- FindStat

Your data matches 138 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000256

Mapping

Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000256: Integer partitions ⟶ 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)
 => [1,1]
 => 0
01 => [1,1] => ([(0,1)],2)
 => [2]
 => 1
10 => [1,1] => ([(0,1)],2)
 => [2]
 => 1
11 => [2] => ([],2)
 => [1,1]
 => 0
000 => [3] => ([],3)
 => [1,1,1]
 => 0
001 => [2,1] => ([(0,2),(1,2)],3)
 => [3]
 => 1
010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1
101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1
110 => [2,1] => ([(0,2),(1,2)],3)
 => [3]
 => 1
111 => [3] => ([],3)
 => [1,1,1]
 => 0
0000 => [4] => ([],4)
 => [1,1,1,1]
 => 0
0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => [4]
 => 1
0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
0110 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
1010 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
1110 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => [4]
 => 1
1111 => [4] => ([],4)
 => [1,1,1,1]
 => 0
00000 => [5] => ([],5)
 => [1,1,1,1,1]
 => 0
00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => 1
00010 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 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)
 => [5]
 => 1
00110 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
01001 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 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)
 => [5]
 => 1
01101 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
01110 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
10010 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 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)
 => [5]
 => 1
10110 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
11001 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 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)
 => [5]
 => 1
11101 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
11110 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => 1
11111 => [5] => ([],5)
 => [1,1,1,1,1]
 => 0
000000 => [6] => ([],6)
 => [1,1,1,1,1,1]
 => 0
000001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [6]
 => 1
000010 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 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)
 => [6]
 => 1
000110 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 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)
 => [6]
 => 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)
 => [6]
 => 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)
 => [6]
 => 1
001110 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 1
010001 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 1

Description

The number of parts from which one can substract 2 and still get an integer partition.

Matching statistic: St000260
(load all 17 compositions to match this statistic)

Mapping

Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000260: 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)
 => 1
1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 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)
 => 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)
 => 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)
 => 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)
 => 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)
 => 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)
 => 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)
 => 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)
 => 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)
 => 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)
 => 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)
 => 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)
 => 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)
 => 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)
 => 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)
 => 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)
 => 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)
 => 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)
 => 1

Description

The radius of a connected graph. This is the minimum eccentricity of any vertex.

Matching statistic: St000473

Mapping

Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000473: Integer partitions ⟶ 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)
 => [1,1]
 => 0
01 => [1,1] => ([(0,1)],2)
 => [2]
 => 1
10 => [1,1] => ([(0,1)],2)
 => [2]
 => 1
11 => [2] => ([],2)
 => [1,1]
 => 0
000 => [3] => ([],3)
 => [1,1,1]
 => 0
001 => [2,1] => ([(0,2),(1,2)],3)
 => [3]
 => 1
010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1
101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1
110 => [2,1] => ([(0,2),(1,2)],3)
 => [3]
 => 1
111 => [3] => ([],3)
 => [1,1,1]
 => 0
0000 => [4] => ([],4)
 => [1,1,1,1]
 => 0
0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => [4]
 => 1
0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
0110 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
1010 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
1110 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => [4]
 => 1
1111 => [4] => ([],4)
 => [1,1,1,1]
 => 0
00000 => [5] => ([],5)
 => [1,1,1,1,1]
 => 0
00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => 1
00010 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 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)
 => [5]
 => 1
00110 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
01001 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 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)
 => [5]
 => 1
01101 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
01110 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
10010 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 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)
 => [5]
 => 1
10110 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
11001 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 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)
 => [5]
 => 1
11101 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
11110 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => 1
11111 => [5] => ([],5)
 => [1,1,1,1,1]
 => 0
000000 => [6] => ([],6)
 => [1,1,1,1,1,1]
 => 0
000001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [6]
 => 1
000010 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 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)
 => [6]
 => 1
000110 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 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)
 => [6]
 => 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)
 => [6]
 => 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)
 => [6]
 => 1
001110 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 1
010001 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 1

Description

The number of parts of a partition that are strictly bigger than the number of ones. This is part of the definition of Dyson's crank of a partition, see [[St000474]].

