19,7),(19,14),(20,6),(20,7)],21)
=> ? = 2 + 1
0011010 => 0001101 => ([(0,4),(0,5),(1,13),(2,1),(2,15),(3,14),(3,16),(4,2),(4,20),(4,21),(5,3),(5,20),(5,21),(7,8),(8,9),(9,11),(10,12),(11,6),(12,6),(13,10),(14,8),(14,18),(15,13),(15,17),(16,17),(16,18),(17,10),(17,19),(18,9),(18,19),(19,11),(19,12),(20,7),(20,14),(21,7),(21,15),(21,16)],22)
=> ? = 2 + 1
0011101 => 1001110 => ([(0,4),(0,5),(1,13),(1,20),(2,3),(2,14),(2,21),(3,8),(3,16),(4,1),(4,17),(4,18),(5,2),(5,17),(5,18),(7,9),(8,10),(9,11),(10,12),(11,6),(12,6),(13,7),(14,8),(15,9),(15,19),(16,10),(16,19),(17,20),(17,21),(18,13),(18,14),(19,11),(19,12),(20,7),(20,15),(21,15),(21,16)],22)
=> ? = 2 + 1
0011110 => 0001111 => ([(0,6),(0,7),(1,4),(1,16),(2,5),(2,15),(3,13),(4,12),(5,3),(5,19),(6,1),(6,17),(7,2),(7,17),(9,11),(10,8),(11,8),(12,9),(13,10),(14,9),(14,18),(15,14),(15,19),(16,12),(16,14),(17,15),(17,16),(18,10),(18,11),(19,13),(19,18)],20)
=> ? = 2 + 1
1111111 => 1111111 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> 1 = 0 + 1
Description
The width of the poset.
This is the size of the poset's longest antichain, also called Dilworth number.
Matching statistic: St000353
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000353: Permutations ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000353: Permutations ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [1,0]
=> [2,1] => 0
1 => [1] => [1,0]
=> [2,1] => 0
00 => [2] => [1,1,0,0]
=> [2,3,1] => 0
01 => [1,1] => [1,0,1,0]
=> [3,1,2] => 1
10 => [1,1] => [1,0,1,0]
=> [3,1,2] => 1
11 => [2] => [1,1,0,0]
=> [2,3,1] => 0
000 => [3] => [1,1,1,0,0,0]
=> [2,3,4,1] => 0
001 => [2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => 1
010 => [1,1,1] => [1,0,1,0,1,0]
=> [4,1,2,3] => 1
101 => [1,1,1] => [1,0,1,0,1,0]
=> [4,1,2,3] => 1
110 => [2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => 1
111 => [3] => [1,1,1,0,0,0]
=> [2,3,4,1] => 0
0000 => [4] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 1
0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 2
1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 2
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1
1110 => [3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 1
1111 => [4] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
00000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 0
00001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 1
00010 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 1
00101 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => 1
00110 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 2
01001 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => 2
01010 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => 1
01101 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => 2
01110 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 2
10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 2
10010 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => 2
10101 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => 1
10110 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => 2
11001 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 2
11010 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => 1
11101 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 1
11110 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 1
11111 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 0
000000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 0
000001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? = 1
000010 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => ? = 1
000101 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => ? = 1
000110 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => ? = 2
001001 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => ? = 2
001010 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,7,1,3,4,5,6] => ? = 1
001101 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => ? = 2
001110 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => ? = 2
010001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => ? = 2
010010 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => ? = 2
010101 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 1
010110 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => ? = 2
011001 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => ? = 2
011010 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => ? = 2
011101 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ? = 2
011110 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 2
100001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 2
100010 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ? = 2
100101 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => ? = 2
100110 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => ? = 2
101001 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => ? = 2
101010 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 1
101101 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => ? = 2
101110 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => ? = 2
110001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => ? = 2
110010 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => ? = 2
110101 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,7,1,3,4,5,6] => ? = 1
110110 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => ? = 2
111001 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => ? = 2
111010 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => ? = 1
111101 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => ? = 1
111110 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? = 1
111111 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 0
0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => ? = 0
0000001 => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? = 1
0000010 => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [2,3,4,5,8,1,6,7] => ? = 1
0000101 => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [2,3,4,8,1,5,6,7] => ? = 1
0000110 => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [2,3,4,6,1,8,5,7] => ? = 2
0001001 => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [2,3,6,1,4,8,5,7] => ? = 2
0001010 => [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [2,3,8,1,4,5,6,7] => ? = 1
0001101 => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [2,3,5,1,8,4,6,7] => ? = 2
0001110 => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [2,3,5,1,6,8,4,7] => ? = 2
0010001 => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [2,5,1,3,6,8,4,7] => ? = 2
0010010 => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,5,1,3,8,4,6,7] => ? = 2
0010101 => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,8,1,3,4,5,6,7] => ? = 1
0010110 => [2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [2,6,1,3,4,8,5,7] => ? = 2
0011001 => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,8,5,7] => ? = 2
0011010 => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,4,1,8,3,5,6,7] => ? = 2
0011101 => [2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [2,4,1,5,8,3,6,7] => ? = 2
Description
The number of inner valleys of a permutation.
