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Your data matches 39 different statistics following compositions of up to 3 maps.
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Matching statistic: St000983
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Mp00234: Binary words —valleys-to-peaks⟶ Binary words
St000983: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000983: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 1 => 1
1 => 1 => 1
00 => 01 => 2
01 => 10 => 2
10 => 11 => 1
11 => 11 => 1
000 => 001 => 2
001 => 010 => 3
010 => 101 => 3
011 => 101 => 3
100 => 101 => 3
101 => 110 => 2
110 => 111 => 1
111 => 111 => 1
0000 => 0001 => 2
0010 => 0101 => 4
0011 => 0101 => 4
0101 => 1010 => 4
0110 => 1011 => 3
0111 => 1011 => 3
1001 => 1010 => 4
1101 => 1110 => 2
1110 => 1111 => 1
1111 => 1111 => 1
00000 => 00001 => 2
00101 => 01010 => 5
00110 => 01011 => 4
00111 => 01011 => 4
01010 => 10101 => 5
01011 => 10101 => 5
01100 => 10101 => 5
01110 => 10111 => 3
01111 => 10111 => 3
10010 => 10101 => 5
10011 => 10101 => 5
11101 => 11110 => 2
11110 => 11111 => 1
11111 => 11111 => 1
000000 => 000001 => 2
001010 => 010101 => 6
001011 => 010101 => 6
001100 => 010101 => 6
001110 => 010111 => 4
001111 => 010111 => 4
010101 => 101010 => 6
010110 => 101011 => 5
010111 => 101011 => 5
011001 => 101010 => 6
011110 => 101111 => 3
011111 => 101111 => 3
Description
The length of the longest alternating subword.
This is the length of the longest consecutive subword of the form $010...$ or of the form $101...$.
Matching statistic: St000691
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Mp00234: Binary words —valleys-to-peaks⟶ Binary words
St000691: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000691: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 1 => 0 = 1 - 1
1 => 1 => 0 = 1 - 1
00 => 01 => 1 = 2 - 1
01 => 10 => 1 = 2 - 1
10 => 11 => 0 = 1 - 1
11 => 11 => 0 = 1 - 1
000 => 001 => 1 = 2 - 1
001 => 010 => 2 = 3 - 1
010 => 101 => 2 = 3 - 1
011 => 101 => 2 = 3 - 1
100 => 101 => 2 = 3 - 1
101 => 110 => 1 = 2 - 1
110 => 111 => 0 = 1 - 1
111 => 111 => 0 = 1 - 1
0000 => 0001 => 1 = 2 - 1
0010 => 0101 => 3 = 4 - 1
0011 => 0101 => 3 = 4 - 1
0101 => 1010 => 3 = 4 - 1
0110 => 1011 => 2 = 3 - 1
0111 => 1011 => 2 = 3 - 1
1001 => 1010 => 3 = 4 - 1
1101 => 1110 => 1 = 2 - 1
1110 => 1111 => 0 = 1 - 1
1111 => 1111 => 0 = 1 - 1
00000 => 00001 => 1 = 2 - 1
00101 => 01010 => 4 = 5 - 1
00110 => 01011 => 3 = 4 - 1
00111 => 01011 => 3 = 4 - 1
01010 => 10101 => 4 = 5 - 1
01011 => 10101 => 4 = 5 - 1
01100 => 10101 => 4 = 5 - 1
01110 => 10111 => 2 = 3 - 1
01111 => 10111 => 2 = 3 - 1
10010 => 10101 => 4 = 5 - 1
10011 => 10101 => 4 = 5 - 1
11101 => 11110 => 1 = 2 - 1
11110 => 11111 => 0 = 1 - 1
11111 => 11111 => 0 = 1 - 1
000000 => 000001 => 1 = 2 - 1
001010 => 010101 => 5 = 6 - 1
001011 => 010101 => 5 = 6 - 1
001100 => 010101 => 5 = 6 - 1
001110 => 010111 => 3 = 4 - 1
001111 => 010111 => 3 = 4 - 1
010101 => 101010 => 5 = 6 - 1
010110 => 101011 => 4 = 5 - 1
010111 => 101011 => 4 = 5 - 1
011001 => 101010 => 5 = 6 - 1
011110 => 101111 => 2 = 3 - 1
011111 => 101111 => 2 = 3 - 1
Description
The number of changes of a binary word.
This is the number of indices $i$ such that $w_i \neq w_{i+1}$.