Matching statistic: St000480

Mapping

Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000480: Integer partitions ⟶ 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)
 => [1,1]
 => 0
01 => [1,1] => ([(0,1)],2)
 => [2]
 => 1
10 => [1,1] => ([(0,1)],2)
 => [2]
 => 1
11 => [2] => ([],2)
 => [1,1]
 => 0
000 => [3] => ([],3)
 => [1,1,1]
 => 0
001 => [2,1] => ([(0,2),(1,2)],3)
 => [3]
 => 1
010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1
101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1
110 => [2,1] => ([(0,2),(1,2)],3)
 => [3]
 => 1
111 => [3] => ([],3)
 => [1,1,1]
 => 0
0000 => [4] => ([],4)
 => [1,1,1,1]
 => 0
0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => [4]
 => 1
0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
0110 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
1010 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
1110 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => [4]
 => 1
1111 => [4] => ([],4)
 => [1,1,1,1]
 => 0
00000 => [5] => ([],5)
 => [1,1,1,1,1]
 => 0
00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => 1
00010 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 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)
 => [5]
 => 1
00110 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
01001 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 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)
 => [5]
 => 1
01101 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
01110 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
10010 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 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)
 => [5]
 => 1
10110 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
11001 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 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)
 => [5]
 => 1
11101 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
11110 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => 1
11111 => [5] => ([],5)
 => [1,1,1,1,1]
 => 0
000000 => [6] => ([],6)
 => [1,1,1,1,1,1]
 => 0
000001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [6]
 => 1
000010 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 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)
 => [6]
 => 1
000110 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 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)
 => [6]
 => 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)
 => [6]
 => 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)
 => [6]
 => 1
001110 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 1
010001 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 1

Description

The number of lower covers of a partition in dominance order. According to [1], Corollary 2.4, the maximum number of elements one element (apparently for

$n\neq 2$ ) can cover is

$\frac{1}{2}(\sqrt{1+8n}-3)$ and an element which covers this number of elements is given by

$(c+i,c,c-1,\dots,3,2,1)$ , where

$1\leq i\leq c+2$ .

Matching statistic: St000535
(load all 18 compositions to match this statistic)

Mapping

Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000535: 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)
 => 1
1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 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)
 => 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)
 => 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)
 => 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)
 => 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)
 => 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)
 => 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)
 => 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)
 => 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)
 => 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)
 => 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)
 => 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)
 => 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)
 => 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)
 => 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)
 => 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)
 => 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)
 => 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)
 => 1

Description

The rank-width of a graph.

Matching statistic: St000875

Mapping

Mp00224: Binary words —runsort⟶ Binary words
Mp00136: Binary words —rotate back-to-front⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
St000875: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

=> 0 => 0 => 0 => 0
=> 1 => 1 => 1 => 0
=> 00 => 00 => 00 => 0
=> 01 => 10 => 10 => 1
=> 01 => 10 => 10 => 1
=> 11 => 11 => 11 => 0
=> 000 => 000 => 000 => 0
=> 001 => 100 => 010 => 1
=> 001 => 100 => 010 => 1
=> 011 => 101 => 101 => 1
=> 011 => 101 => 101 => 1
=> 111 => 111 => 111 => 0
=> 0000 => 0000 => 0000 => 0
=> 0001 => 1000 => 0010 => 1
=> 0001 => 1000 => 0010 => 1
=> 0101 => 1010 => 0110 => 1
=> 0011 => 1001 => 0101 => 1
=> 0011 => 1001 => 0101 => 1
=> 0011 => 1001 => 0101 => 1
=> 0111 => 1011 => 1011 => 1
=> 0111 => 1011 => 1011 => 1
=> 1111 => 1111 => 1111 => 0
=> 00000 => 00000 => 00000 => 0
=> 00001 => 10000 => 00010 => 1
=> 00001 => 10000 => 00010 => 1
=> 00101 => 10010 => 00110 => 1
=> 00011 => 10001 => 00101 => 1
=> 00101 => 10010 => 00110 => 1
=> 00101 => 10010 => 00110 => 1
=> 01011 => 10101 => 01101 => 1
=> 00111 => 10011 => 01011 => 1
=> 00011 => 10001 => 00101 => 1
=> 00011 => 10001 => 00101 => 1
=> 01011 => 10101 => 01101 => 1
=> 00111 => 10011 => 01011 => 1
=> 00111 => 10011 => 01011 => 1
=> 00111 => 10011 => 01011 => 1
=> 01111 => 10111 => 10111 => 1
=> 01111 => 10111 => 10111 => 1
=> 11111 => 11111 => 11111 => 0
000000 => 000000 => 000000 => 000000 => 0
000001 => 000001 => 100000 => 000010 => 1
000010 => 000001 => 100000 => 000010 => 1
000101 => 000101 => 100010 => 000110 => 1
000110 => 000011 => 100001 => 000101 => 1
001001 => 001001 => 100100 => 100010 => 1
001010 => 000101 => 100010 => 000110 => 1
001101 => 001101 => 100110 => 001110 => 1
001110 => 000111 => 100011 => 001011 => 1
010001 => 000101 => 100010 => 000110 => 1

