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Your data matches 17 different statistics following compositions of up to 3 maps.
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Matching statistic: St001232
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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 => [2] => [1,1,0,0]
=> 0
1 => [1,1] => [1,0,1,0]
=> 1
00 => [3] => [1,1,1,0,0,0]
=> 0
01 => [2,1] => [1,1,0,0,1,0]
=> 1
10 => [1,2] => [1,0,1,1,0,0]
=> 2
000 => [4] => [1,1,1,1,0,0,0,0]
=> 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 4
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 4
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 4
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 4
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 5
10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 5
10101 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5
000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 0
000001 => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1
000010 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> 2
000100 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> 3
000101 => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> 3
001000 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> 4
001001 => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> 4
001010 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> 4
010000 => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> 5
010001 => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> 5
010010 => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> 5
010100 => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> 5
010101 => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> 5
100000 => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
100001 => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
100010 => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> 6
100100 => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> 6
100101 => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> 6
101000 => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> 6
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)
(load all 2 compositions to match this statistic)
Mp00104: Binary words —reverse⟶ Binary words
Mp00178: Binary words —to composition⟶ 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%
Mp00178: Binary words —to composition⟶ 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 => 0 => [2] => ([],2)
=> 0
1 => 1 => [1,1] => ([(0,1)],2)
=> 1
00 => 00 => [3] => ([],3)
=> 0
01 => 10 => [1,2] => ([(1,2)],3)
=> 1
10 => 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2
000 => 000 => [4] => ([],4)
=> 0
001 => 100 => [1,3] => ([(2,3)],4)
=> 1
010 => 010 => [2,2] => ([(1,3),(2,3)],4)
=> 2
100 => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
101 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
0000 => 0000 => [5] => ([],5)
=> 0
0001 => 1000 => [1,4] => ([(3,4)],5)
=> 1
0010 => 0100 => [2,3] => ([(2,4),(3,4)],5)
=> 2
0100 => 0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
0101 => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
1000 => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
1001 => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
1010 => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
00000 => 00000 => [6] => ([],6)
=> 0
00001 => 10000 => [1,5] => ([(4,5)],6)
=> 1
00010 => 01000 => [2,4] => ([(3,5),(4,5)],6)
=> 2
00100 => 00100 => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 3
00101 => 10100 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3
01000 => 00010 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
01001 => 10010 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
01010 => 01010 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
10000 => 00001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 5
10001 => 10001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
10010 => 01001 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
10100 => 00101 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
10101 => 10101 => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
000000 => 000000 => [7] => ([],7)
=> 0
000001 => 100000 => [1,6] => ([(5,6)],7)
=> 1
000010 => 010000 => [2,5] => ([(4,6),(5,6)],7)
=> 2
000100 => 001000 => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 3
000101 => 101000 => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> 3
001000 => 000100 => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 4
001001 => 100100 => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
001010 => 010100 => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
010000 => 000010 => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 5
010001 => 100010 => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
010010 => 010010 => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
010100 => 001010 => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
010101 => 101010 => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
100000 => 000001 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 6
100001 => 100001 => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
100010 => 010001 => [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
100100 => 001001 => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
100101 => 101001 => [1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
101000 => 000101 => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
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)
(load all 2 compositions to match this statistic)
Mp00104: Binary words —reverse⟶ Binary words
Mp00178: Binary words —to composition⟶ 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%
Mp00178: Binary words —to composition⟶ 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 => 0 => [2] => ([],2)
=> 0
1 => 1 => [1,1] => ([(0,1)],2)
=> 1
00 => 00 => [3] => ([],3)
=> 0
01 => 10 => [1,2] => ([(1,2)],3)
=> 1
10 => 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2
000 => 000 => [4] => ([],4)
=> 0
001 => 100 => [1,3] => ([(2,3)],4)