The number of valleys including the boundary is [[St000099]].
Matching statistic: St000711
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000711: Permutations ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000711: Permutations ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [1,0]
=> [2,1] => 0
1 => [1] => [1,0]
=> [2,1] => 0
00 => [2] => [1,1,0,0]
=> [2,3,1] => 0
01 => [1,1] => [1,0,1,0]
=> [3,1,2] => 1
10 => [1,1] => [1,0,1,0]
=> [3,1,2] => 1
11 => [2] => [1,1,0,0]
=> [2,3,1] => 0
000 => [3] => [1,1,1,0,0,0]
=> [2,3,4,1] => 0
001 => [2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => 1
010 => [1,1,1] => [1,0,1,0,1,0]
=> [4,1,2,3] => 1
101 => [1,1,1] => [1,0,1,0,1,0]
=> [4,1,2,3] => 1
110 => [2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => 1
111 => [3] => [1,1,1,0,0,0]
=> [2,3,4,1] => 0
0000 => [4] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 1
0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 2
1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 2
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1
1110 => [3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 1
1111 => [4] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
00000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 0
00001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 1
00010 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 1
00101 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => 1
00110 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 2
01001 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => 2
01010 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => 1
01101 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => 2
01110 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 2
10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 2
10010 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => 2
10101 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => 1
10110 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => 2
11001 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 2
11010 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => 1
11101 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 1
11110 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 1
11111 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 0
000000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 0
000001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? = 1
000010 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => ? = 1
000101 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => ? = 1
000110 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => ? = 2
001001 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => ? = 2
001010 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,7,1,3,4,5,6] => ? = 1
001101 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => ? = 2
001110 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => ? = 2
010001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => ? = 2
010010 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => ? = 2
010101 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 1
010110 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => ? = 2
011001 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => ? = 2
011010 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => ? = 2
011101 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ? = 2
011110 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 2
100001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 2
100010 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ? = 2
100101 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => ? = 2
100110 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => ? = 2
101001 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => ? = 2
101010 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 1
101101 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => ? = 2
101110 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => ? = 2
110001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => ? = 2
110010 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => ? = 2
110101 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,7,1,3,4,5,6] => ? = 1
110110 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => ? = 2
111001 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => ? = 2
111010 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => ? = 1
111101 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => ? = 1
111110 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? = 1
111111 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 0
0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => ? = 0
0000001 => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? = 1
0000010 => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [2,3,4,5,8,1,6,7] => ? = 1
0000101 => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [2,3,4,8,1,5,6,7] => ? = 1
0000110 => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [2,3,4,6,1,8,5,7] => ? = 2
0001001 => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [2,3,6,1,4,8,5,7] => ? = 2
0001010 => [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [2,3,8,1,4,5,6,7] => ? = 1
0001101 => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [2,3,5,1,8,4,6,7] => ? = 2
0001110 => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [2,3,5,1,6,8,4,7] => ? = 2
0010001 => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [2,5,1,3,6,8,4,7] => ? = 2
0010010 => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,5,1,3,8,4,6,7] => ? = 2
0010101 => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,8,1,3,4,5,6,7] => ? = 1
0010110 => [2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [2,6,1,3,4,8,5,7] => ? = 2
0011001 => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,8,5,7] => ? = 2
0011010 => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,4,1,8,3,5,6,7] => ? = 2
0011101 => [2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [2,4,1,5,8,3,6,7] => ? = 2
Description
The number of big exceedences of a permutation.
A big exceedence of a permutation $\pi$ is an index $i$ such that $\pi(i) - i > 1$.
This statistic is equidistributed with either of the numbers of big descents, big ascents, and big deficiencies.
Matching statistic: St001212
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001212: Dyck paths ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001212: Dyck paths ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [1,0]
=> [1,1,0,0]
=> 0
1 => [1] => [1,0]
=> [1,1,0,0]
=> 0
00 => [2] => [1,1,0,0]
=> [1,1,1,0,0,0]
=> 0
01 => [1,1] => [1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
10 => [1,1] => [1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
11 => [2] => [1,1,0,0]
=> [1,1,1,0,0,0]
=> 0
000 => [3] => [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
001 => [2,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
010 => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
101 => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
110 => [2,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
111 => [3] => [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
0000 => [4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
1110 => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
1111 => [4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
00000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
00001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1
00010 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 1
00101 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 1
00110 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 2
01001 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 2
01010 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 1
01101 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 2
01110 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 2
10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 2
10010 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 2
10101 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 1
10110 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 2
11001 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 2
11010 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 1
11101 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 1
11110 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1
11111 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
000000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
000001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 1
000010 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 1
000101 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 1
000110 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> ? = 2
001001 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> ? = 2
001010 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
001101 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> ? = 2
001110 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> ? = 2
010001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> ? = 2
010010 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 2
010101 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
010110 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 2
011001 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 2
011010 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 2
011101 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 2
011110 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 2
100001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 2
100010 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 2
100101 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 2
100110 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 2
101001 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 2
101010 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
101101 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 2
101110 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> ? = 2
110001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> ? = 2
110010 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> ? = 2
110101 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
110110 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> ? = 2
111001 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> ? = 2
111010 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 1
111101 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 1
111110 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 1
111111 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
0000001 => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1
0000010 => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ? = 1
0000101 => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 1
0000110 => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,1,0,0]
=> ? = 2
0001001 => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,1,0,0]
=> ? = 2
0001010 => [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
0001101 => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,1,0,0]
=> ? = 2
0001110 => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,1,0,0]
=> ? = 2
0010001 => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,1,0,0]
=> ? = 2
0010010 => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 2
0010101 => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
0010110 => [2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 2
0011001 => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 2
0011010 => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 2
0011101 => [2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 2
Description
The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module.
