Your data matches 27 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001232
(load all 2 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [1,0]
 => 0
1 => [1] => [1,0]
 => 0
00 => [2] => [1,1,0,0]
 => 0
01 => [1,1] => [1,0,1,0]
 => 1
10 => [1,1] => [1,0,1,0]
 => 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]
 => 1
011 => [1,2] => [1,0,1,1,0,0]
 => 2
100 => [1,2] => [1,0,1,1,0,0]
 => 2
110 => [2,1] => [1,1,0,0,1,0]
 => 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]
 => 1
0011 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 3
0111 => [1,3] => [1,0,1,1,1,0,0,0]
 => 3
1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => 3
1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 3
1100 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
1110 => [3,1] => [1,1,1,0,0,0,1,0]
 => 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]
 => 1
00011 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
00110 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3
00111 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 3
01100 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 4
01110 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 4
01111 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 4
10000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 4
10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 4
10011 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 4
11000 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 3
11001 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3
11100 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
11110 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 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]
 => 1
000011 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 2
000110 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 3
000111 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 3
001100 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 4
001110 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 4
001111 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 4
011000 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 5
011001 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 5
011100 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 5
011110 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 5
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000171
(load all 2 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000171: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [1] => ([],1)
 => 0
1 => [1] => [1] => ([],1)
 => 0
00 => [2] => [2] => ([],2)
 => 0
01 => [1,1] => [1,1] => ([(0,1)],2)
 => 1
10 => [1,1] => [1,1] => ([(0,1)],2)
 => 1
11 => [2] => [2] => ([],2)
 => 0
000 => [3] => [3] => ([],3)
 => 0
001 => [2,1] => [1,2] => ([(1,2)],3)
 => 1
011 => [1,2] => [2,1] => ([(0,2),(1,2)],3)
 => 2
100 => [1,2] => [2,1] => ([(0,2),(1,2)],3)
 => 2
110 => [2,1] => [1,2] => ([(1,2)],3)
 => 1
111 => [3] => [3] => ([],3)
 => 0
0000 => [4] => [4] => ([],4)
 => 0
0001 => [3,1] => [1,3] => ([(2,3)],4)
 => 1
0011 => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => 2
0110 => [1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
0111 => [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
1000 => [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
1001 => [1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
1100 => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => 2
1110 => [3,1] => [1,3] => ([(2,3)],4)
 => 1
1111 => [4] => [4] => ([],4)
 => 0
00000 => [5] => [5] => ([],5)
 => 0
00001 => [4,1] => [1,4] => ([(3,4)],5)
 => 1
00011 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 2
00110 => [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
00111 => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
01100 => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
01110 => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
01111 => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
10000 => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
10001 => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
10011 => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
11000 => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
11001 => [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
11100 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 2
11110 => [4,1] => [1,4] => ([(3,4)],5)
 => 1
11111 => [5] => [5] => ([],5)
 => 0
000000 => [6] => [6] => ([],6)
 => 0
000001 => [5,1] => [1,5] => ([(4,5)],6)
 => 1
000011 => [4,2] => [2,4] => ([(3,5),(4,5)],6)
 => 2
000110 => [3,2,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
000111 => [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 3
001100 => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
001110 => [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
001111 => [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 4
011000 => [1,2,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
011001 => [1,2,2,1] => [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)
 => 5
011100 => [1,3,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
011110 => [1,4,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
Description
The degree of the graph. This is the maximal vertex degree of a graph.