Matching statistic: St000982
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Mp00234: Binary words —valleys-to-peaks⟶ Binary words
Mp00158: Binary words —alternating inverse⟶ Binary words
St000982: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00158: Binary words —alternating inverse⟶ Binary words
St000982: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 1 => 1 => 1
1 => 1 => 1 => 1
00 => 01 => 00 => 2
01 => 10 => 11 => 2
10 => 11 => 10 => 1
11 => 11 => 10 => 1
000 => 001 => 011 => 2
001 => 010 => 000 => 3
010 => 101 => 111 => 3
011 => 101 => 111 => 3
100 => 101 => 111 => 3
101 => 110 => 100 => 2
110 => 111 => 101 => 1
111 => 111 => 101 => 1
0000 => 0001 => 0100 => 2
0010 => 0101 => 0000 => 4
0011 => 0101 => 0000 => 4
0101 => 1010 => 1111 => 4
0110 => 1011 => 1110 => 3
0111 => 1011 => 1110 => 3
1001 => 1010 => 1111 => 4
1101 => 1110 => 1011 => 2
1110 => 1111 => 1010 => 1
1111 => 1111 => 1010 => 1
00000 => 00001 => 01011 => 2
00101 => 01010 => 00000 => 5
00110 => 01011 => 00001 => 4
00111 => 01011 => 00001 => 4
01010 => 10101 => 11111 => 5
01011 => 10101 => 11111 => 5
01100 => 10101 => 11111 => 5
01110 => 10111 => 11101 => 3
01111 => 10111 => 11101 => 3
10010 => 10101 => 11111 => 5
10011 => 10101 => 11111 => 5
11101 => 11110 => 10100 => 2
11110 => 11111 => 10101 => 1
11111 => 11111 => 10101 => 1
000000 => 000001 => 010100 => 2
001010 => 010101 => 000000 => 6
001011 => 010101 => 000000 => 6
001100 => 010101 => 000000 => 6
001110 => 010111 => 000010 => 4
001111 => 010111 => 000010 => 4
010101 => 101010 => 111111 => 6
010110 => 101011 => 111110 => 5
010111 => 101011 => 111110 => 5
011001 => 101010 => 111111 => 6
011110 => 101111 => 111010 => 3
011111 => 101111 => 111010 => 3
Description
The length of the longest constant subword.
Matching statistic: St001486
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Mp00234: Binary words —valleys-to-peaks⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St001486: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
St001486: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 1 => [1,1] => 2 = 1 + 1
1 => 1 => [1,1] => 2 = 1 + 1
00 => 01 => [2,1] => 3 = 2 + 1
01 => 10 => [1,2] => 3 = 2 + 1
10 => 11 => [1,1,1] => 2 = 1 + 1
11 => 11 => [1,1,1] => 2 = 1 + 1
000 => 001 => [3,1] => 3 = 2 + 1
001 => 010 => [2,2] => 4 = 3 + 1
010 => 101 => [1,2,1] => 4 = 3 + 1
011 => 101 => [1,2,1] => 4 = 3 + 1
100 => 101 => [1,2,1] => 4 = 3 + 1
101 => 110 => [1,1,2] => 3 = 2 + 1
110 => 111 => [1,1,1,1] => 2 = 1 + 1
111 => 111 => [1,1,1,1] => 2 = 1 + 1
0000 => 0001 => [4,1] => 3 = 2 + 1
0010 => 0101 => [2,2,1] => 5 = 4 + 1
0011 => 0101 => [2,2,1] => 5 = 4 + 1
0101 => 1010 => [1,2,2] => 5 = 4 + 1
0110 => 1011 => [1,2,1,1] => 4 = 3 + 1
0111 => 1011 => [1,2,1,1] => 4 = 3 + 1
1001 => 1010 => [1,2,2] => 5 = 4 + 1
1101 => 1110 => [1,1,1,2] => 3 = 2 + 1
1110 => 1111 => [1,1,1,1,1] => 2 = 1 + 1
1111 => 1111 => [1,1,1,1,1] => 2 = 1 + 1
00000 => 00001 => [5,1] => 3 = 2 + 1
00101 => 01010 => [2,2,2] => 6 = 5 + 1
00110 => 01011 => [2,2,1,1] => 5 = 4 + 1
00111 => 01011 => [2,2,1,1] => 5 = 4 + 1
01010 => 10101 => [1,2,2,1] => 6 = 5 + 1
01011 => 10101 => [1,2,2,1] => 6 = 5 + 1
01100 => 10101 => [1,2,2,1] => 6 = 5 + 1
01110 => 10111 => [1,2,1,1,1] => 4 = 3 + 1
01111 => 10111 => [1,2,1,1,1] => 4 = 3 + 1
10010 => 10101 => [1,2,2,1] => 6 = 5 + 1
10011 => 10101 => [1,2,2,1] => 6 = 5 + 1
11101 => 11110 => [1,1,1,1,2] => 3 = 2 + 1
11110 => 11111 => [1,1,1,1,1,1] => 2 = 1 + 1
11111 => 11111 => [1,1,1,1,1,1] => 2 = 1 + 1
000000 => 000001 => [6,1] => 3 = 2 + 1
001010 => 010101 => [2,2,2,1] => 7 = 6 + 1
001011 => 010101 => [2,2,2,1] => 7 = 6 + 1
001100 => 010101 => [2,2,2,1] => 7 = 6 + 1
001110 => 010111 => [2,2,1,1,1] => 5 = 4 + 1
001111 => 010111 => [2,2,1,1,1] => 5 = 4 + 1
010101 => 101010 => [1,2,2,2] => 7 = 6 + 1
010110 => 101011 => [1,2,2,1,1] => 6 = 5 + 1
010111 => 101011 => [1,2,2,1,1] => 6 = 5 + 1
011001 => 101010 => [1,2,2,2] => 7 = 6 + 1
011110 => 101111 => [1,2,1,1,1,1] => 4 = 3 + 1
011111 => 101111 => [1,2,1,1,1,1] => 4 = 3 + 1
Description
The number of corners of the ribbon associated with an integer composition.