Description

The semilength of the longest Dyck word in the Catalan factorisation of a binary word. Every binary word can be written in a unique way as

$(\mathcal D 0)^\ell \mathcal D (1 \mathcal D)^m$ , where

$\mathcal D$ is the set of Dyck words. This is the Catalan factorisation, see [1, sec.9.1.2]. This statistic records the semilength of the longest Dyck word in this factorisation.

Matching statistic: St001271
(load all 7 compositions to match this statistic)

Mapping

Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001271: 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)
 => 1
1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 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)
 => 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)
 => 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)
 => 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)
 => 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)
 => 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)
 => 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)
 => 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)
 => 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)
 => 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)
 => 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)
 => 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)
 => 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)
 => 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)
 => 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)
 => 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)
 => 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)
 => 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)
 => 1

Description

The competition number of a graph. The competition graph of a digraph

$D$ is a (simple undirected) graph which has the same vertex set as

$D$ and has an edge between

$x$ and

$y$ if and only if there exists a vertex

$v$ in

$D$ such that

$(x, v)$ and

$(y, v)$ are arcs of

$D$ . For any graph,

$G$ together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number

$k(G)$ is the smallest number of such isolated vertices.

Matching statistic: St001280

Mapping

Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001280: Integer partitions ⟶ 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)
 => [1,1]
 => 0
01 => [1,1] => ([(0,1)],2)
 => [2]
 => 1
10 => [1,1] => ([(0,1)],2)
 => [2]
 => 1
11 => [2] => ([],2)
 => [1,1]
 => 0
000 => [3] => ([],3)
 => [1,1,1]
 => 0
001 => [2,1] => ([(0,2),(1,2)],3)
 => [3]
 => 1
010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1
101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1
110 => [2,1] => ([(0,2),(1,2)],3)
 => [3]
 => 1
111 => [3] => ([],3)
 => [1,1,1]
 => 0
0000 => [4] => ([],4)
 => [1,1,1,1]
 => 0
0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => [4]
 => 1
0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
0110 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
1010 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
1110 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => [4]
 => 1
1111 => [4] => ([],4)
 => [1,1,1,1]
 => 0
00000 => [5] => ([],5)
 => [1,1,1,1,1]
 => 0
00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => 1
00010 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 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)
 => [5]
 => 1
00110 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
01001 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 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)
 => [5]
 => 1
01101 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
01110 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
10010 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 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)
 => [5]
 => 1
10110 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
11001 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 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)
 => [5]
 => 1
11101 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
11110 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => 1
11111 => [5] => ([],5)
 => [1,1,1,1,1]
 => 0
000000 => [6] => ([],6)
 => [1,1,1,1,1,1]
 => 0
000001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [6]
 => 1
000010 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 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)
 => [6]
 => 1
000110 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 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)
 => [6]
 => 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)
 => [6]
 => 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)
 => [6]
 => 1
001110 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 1
010001 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 1

Description

The number of parts of an integer partition that are at least two.

Matching statistic: St000013

Mapping

Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

0 => [1] => [1,0]
 => [1,0]
 => 1 = 0 + 1
1 => [1] => [1,0]
 => [1,0]
 => 1 = 0 + 1
00 => [2] => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 0 + 1
01 => [1,1] => [1,0,1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
10 => [1,1] => [1,0,1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
11 => [2] => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 0 + 1
000 => [3] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 1 = 0 + 1
001 => [2,1] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
010 => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
101 => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
110 => [2,1] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
111 => [3] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 1 = 0 + 1
0000 => [4] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
0001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 2 = 1 + 1
0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 2 = 1 + 1
1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
1110 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
1111 => [4] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
00000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
00001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
00010 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
00101 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2 = 1 + 1
00110 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
01001 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2 = 1 + 1
01010 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 2 = 1 + 1
01101 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2 = 1 + 1
01110 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2 = 1 + 1
10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2 = 1 + 1
10010 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2 = 1 + 1
10101 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 2 = 1 + 1
10110 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2 = 1 + 1
11001 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
11010 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2 = 1 + 1
11101 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
11110 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
11111 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
000000 => [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]
 => 1 = 0 + 1
000001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
000010 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
000101 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 2 = 1 + 1
000110 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
001001 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => 2 = 1 + 1
001010 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 2 = 1 + 1
001101 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => 2 = 1 + 1
001110 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 2 = 1 + 1
010001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => 2 = 1 + 1