Matching statistic: St001553
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001553: Dyck paths ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 100%
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001553: Dyck paths ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [1] => [1,0]
=> 0
1 => [1] => [1] => [1,0]
=> 0
00 => [2] => [1,1] => [1,0,1,0]
=> 0
01 => [1,1] => [2] => [1,1,0,0]
=> 1
10 => [1,1] => [2] => [1,1,0,0]
=> 1
11 => [2] => [1,1] => [1,0,1,0]
=> 0
000 => [3] => [1,1,1] => [1,0,1,0,1,0]
=> 0
001 => [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
010 => [1,1,1] => [3] => [1,1,1,0,0,0]
=> 1
101 => [1,1,1] => [3] => [1,1,1,0,0,0]
=> 1
110 => [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
111 => [3] => [1,1,1] => [1,0,1,0,1,0]
=> 0
0000 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
0001 => [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
0010 => [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
0101 => [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 1
0110 => [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
1001 => [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
1010 => [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 1
1101 => [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
1110 => [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
1111 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
00000 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0
00001 => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
00010 => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
00101 => [2,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
00110 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
01001 => [1,1,2,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
01010 => [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
01101 => [1,2,1,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
01110 => [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
10001 => [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
10010 => [1,2,1,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
10101 => [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
10110 => [1,1,2,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
11001 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
11010 => [2,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
11101 => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
11110 => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
11111 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0
000000 => [6] => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
000001 => [5,1] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
000010 => [4,1,1] => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 1
000101 => [3,1,1,1] => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
000110 => [3,2,1] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 2
001001 => [2,1,2,1] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 2
001010 => [2,1,1,1,1] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
001101 => [2,2,1,1] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 2
001110 => [2,3,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 2
010001 => [1,1,3,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 2
010010 => [1,1,2,1,1] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 2
010101 => [1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
010110 => [1,1,1,2,1] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 2
011001 => [1,2,2,1] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 2
011010 => [1,2,1,1,1] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 2
011101 => [1,3,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 2
011110 => [1,4,1] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 2
100001 => [1,4,1] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 2
100010 => [1,3,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 2
100101 => [1,2,1,1,1] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 2
100110 => [1,2,2,1] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 2
101001 => [1,1,1,2,1] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 2
101010 => [1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
101101 => [1,1,2,1,1] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 2
101110 => [1,1,3,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 2
110001 => [2,3,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 2
110010 => [2,2,1,1] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 2
110101 => [2,1,1,1,1] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
110110 => [2,1,2,1] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 2
111001 => [3,2,1] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 2
111010 => [3,1,1,1] => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
111101 => [4,1,1] => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 1
111110 => [5,1] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
111111 => [6] => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
0000000 => [7] => [1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
0000001 => [6,1] => [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
0000010 => [5,1,1] => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 1
0000101 => [4,1,1,1] => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
0000110 => [4,2,1] => [1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 2
0001001 => [3,1,2,1] => [1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 2
0001010 => [3,1,1,1,1] => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
0001101 => [3,2,1,1] => [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 2
0001110 => [3,3,1] => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 2
0010001 => [2,1,3,1] => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 2
0010010 => [2,1,2,1,1] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 2
0010101 => [2,1,1,1,1,1] => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
0010110 => [2,1,1,2,1] => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 2
0011001 => [2,2,2,1] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 2
0011010 => [2,2,1,1,1] => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 2
0011101 => [2,3,1,1] => [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 2
Description
The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path.
The statistic returns zero in case that bimodule is the zero module.
The following 51 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000092The number of outer peaks of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000097The order of the largest clique of the graph. St000093The cardinality of a maximal independent set of vertices of a graph. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St001335The cardinality of a minimal cycle-isolating set of a graph. St000098The chromatic number of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000298The order dimension or Dushnik-Miller dimension of 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. 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. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph 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. St001792The arboricity of a graph. St001826The maximal number of leaves on a vertex of a graph. St001962The proper pathwidth of a graph. St000307The number of rowmotion orbits of a poset. St000544The cop number of a graph. St000785The number of distinct colouring schemes of a graph. St001029The size of the core of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number 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. St001746The coalition number of a graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001057The Grundy value of the game of creating an independent set in a graph. St000258The burning number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St000640The rank of the largest boolean interval in a poset. St000454The largest eigenvalue of a graph if it is integral. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000822The Hadwiger number of the graph. St001330The hat guessing number of a graph. St001642The Prague dimension of a graph. St001323The independence gap of 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.
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