=> 1
010 => 010 => [2,2] => ([(1,3),(2,3)],4)
=> 2
100 => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
101 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
0000 => 0000 => [5] => ([],5)
=> 0
0001 => 1000 => [1,4] => ([(3,4)],5)
=> 1
0010 => 0100 => [2,3] => ([(2,4),(3,4)],5)
=> 2
0100 => 0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
0101 => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
1000 => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
1001 => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
1010 => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
00000 => 00000 => [6] => ([],6)
=> 0
00001 => 10000 => [1,5] => ([(4,5)],6)
=> 1
00010 => 01000 => [2,4] => ([(3,5),(4,5)],6)
=> 2
00100 => 00100 => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 3
00101 => 10100 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3
01000 => 00010 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
01001 => 10010 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
01010 => 01010 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
10000 => 00001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 5
10001 => 10001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
10010 => 01001 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
10100 => 00101 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
10101 => 10101 => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
000000 => 000000 => [7] => ([],7)
=> 0
000001 => 100000 => [1,6] => ([(5,6)],7)
=> 1
000010 => 010000 => [2,5] => ([(4,6),(5,6)],7)
=> 2
000100 => 001000 => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 3
000101 => 101000 => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> 3
001000 => 000100 => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 4
001001 => 100100 => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
001010 => 010100 => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
010000 => 000010 => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 5
010001 => 100010 => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
010010 => 010010 => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
010100 => 001010 => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
010101 => 101010 => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
100000 => 000001 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 6
100001 => 100001 => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
100010 => 010001 => [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
100100 => 001001 => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
100101 => 101001 => [1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
101000 => 000101 => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
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
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00105: Binary words —complement⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St001721: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St001721: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 1 => [1,1] => 11 => 0
1 => 0 => [2] => 10 => 1
00 => 11 => [1,1,1] => 111 => 0
01 => 10 => [1,2] => 110 => 1
10 => 01 => [2,1] => 101 => 2
000 => 111 => [1,1,1,1] => 1111 => 0
001 => 110 => [1,1,2] => 1110 => 1
010 => 101 => [1,2,1] => 1101 => 2
100 => 011 => [2,1,1] => 1011 => 3
101 => 010 => [2,2] => 1010 => 3
0000 => 1111 => [1,1,1,1,1] => 11111 => 0
0001 => 1110 => [1,1,1,2] => 11110 => 1
0010 => 1101 => [1,1,2,1] => 11101 => 2
0100 => 1011 => [1,2,1,1] => 11011 => 3
0101 => 1010 => [1,2,2] => 11010 => 3
1000 => 0111 => [2,1,1,1] => 10111 => 4
1001 => 0110 => [2,1,2] => 10110 => 4
1010 => 0101 => [2,2,1] => 10101 => 4
00000 => 11111 => [1,1,1,1,1,1] => 111111 => 0
00001 => 11110 => [1,1,1,1,2] => 111110 => 1
00010 => 11101 => [1,1,1,2,1] => 111101 => 2
00100 => 11011 => [1,1,2,1,1] => 111011 => 3
00101 => 11010 => [1,1,2,2] => 111010 => 3
01000 => 10111 => [1,2,1,1,1] => 110111 => 4
01001 => 10110 => [1,2,1,2] => 110110 => 4
01010 => 10101 => [1,2,2,1] => 110101 => 4
10000 => 01111 => [2,1,1,1,1] => 101111 => 5
10001 => 01110 => [2,1,1,2] => 101110 => 5
10010 => 01101 => [2,1,2,1] => 101101 => 5
10100 => 01011 => [2,2,1,1] => 101011 => 5
10101 => 01010 => [2,2,2] => 101010 => 5
000000 => 111111 => [1,1,1,1,1,1,1] => 1111111 => 0
000001 => 111110 => [1,1,1,1,1,2] => 1111110 => 1
000010 => 111101 => [1,1,1,1,2,1] => 1111101 => 2
000100 => 111011 => [1,1,1,2,1,1] => 1111011 => 3
000101 => 111010 => [1,1,1,2,2] => 1111010 => 3
001000 => 110111 => [1,1,2,1,1,1] => 1110111 => 4
001001 => 110110 => [1,1,2,1,2] => 1110110 => 4
001010 => 110101 => [1,1,2,2,1] => 1110101 => 4
010000 => 101111 => [1,2,1,1,1,1] => 1101111 => 5
010001 => 101110 => [1,2,1,1,2] => 1101110 => 5
010010 => 101101 => [1,2,1,2,1] => 1101101 => 5
010100 => 101011 => [1,2,2,1,1] => 1101011 => 5
010101 => 101010 => [1,2,2,2] => 1101010 => 5
100000 => 011111 => [2,1,1,1,1,1] => 1011111 => 6
100001 => 011110 => [2,1,1,1,2] => 1011110 => 6
100010 => 011101 => [2,1,1,2,1] => 1011101 => 6
100100 => 011011 => [2,1,2,1,1] => 1011011 => 6
100101 => 011010 => [2,1,2,2] => 1011010 => 6
101000 => 010111 => [2,2,1,1,1] => 1010111 => 6
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
Mp00178: Binary words —to composition⟶ 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%
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 => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
1 => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 2 = 1 + 1
00 => [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
01 => [2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
10 => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 3 = 2 + 1
000 => [4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 4 = 3 + 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 4 = 3 + 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
0001 => [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
0010 => [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
0100 => [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