We associate a ribbon shape to a composition $c=(c_1,\dots,c_n)$ with $c_i$ cells in the $i$-th row from bottom to top, such that the cells in two rows overlap in precisely one cell.
This statistic records the total number of corners of the ribbon shape.
Matching statistic: St000010
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 1 => [1] => [1]
=> 1
1 => 1 => [1] => [1]
=> 1
00 => 01 => [1,1] => [1,1]
=> 2
01 => 10 => [1,1] => [1,1]
=> 2
10 => 11 => [2] => [2]
=> 1
11 => 11 => [2] => [2]
=> 1
000 => 001 => [2,1] => [2,1]
=> 2
001 => 010 => [1,1,1] => [1,1,1]
=> 3
010 => 101 => [1,1,1] => [1,1,1]
=> 3
011 => 101 => [1,1,1] => [1,1,1]
=> 3
100 => 101 => [1,1,1] => [1,1,1]
=> 3
101 => 110 => [2,1] => [2,1]
=> 2
110 => 111 => [3] => [3]
=> 1
111 => 111 => [3] => [3]
=> 1
0000 => 0001 => [3,1] => [3,1]
=> 2
0010 => 0101 => [1,1,1,1] => [1,1,1,1]
=> 4
0011 => 0101 => [1,1,1,1] => [1,1,1,1]
=> 4
0101 => 1010 => [1,1,1,1] => [1,1,1,1]
=> 4
0110 => 1011 => [1,1,2] => [2,1,1]
=> 3
0111 => 1011 => [1,1,2] => [2,1,1]
=> 3
1001 => 1010 => [1,1,1,1] => [1,1,1,1]
=> 4
1101 => 1110 => [3,1] => [3,1]
=> 2
1110 => 1111 => [4] => [4]
=> 1
1111 => 1111 => [4] => [4]
=> 1
00000 => 00001 => [4,1] => [4,1]
=> 2
00101 => 01010 => [1,1,1,1,1] => [1,1,1,1,1]
=> 5
00110 => 01011 => [1,1,1,2] => [2,1,1,1]
=> 4
00111 => 01011 => [1,1,1,2] => [2,1,1,1]
=> 4
01010 => 10101 => [1,1,1,1,1] => [1,1,1,1,1]
=> 5
01011 => 10101 => [1,1,1,1,1] => [1,1,1,1,1]
=> 5
01100 => 10101 => [1,1,1,1,1] => [1,1,1,1,1]
=> 5
01110 => 10111 => [1,1,3] => [3,1,1]
=> 3
01111 => 10111 => [1,1,3] => [3,1,1]
=> 3
10010 => 10101 => [1,1,1,1,1] => [1,1,1,1,1]
=> 5
10011 => 10101 => [1,1,1,1,1] => [1,1,1,1,1]
=> 5
11101 => 11110 => [4,1] => [4,1]
=> 2
11110 => 11111 => [5] => [5]
=> 1
11111 => 11111 => [5] => [5]
=> 1
000000 => 000001 => [5,1] => [5,1]
=> 2
001010 => 010101 => [1,1,1,1,1,1] => [1,1,1,1,1,1]
=> 6
001011 => 010101 => [1,1,1,1,1,1] => [1,1,1,1,1,1]
=> 6
001100 => 010101 => [1,1,1,1,1,1] => [1,1,1,1,1,1]
=> 6
001110 => 010111 => [1,1,1,3] => [3,1,1,1]
=> 4
001111 => 010111 => [1,1,1,3] => [3,1,1,1]
=> 4
010101 => 101010 => [1,1,1,1,1,1] => [1,1,1,1,1,1]
=> 6
010110 => 101011 => [1,1,1,1,2] => [2,1,1,1,1]
=> 5
010111 => 101011 => [1,1,1,1,2] => [2,1,1,1,1]
=> 5
011001 => 101010 => [1,1,1,1,1,1] => [1,1,1,1,1,1]
=> 6
011110 => 101111 => [1,1,4] => [4,1,1]
=> 3
011111 => 101111 => [1,1,4] => [4,1,1]
=> 3
Description
The length of the partition.