Matching statistic: St000987
(load all 2 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000987: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [1] => ([],1)
 => 0
1 => [1] => [1] => ([],1)
 => 0
00 => [2] => [2] => ([],2)
 => 0
01 => [1,1] => [1,1] => ([(0,1)],2)
 => 1
10 => [1,1] => [1,1] => ([(0,1)],2)
 => 1
11 => [2] => [2] => ([],2)
 => 0
000 => [3] => [3] => ([],3)
 => 0
001 => [2,1] => [1,2] => ([(1,2)],3)
 => 1
011 => [1,2] => [2,1] => ([(0,2),(1,2)],3)
 => 2
100 => [1,2] => [2,1] => ([(0,2),(1,2)],3)
 => 2
110 => [2,1] => [1,2] => ([(1,2)],3)
 => 1
111 => [3] => [3] => ([],3)
 => 0
0000 => [4] => [4] => ([],4)
 => 0
0001 => [3,1] => [1,3] => ([(2,3)],4)
 => 1
0011 => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => 2
0110 => [1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
0111 => [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
1000 => [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
1001 => [1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
1100 => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => 2
1110 => [3,1] => [1,3] => ([(2,3)],4)
 => 1
1111 => [4] => [4] => ([],4)
 => 0
00000 => [5] => [5] => ([],5)
 => 0
00001 => [4,1] => [1,4] => ([(3,4)],5)
 => 1
00011 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 2
00110 => [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
00111 => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
01100 => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
01110 => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
01111 => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
10000 => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
10001 => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
10011 => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
11000 => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
11001 => [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
11100 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 2
11110 => [4,1] => [1,4] => ([(3,4)],5)
 => 1
11111 => [5] => [5] => ([],5)
 => 0
000000 => [6] => [6] => ([],6)
 => 0
000001 => [5,1] => [1,5] => ([(4,5)],6)
 => 1
000011 => [4,2] => [2,4] => ([(3,5),(4,5)],6)
 => 2
000110 => [3,2,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
000111 => [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 3
001100 => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
001110 => [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
001111 => [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 4
011000 => [1,2,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
011001 => [1,2,2,1] => [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)
 => 5
011100 => [1,3,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
011110 => [1,4,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
Description
The number of positive eigenvalues of the Laplacian matrix of the graph. This is the number of vertices minus the number of connected components of the graph.
Matching statistic: St001721
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St001721: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [1] => 1 => 0
1 => [1] => [1] => 1 => 0
00 => [2] => [1,1] => 11 => 0
01 => [1,1] => [2] => 10 => 1
10 => [1,1] => [2] => 10 => 1
11 => [2] => [1,1] => 11 => 0
000 => [3] => [1,1,1] => 111 => 0
001 => [2,1] => [1,2] => 110 => 1
011 => [1,2] => [2,1] => 101 => 2
100 => [1,2] => [2,1] => 101 => 2
110 => [2,1] => [1,2] => 110 => 1
111 => [3] => [1,1,1] => 111 => 0
0000 => [4] => [1,1,1,1] => 1111 => 0
0001 => [3,1] => [1,1,2] => 1110 => 1
0011 => [2,2] => [1,2,1] => 1101 => 2
0110 => [1,2,1] => [2,2] => 1010 => 3
0111 => [1,3] => [2,1,1] => 1011 => 3
1000 => [1,3] => [2,1,1] => 1011 => 3
1001 => [1,2,1] => [2,2] => 1010 => 3
1100 => [2,2] => [1,2,1] => 1101 => 2
1110 => [3,1] => [1,1,2] => 1110 => 1
1111 => [4] => [1,1,1,1] => 1111 => 0
00000 => [5] => [1,1,1,1,1] => 11111 => 0
00001 => [4,1] => [1,1,1,2] => 11110 => 1
00011 => [3,2] => [1,1,2,1] => 11101 => 2
00110 => [2,2,1] => [1,2,2] => 11010 => 3
00111 => [2,3] => [1,2,1,1] => 11011 => 3
01100 => [1,2,2] => [2,2,1] => 10101 => 4
01110 => [1,3,1] => [2,1,2] => 10110 => 4