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4 = 3 + 1
1000 => [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
1001 => [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
1010 => [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
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 3 = 2 + 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 4 = 3 + 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> 4 = 3 + 1
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 5 = 4 + 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> 5 = 4 + 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> 5 = 4 + 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 6 = 5 + 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 6 = 5 + 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 6 = 5 + 1
10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 6 = 5 + 1
10101 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> 6 = 5 + 1
000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
000001 => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
000010 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
000100 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> 4 = 3 + 1
000101 => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> 4 = 3 + 1
001000 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> 5 = 4 + 1
001001 => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> 5 = 4 + 1
001010 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> 5 = 4 + 1
010000 => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> 6 = 5 + 1
010001 => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> 6 = 5 + 1
010010 => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> 6 = 5 + 1
010100 => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> 6 = 5 + 1
010101 => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> 6 = 5 + 1
100000 => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 7 = 6 + 1
100001 => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> 7 = 6 + 1
100010 => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> 7 = 6 + 1
100100 => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> 7 = 6 + 1
100101 => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> 7 = 6 + 1
101000 => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> 7 = 6 + 1
Description
The position of the first return of a Dyck path.
Matching statistic: St000476
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00104: Binary words —reverse⟶ Binary words
Mp00178: Binary words —to composition⟶ 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%
Mp00178: Binary words —to composition⟶ 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 => 0 => [2] => [1,1,0,0]
=> 0
1 => 1 => [1,1] => [1,0,1,0]
=> 1
00 => 00 => [3] => [1,1,1,0,0,0]
=> 0
01 => 10 => [1,2] => [1,0,1,1,0,0]
=> 1
10 => 01 => [2,1] => [1,1,0,0,1,0]
=> 2
000 => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 0
001 => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
010 => 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
100 => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
101 => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3
0000 => 0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
0001 => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
0010 => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
0100 => 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
0101 => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 3
1000 => 0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
1001 => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 4
1010 => 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 4
00000 => 00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
00001 => 10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
00010 => 01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
00100 => 00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
00101 => 10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 3
01000 => 00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4
01001 => 10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 4
01010 => 01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 4
10000 => 00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 5
10001 => 10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5
10010 => 01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 5
10100 => 00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 5
10101 => 10101 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5
000000 => 000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 0
000001 => 100000 => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
000010 => 010000 => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> 2
000100 => 001000 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> 3
000101 => 101000 => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> 3
001000 => 000100 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> 4
001001 => 100100 => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> 4
001010 => 010100 => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> 4
010000 => 000010 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> 5
010001 => 100010 => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> 5
010010 => 010010 => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> 5
010100 => 001010 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> 5
010101 => 101010 => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> 5
100000 => 000001 => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 6
100001 => 100001 => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
100010 => 010001 => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> 6
100100 => 001001 => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> 6
100101 => 101001 => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> 6
101000 => 000101 => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> 6
=> => [1] => [1,0]
=> ? = 0
Description
The sum of the semi-lengths of tunnels before a valley of a Dyck path.