Description

The height of a Dyck path. The height of a Dyck path

$D$ of semilength

$n$ is defined as the maximal height of a peak of

$D$ . The height of

$D$ at position

$i$ is the number of up-steps minus the number of down-steps before position

$i$ .

Matching statistic: St000258
(load all 3 compositions to match this statistic)

Mapping

Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000258: 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)
 => 2 = 1 + 1
1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 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)
 => 2 = 1 + 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)
 => 2 = 1 + 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)
 => 2 = 1 + 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)
 => 2 = 1 + 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)
 => 2 = 1 + 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)
 => 2 = 1 + 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)
 => 2 = 1 + 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)
 => 2 = 1 + 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)
 => 2 = 1 + 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)
 => 2 = 1 + 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)
 => 2 = 1 + 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)
 => 2 = 1 + 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)
 => 2 = 1 + 1

Description

The burning number of a graph. This is the minimum number of rounds needed to burn all vertices of a graph. In each round, the neighbours of all burned vertices are burnt. Additionally, an unburned vertex may be chosen to be burned.

The following 128 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000378The diagonal inversion number of an integer partition. St000758The length of the longest staircase fitting into an integer composition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St000918The 2-limited packing number of a graph. St000442The maximal area to the right of an up step of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001335The cardinality of a minimal cycle-isolating set of a graph. St000397The Strahler number of a rooted tree. St000444The length of the maximal rise of a Dyck path. St000919The number of maximal left branches of a binary tree. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001333The cardinality of a minimal edge-isolating set of a graph. St001393The induced matching number of a graph. St001261The Castelnuovo-Mumford regularity of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St000667The greatest common divisor of the parts of the partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St000699The toughness times the least common multiple of 1,. St001198The number of simple modules in the algebra

$eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001206The maximal dimension of an indecomposable projective

$eAe$ -module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ . St000662The staircase size of the code of a permutation. St000451The length of the longest pattern of the form k 1 2. St000264The girth of a graph, which is not a tree. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St000742The number of big ascents of a permutation after prepending zero. St000996The number of exclusive left-to-right maxima of a permutation. St000141The maximum drop size of a permutation. St000007The number of saliances of the permutation. St000352The Elizalde-Pak rank of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000864The number of circled entries of the shifted recording tableau of a permutation. St001737The number of descents of type 2 in a permutation. St000542The number of left-to-right-minima of a permutation. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001192The maximal dimension of

$Ext_A^2(S,A)$ for a simple module

$S$ over the corresponding Nakayama algebra

$A$ . St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St000259The diameter of a connected graph. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000769The major index of a composition regarded as a word. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001194The injective dimension of

$A/AfA$ in the corresponding Nakayama algebra

$A$ when

$Af$ is the minimal faithful projective-injective left

$A$ -module St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001512The minimum rank of a graph. St001673The degree of asymmetry of an integer composition. St000299The number of nonisomorphic vertex-induced subtrees. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000767The number of runs in an integer composition. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000903The number of different parts of an integer composition. St000990The first ascent of a permutation. St001093The detour number of a graph. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001674The number of vertices of the largest induced star graph in the graph. St001486The number of corners of the ribbon associated with an integer composition. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n−1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a special CNakayama algebra. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000862The number of parts of the shifted shape of a permutation. St001741The largest integer such that all patterns of this size are contained in the permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St000993The multiplicity of the largest part of an integer partition. St000929The constant term of the character polynomial of an integer partition. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St001092The number of distinct even parts of a partition. St000298The order dimension or Dushnik-Miller dimension of a poset. St001734The lettericity of a graph. St001174The Gorenstein dimension of the algebra

$A/I$ when

$I$ is the tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000955Number of times one has

$Ext^i(D(A),A)>0$ for

$i>0$ for the corresponding LNakayama algebra. 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. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000098The chromatic number of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. 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. 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. 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.