01111 => [1,4] => [2,1,1,1] => 10111 => 4
10000 => [1,4] => [2,1,1,1] => 10111 => 4
10001 => [1,3,1] => [2,1,2] => 10110 => 4
10011 => [1,2,2] => [2,2,1] => 10101 => 4
11000 => [2,3] => [1,2,1,1] => 11011 => 3
11001 => [2,2,1] => [1,2,2] => 11010 => 3
11100 => [3,2] => [1,1,2,1] => 11101 => 2
11110 => [4,1] => [1,1,1,2] => 11110 => 1
11111 => [5] => [1,1,1,1,1] => 11111 => 0
000000 => [6] => [1,1,1,1,1,1] => 111111 => 0
000001 => [5,1] => [1,1,1,1,2] => 111110 => 1
000011 => [4,2] => [1,1,1,2,1] => 111101 => 2
000110 => [3,2,1] => [1,1,2,2] => 111010 => 3
000111 => [3,3] => [1,1,2,1,1] => 111011 => 3
001100 => [2,2,2] => [1,2,2,1] => 110101 => 4
001110 => [2,3,1] => [1,2,1,2] => 110110 => 4
001111 => [2,4] => [1,2,1,1,1] => 110111 => 4
011000 => [1,2,3] => [2,2,1,1] => 101011 => 5
011001 => [1,2,2,1] => [2,2,2] => 101010 => 5
011100 => [1,3,2] => [2,1,2,1] => 101101 => 5
011110 => [1,4,1] => [2,1,1,2] => 101110 => 5
Description
The degree of a binary word. A valley in a binary word is a letter $0$ which is not immediately followed by a $1$. A peak is a letter $1$ which is not immediately followed by a $0$. Let $f$ be the map that replaces every valley with a peak. The degree of a binary word $w$ is the number of times $f$ has to be applied to obtain a binary word without zeros.
Matching statistic: St000026
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000026: 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]
 => [1,1,0,0,1,0]
 => 2 = 1 + 1
011 => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 3 = 2 + 1
100 => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 3 = 2 + 1
110 => [2,1] => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 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]
 => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
0011 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 3 = 2 + 1
0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 4 = 3 + 1
0111 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 4 = 3 + 1
1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 4 = 3 + 1
1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 4 = 3 + 1
1100 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 3 = 2 + 1
1110 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 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]
 => [1,1,0,0,1,0,1,0,1,0]
 => 2 = 1 + 1
00011 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 3 = 2 + 1
00110 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 4 = 3 + 1
00111 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 4 = 3 + 1
01100 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 5 = 4 + 1
01110 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 5 = 4 + 1
01111 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 5 = 4 + 1
10000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 5 = 4 + 1
10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 5 = 4 + 1
10011 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 5 = 4 + 1
11000 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 4 = 3 + 1
11001 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 4 = 3 + 1
11100 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 3 = 2 + 1
11110 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 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]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 2 = 1 + 1
000011 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => 3 = 2 + 1
000110 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => 4 = 3 + 1
000111 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => 4 = 3 + 1
001100 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => 5 = 4 + 1
001110 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => 5 = 4 + 1
001111 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => 5 = 4 + 1
011000 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => 6 = 5 + 1
011001 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => 6 = 5 + 1
011100 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => 6 = 5 + 1
011110 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 6 = 5 + 1
Description
The position of the first return of a Dyck path.