For each valley $v$ in a Dyck path $D$ there is a corresponding tunnel, which
is the factor $T_v = s_i\dots s_j$ of $D$ where $s_i$ is the step after the first intersection of $D$ with the line $y = ht(v)$ to the left of $s_j$. This statistic is
$$
\sum_v (j_v-i_v)/2.
$$
Matching statistic: St000010
Mp00178: Binary words —to composition⟶ 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%
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 => [2] => [1,1,0,0]
=> []
=> 0
1 => [1,1] => [1,0,1,0]
=> [1]
=> 1
00 => [3] => [1,1,1,0,0,0]
=> []
=> 0
01 => [2,1] => [1,1,0,0,1,0]
=> [2]
=> 1
10 => [1,2] => [1,0,1,1,0,0]
=> [1,1]
=> 2
000 => [4] => [1,1,1,1,0,0,0,0]
=> []
=> 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [3]
=> 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 2
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 3
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 3
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 2
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 3
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 3
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 4
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 4
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 4
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> 0
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> 1
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> 2
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> 3
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> 3
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> 4
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> 4
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> 4
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> 5
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> 5
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> 5
10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> 5
10101 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> 5
000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> 0
000001 => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6]
=> 1
000010 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,5]
=> 2
000100 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,4,4]
=> 3
000101 => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [6,4,4]
=> 3
001000 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3]
=> 4
001001 => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3]
=> 4
001010 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3]
=> 4
010000 => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2,2]
=> 5
010001 => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,2,2,2]
=> 5
010010 => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [5,5,2,2,2]
=> 5
010100 => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [4,4,4,2,2]
=> 5
010101 => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2]
=> ? = 5
100000 => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1]
=> 6
100001 => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,1,1,1,1]
=> 6
100010 => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,5,1,1,1,1]
=> 6
100100 => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,4,4,1,1,1]
=> 6
100101 => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [6,4,4,1,1,1]
=> 6
101000 => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,1,1]
=> 6
101001 => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3,1,1]
=> 6
101010 => [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
Description
The length of the partition.
Matching statistic: St001725
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00104: Binary words —reverse⟶ Binary words
Mp00178: Binary words —to composition⟶ 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%
Mp00178: Binary words —to composition⟶ 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 => 0 => [2] => ([],2)
=> 1 = 0 + 1
1 => 1 => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
00 => 00 => [3] => ([],3)
=> 1 = 0 + 1
01 => 10 => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
10 => 01 => [2,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
000 => 000 => [4] => ([],4)
=> 1 = 0 + 1
001 => 100 => [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
010 => 010 => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
100 => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 4 = 3 + 1
101 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
0000 => 0000 => [5] => ([],5)
=> 1 = 0 + 1
0001 => 1000 => [1,4] => ([(3,4)],5)
=> 2 = 1 + 1
0010 => 0100 => [2,3] => ([(2,4),(3,4)],5)
=> 3 = 2 + 1
0100 => 0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 4 = 3 + 1
0101 => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
1000 => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 5 = 4 + 1
1001 => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
1010 => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
00000 => 00000 => [6] => ([],6)
=> 1 = 0 + 1
00001 => 10000 => [1,5] => ([(4,5)],6)
=> 2 = 1 + 1
00010 => 01000 => [2,4] => ([(3,5),(4,5)],6)
=> 3 = 2 + 1
00100 => 00100 => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 4 = 3 + 1
00101 => 10100 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
01000 => 00010 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 5 = 4 + 1
01001 => 10010 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
01010 => 01010 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
10000 => 00001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 6 = 5 + 1
10001 => 10001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
10010 => 01001 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
10100 => 00101 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
10101 => 10101 => [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
000000 => 000000 => [7] => ([],7)
=> 1 = 0 + 1
000001 => 100000 => [1,6] => ([(5,6)],7)
=> 2 = 1 + 1
000010 => 010000 => [2,5] => ([(4,6),(5,6)],7)
=> 3 = 2 + 1
000100 => 001000 => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 4 = 3 + 1
000101 => 101000 => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 3 + 1
001000 => 000100 => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 5 = 4 + 1
001001 => 100100 => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 5 = 4 + 1
001010 => 010100 => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5 = 4 + 1
010000 => 000010 => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 6 = 5 + 1
010001 => 100010 => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 6 = 5 + 1
010010 => 010010 => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6 = 5 + 1
010100 => 001010 => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6 = 5 + 1
010101 => 101010 => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6 = 5 + 1
100000 => 000001 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 7 = 6 + 1
100001 => 100001 => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
100010 => 010001 => [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
100100 => 001001 => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
100101 => 101001 => [1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
101000 => 000101 => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
101001 => 100101 => [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
101010 => 010101 => [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
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
Mp00104: Binary words —reverse⟶ Binary words
Mp00178: Binary words —to composition⟶ 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%
Mp00178: Binary words —to composition⟶ 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 => 0 => [2] => ([],2)
=> ? = 0
1 => 1 => [1,1] => ([(0,1)],2)
=> 1
00 => 00 => [3] => ([],3)
=> ? = 0
01 => 10 => [1,2] => ([(1,2)],3)
=> 1
10 => 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2
000 => 000 => [4] => ([],4)
=> ? = 0
001 => 100 => [1,3] => ([(2,3)],4)
=> 1
010 => 010 => [2,2] => ([(1,3),(2,3)],4)
=> 2
100 => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
101 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
0000 => 0000 => [5] => ([],5)
=> ? = 0
0001 => 1000 => [1,4] => ([(3,4)],5)
=> 1
0010 => 0100 => [2,3] => ([(2,4),(3,4)],5)
=> 2
0100 => 0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
0101 => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
1000 => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
1001 => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
1010 => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
00000 => 00000 => [6] => ([],6)
=> ? = 0
00001 => 10000 => [1,5] => ([(4,5)],6)
=> 1
00010 => 01000 => [2,4] => ([(3,5),(4,5)],6)
=> 2
00100 => 00100 => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 3
00101 => 10100 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3
01000 => 00010 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
01001 => 10010 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
01010 => 01010 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
10000 => 00001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 5
10001 => 10001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
10010 => 01001 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
10100 => 00101 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
10101 => 10101 => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
000000 => 000000 => [7] => ([],7)
=> ? = 0
000001 => 100000 => [1,6] => ([(5,6)],7)
=> 1
000010 => 010000 => [2,5] => ([(4,6),(5,6)],7)
=> 2
000100 => 001000 => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 3
000101 => 101000 => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> 3
001000 => 000100 => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 4
001001 => 100100 => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
001010 => 010100 => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
010000 => 000010 => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 5
010001 => 100010 => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
010010 => 010010 => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
010100 => 001010 => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
010101 => 101010 => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
100000 => 000001 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 6
100001 => 100001 => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
100010 => 010001 => [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
100100 => 001001 => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
100101 => 101001 => [1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
101000 => 000101 => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
101001 => 100101 => [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
101010 => 010101 => [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] => ([],1)
=> ? = 0
Description
The acyclic chromatic index of a graph.
An acyclic edge coloring of a graph is a proper colouring of the edges of a graph such that the union of the edges colored with any two given colours is a forest.
The smallest number of colours such that such a colouring exists is the acyclic chromatic index.
Matching statistic: St001227
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001227: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 86%
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 => [2] => [1,1,0,0]
=> 0
1 => [1,1] => [1,0,1,0]
=> 1
00 => [3] => [1,1,1,0,0,0]
=> 0
01 => [2,1] => [1,1,0,0,1,0]
=> 1
10 => [1,2] => [1,0,1,1,0,0]
=> 2
000 => [4] => [1,1,1,1,0,0,0,0]
=> 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 4
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 4
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 4
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 4
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 5
10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 5
10101 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5
000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
000001 => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
000010 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 2
000100 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 3
000101 => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
001000 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
001001 => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 4
001010 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 4
010000 => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5
010001 => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 5
010010 => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 5
010100 => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 5
010101 => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 5
100000 => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 6
100001 => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 6
100010 => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 6
100100 => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 6
100101 => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 6
101000 => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 6
101001 => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 6
101010 => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 6
=> [1] => [1,0]
=> 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 7 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. St001645The pebbling number of a connected graph. St001330The hat guessing number of a graph.
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