Matching statistic: St000476
(load all 2 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000476: Dyck paths ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [1] => [1,0]
 => ? = 0
1 => [1] => [1] => [1,0]
 => ? = 0
00 => [2] => [2] => [1,1,0,0]
 => 0
01 => [1,1] => [1,1] => [1,0,1,0]
 => 1
10 => [1,1] => [1,1] => [1,0,1,0]
 => 1
11 => [2] => [2] => [1,1,0,0]
 => 0
000 => [3] => [3] => [1,1,1,0,0,0]
 => 0
001 => [2,1] => [1,2] => [1,0,1,1,0,0]
 => 1
011 => [1,2] => [2,1] => [1,1,0,0,1,0]
 => 2
100 => [1,2] => [2,1] => [1,1,0,0,1,0]
 => 2
110 => [2,1] => [1,2] => [1,0,1,1,0,0]
 => 1
111 => [3] => [3] => [1,1,1,0,0,0]
 => 0
0000 => [4] => [4] => [1,1,1,1,0,0,0,0]
 => 0
0001 => [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
0011 => [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
0110 => [1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 3
0111 => [1,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => 3
1000 => [1,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => 3
1001 => [1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 3
1100 => [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
1110 => [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
1111 => [4] => [4] => [1,1,1,1,0,0,0,0]
 => 0
00000 => [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 0
00001 => [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
00011 => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
00110 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 3
00111 => [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
01100 => [1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 4
01110 => [1,3,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 4
01111 => [1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 4
10000 => [1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 4
10001 => [1,3,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 4
10011 => [1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 4
11000 => [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
11001 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 3
11100 => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
11110 => [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
11111 => [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 0
000000 => [6] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
000001 => [5,1] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1
000011 => [4,2] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 2
000110 => [3,2,1] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 3
000111 => [3,3] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 3
001100 => [2,2,2] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 4
001110 => [2,3,1] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 4
001111 => [2,4] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 4
011000 => [1,2,3] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 5
011001 => [1,2,2,1] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 5
011100 => [1,3,2] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 5
011110 => [1,4,1] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 5
011111 => [1,5] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 5
100000 => [1,5] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 5
Description
The sum of the semi-lengths of tunnels before a valley of a Dyck path. For each valley $v$ in a Dyck path $D$ there is a corresponding tunnel, which is the factor $T_v = s_i\dots s_j$ of $D$ where $s_i$ is the step after the first intersection of $D$ with the line $y = ht(v)$ to the left of $s_j$. This statistic is $$ \sum_v (j_v-i_v)/2. $$
Matching statistic: St000010
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 96% ●values known / values provided: 96%●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
011 => [1,2] => [1,0,1,1,0,0]
 => [1,1]
 => 2
100 => [1,2] => [1,0,1,1,0,0]
 => [1,1]
 => 2
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
0011 => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 2
0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 3
0111 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 3
1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 3
1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 3
1100 => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 2
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
00011 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => 2
00110 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => 3
00111 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 3
01100 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => 4
01110 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => 4
01111 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 4
10000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 4
10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => 4
10011 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => 4
11000 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 3
11001 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => 3
11100 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => 2
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
000011 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [4,4]
 => 2
000110 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [5,3,3]
 => 3
000111 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [3,3,3]
 => 3
001100 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [4,4,2,2]
 => 4
001110 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [5,2,2,2]
 => 4
001111 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,2,2,2]
 => 4
011000 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [3,3,3,1,1]
 => 5
011001 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1,1]
 => 5
011100 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [4,4,1,1,1]
 => 5
011110 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [5,1,1,1,1]
 => 5
0011001 => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [6,4,4,2,2]
 => ? = 5
0110011 => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [5,5,3,3,1,1]
 => ? = 6
1001100 => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [5,5,3,3,1,1]
 => ? = 6
1100110 => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [6,4,4,2,2]
 => ? = 5
Description
The length of the partition.
Matching statistic: St001725
(load all 2 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001725: Graphs ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [1] => ([],1)
 => 1 = 0 + 1
1 => [1] => [1] => ([],1)
 => 1 = 0 + 1
00 => [2] => [2] => ([],2)
 => 1 = 0 + 1
01 => [1,1] => [1,1] => ([(0,1)],2)
 => 2 = 1 + 1
10 => [1,1] => [1,1] => ([(0,1)],2)
 => 2 = 1 + 1
11 => [2] => [2] => ([],2)
 => 1 = 0 + 1
000 => [3] => [3] => ([],3)
 => 1 = 0 + 1
001 => [2,1] => [1,2] => ([(1,2)],3)
 => 2 = 1 + 1
011 => [1,2] => [2,1] => ([(0,2),(1,2)],3)
 => 3 = 2 + 1
100 => [1,2] => [2,1] => ([(0,2),(1,2)],3)
 => 3 = 2 + 1
110 => [2,1] => [1,2] => ([(1,2)],3)
 => 2 = 1 + 1
111 => [3] => [3] => ([],3)
 => 1 = 0 + 1
0000 => [4] => [4] => ([],4)
 => 1 = 0 + 1
0001 => [3,1] => [1,3] => ([(2,3)],4)
 => 2 = 1 + 1
0011 => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
0110 => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
0111 => [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 4 = 3 + 1
1000 => [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 4 = 3 + 1
1001 => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
1100 => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
1110 => [3,1] => [1,3] => ([(2,3)],4)
 => 2 = 1 + 1
1111 => [4] => [4] => ([],4)
 => 1 = 0 + 1
00000 => [5] => [5] => ([],5)
 => 1 = 0 + 1
00001 => [4,1] => [1,4] => ([(3,4)],5)
 => 2 = 1 + 1
00011 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 3 = 2 + 1
00110 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
00111 => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
01100 => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
01110 => [1,3,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
01111 => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 5 = 4 + 1
10000 => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 5 = 4 + 1
10001 => [1,3,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
10011 => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
11000 => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
11001 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
11100 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 3 = 2 + 1
11110 => [4,1] => [1,4] => ([(3,4)],5)
 => 2 = 1 + 1
11111 => [5] => [5] => ([],5)
 => 1 = 0 + 1
000000 => [6] => [6] => ([],6)
 => 1 = 0 + 1
000001 => [5,1] => [1,5] => ([(4,5)],6)
 => 2 = 1 + 1
000011 => [4,2] => [2,4] => ([(3,5),(4,5)],6)
 => 3 = 2 + 1
000110 => [3,2,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 4 = 3 + 1
000111 => [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 4 = 3 + 1
001100 => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5 = 4 + 1
001110 => [2,3,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 5 = 4 + 1
001111 => [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 5 = 4 + 1
011000 => [1,2,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
011001 => [1,2,2,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
011100 => [1,3,2] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
011110 => [1,4,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
0110001 => [1,2,3,1] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6 + 1
0110011 => [1,2,2,2] => [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6 + 1
1001100 => [1,2,2,2] => [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6 + 1
1001110 => [1,2,3,1] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6 + 1
Description
The harmonious chromatic number of a graph. A harmonious colouring is a proper vertex colouring such that any pair of colours appears at most once on adjacent vertices.
Matching statistic: St001118
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001118: Graphs ⟶ ℤResult quality: 86% ●values known / values provided: 87%●distinct values known / distinct values provided: 86%
Values
0 => [1] => [1] => ([],1)
 => ? = 0
1 => [1] => [1] => ([],1)
 => ? = 0
00 => [2] => [2] => ([],2)
 => ? = 0
01 => [1,1] => [1,1] => ([(0,1)],2)
 => 1
10 => [1,1] => [1,1] => ([(0,1)],2)
 => 1
11 => [2] => [2] => ([],2)
 => ? = 0
000 => [3] => [3] => ([],3)
 => ? = 0
001 => [2,1] => [1,2] => ([(1,2)],3)
 => 1
011 => [1,2] => [2,1] => ([(0,2),(1,2)],3)
 => 2
100 => [1,2] => [2,1] => ([(0,2),(1,2)],3)
 => 2
110 => [2,1] => [1,2] => ([(1,2)],3)
 => 1
111 => [3] => [3] => ([],3)
 => ? = 0
0000 => [4] => [4] => ([],4)
 => ? = 0
0001 => [3,1] => [1,3] => ([(2,3)],4)
 => 1
0011 => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => 2
0110 => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
0111 => [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
1000 => [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
1001 => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
1100 => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => 2
1110 => [3,1] => [1,3] => ([(2,3)],4)
 => 1
1111 => [4] => [4] => ([],4)
 => ? = 0
00000 => [5] => [5] => ([],5)
 => ? = 0
00001 => [4,1] => [1,4] => ([(3,4)],5)
 => 1
00011 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 2
00110 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
00111 => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
01100 => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
01110 => [1,3,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
01111 => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
10000 => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
10001 => [1,3,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
10011 => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
11000 => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
11001 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
11100 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 2
11110 => [4,1] => [1,4] => ([(3,4)],5)
 => 1
11111 => [5] => [5] => ([],5)
 => ? = 0
000000 => [6] => [6] => ([],6)
 => ? = 0
000001 => [5,1] => [1,5] => ([(4,5)],6)
 => 1
000011 => [4,2] => [2,4] => ([(3,5),(4,5)],6)
 => 2
000110 => [3,2,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 3
000111 => [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 3
001100 => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
001110 => [2,3,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
001111 => [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 4
011000 => [1,2,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
011001 => [1,2,2,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
011100 => [1,3,2] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
011110 => [1,4,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
011111 => [1,5] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 5
100000 => [1,5] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 5
100001 => [1,4,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
100011 => [1,3,2] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
100110 => [1,2,2,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
100111 => [1,2,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
110000 => [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 4
110001 => [2,3,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
110011 => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
111000 => [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 3
111001 => [3,2,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 3
111111 => [6] => [6] => ([],6)
 => ? = 0
0000000 => [7] => [7] => ([],7)
 => ? = 0
1111111 => [7] => [7] => ([],7)
 => ? = 0
Description
The acyclic chromatic index of a graph. An acyclic edge coloring of a graph is a proper colouring of the edges of a graph such that the union of the edges colored with any two given colours is a forest. The smallest number of colours such that such a colouring exists is the acyclic chromatic index.
Matching statistic: St001227
(load all 2 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001227: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 86%
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
10 => [1,1] => [1,0,1,0]
 => 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]
 => 1
011 => [1,2] => [1,0,1,1,0,0]
 => 2
100 => [1,2] => [1,0,1,1,0,0]
 => 2
110 => [2,1] => [1,1,0,0,1,0]
 => 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]
 => 1
0011 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 3
0111 => [1,3] => [1,0,1,1,1,0,0,0]
 => 3
1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => 3
1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 3
1100 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
1110 => [3,1] => [1,1,1,0,0,0,1,0]
 => 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]
 => 1
00011 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
00110 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3
00111 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 3
01100 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 4
01110 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 4
01111 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 4
10000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 4
10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 4
10011 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 4
11000 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 3
11001 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3
11100 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
11110 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 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]
 => 1
000011 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 2
000110 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 3
000111 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 3
001100 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 4
001110 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 4
001111 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 4
011000 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 5
011001 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 5
011100 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 5
011110 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 5
0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
0000001 => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 1
0000011 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 2
0000110 => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => ? = 3
0000111 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 3
0001100 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 4
0001110 => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 4
0001111 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 4
0011000 => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 5
0011001 => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 5
0011100 => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 5
0011110 => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 5
0011111 => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 5
0110000 => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 6
0110001 => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => ? = 6
0110011 => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 6
0111000 => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 6
0111001 => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => ? = 6
0111100 => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => ? = 6
0111110 => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 6
0111111 => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6
1000000 => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6
1000001 => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 6
1000011 => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => ? = 6
1000110 => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => ? = 6
1000111 => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 6
1001100 => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 6
1001110 => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => ? = 6
1001111 => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 6
1100000 => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 5
1100001 => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 5
1100011 => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 5
1100110 => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 5
1100111 => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 5
1110000 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 4
1110001 => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 4
1110011 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 4
1111000 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 3
1111001 => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => ? = 3
1111100 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 2
1111110 => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 1
1111111 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
Description
The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra.
The following 17 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000653The last descent of a permutation. St001117The game chromatic index of a graph. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001480The number of simple summands of the module J^2/J^3. St001742The difference of the maximal and the minimal degree in a graph. St001645The pebbling number of a connected graph. St000632The jump number of the poset. St000307The number of rowmotion orbits of a poset. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. St001644The dimension of a graph. St000455The second largest eigenvalue of a graph if it is integral. St001877Number of indecomposable injective modules with projective dimension 2. St001624The breadth of a lattice. St001738The minimal order of a graph which is not an induced subgraph of the given graph.