Matching statistic: St000097
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 1 => [1] => ([],1)
=> 1
1 => 1 => [1] => ([],1)
=> 1
00 => 01 => [1,1] => ([(0,1)],2)
=> 2
01 => 10 => [1,1] => ([(0,1)],2)
=> 2
10 => 11 => [2] => ([],2)
=> 1
11 => 11 => [2] => ([],2)
=> 1
000 => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2
001 => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
010 => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
011 => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
100 => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
101 => 110 => [2,1] => ([(0,2),(1,2)],3)
=> 2
110 => 111 => [3] => ([],3)
=> 1
111 => 111 => [3] => ([],3)
=> 1
0000 => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
0010 => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
0011 => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
0101 => 1010 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
0110 => 1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
0111 => 1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
1001 => 1010 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
1101 => 1110 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
1110 => 1111 => [4] => ([],4)
=> 1
1111 => 1111 => [4] => ([],4)
=> 1
00000 => 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
00101 => 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
00110 => 01011 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
00111 => 01011 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
01010 => 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
01011 => 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
01100 => 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
01110 => 10111 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
01111 => 10111 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
10010 => 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
10011 => 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
11101 => 11110 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
11110 => 11111 => [5] => ([],5)
=> 1
11111 => 11111 => [5] => ([],5)
=> 1
000000 => 000001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
001010 => 010101 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
001011 => 010101 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
001100 => 010101 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
001110 => 010111 => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
001111 => 010111 => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
010101 => 101010 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
010110 => 101011 => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
010111 => 101011 => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
011001 => 101010 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
011110 => 101111 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 3
011111 => 101111 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 3
Description
The order of the largest clique of the graph.
A clique in a graph $G$ is a subset $U \subseteq V(G)$ such that any pair of vertices in $U$ are adjacent. I.e. the subgraph induced by $U$ is a complete graph.
Matching statistic: St000098
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000098: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000098: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 1 => [1] => ([],1)
=> 1
1 => 1 => [1] => ([],1)
=> 1
00 => 01 => [1,1] => ([(0,1)],2)
=> 2
01 => 10 => [1,1] => ([(0,1)],2)
=> 2
10 => 11 => [2] => ([],2)
=> 1
11 => 11 => [2] => ([],2)
=> 1
000 => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2
001 => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
010 => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
011 => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
100 => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
101 => 110 => [2,1] => ([(0,2),(1,2)],3)
=> 2
110 => 111 => [3] => ([],3)
=> 1
111 => 111 => [3] => ([],3)
=> 1
0000 => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
0010 => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
0011 => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
0101 => 1010 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
0110 => 1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
0111 => 1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
1001 => 1010 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
1101 => 1110 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
1110 => 1111 => [4] => ([],4)
=> 1
1111 => 1111 => [4] => ([],4)
=> 1
00000 => 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
00101 => 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
00110 => 01011 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
00111 => 01011 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
01010 => 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
01011 => 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
01100 => 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
01110 => 10111 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
01111 => 10111 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
10010 => 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
10011 => 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
11101 => 11110 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
11110 => 11111 => [5] => ([],5)
=> 1
11111 => 11111 => [5] => ([],5)
=> 1
000000 => 000001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
001010 => 010101 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
001011 => 010101 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
001100 => 010101 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
001110 => 010111 => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
001111 => 010111 => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
010101 => 101010 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
010110 => 101011 => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
010111 => 101011 => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
011001 => 101010 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
011110 => 101111 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 3
011111 => 101111 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 3
Description
The chromatic number of a graph.
The minimal number of colors needed to color the vertices of the graph such that no two vertices which share an edge have the same color.
Matching statistic: St000288
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 1 => [1] => 1 => 1
1 => 1 => [1] => 1 => 1
00 => 01 => [1,1] => 11 => 2
01 => 10 => [1,1] => 11 => 2
10 => 11 => [2] => 10 => 1
11 => 11 => [2] => 10 => 1
000 => 001 => [2,1] => 101 => 2
001 => 010 => [1,1,1] => 111 => 3
010 => 101 => [1,1,1] => 111 => 3
011 => 101 => [1,1,1] => 111 => 3
100 => 101 => [1,1,1] => 111 => 3
101 => 110 => [2,1] => 101 => 2
110 => 111 => [3] => 100 => 1
111 => 111 => [3] => 100 => 1
0000 => 0001 => [3,1] => 1001 => 2
0010 => 0101 => [1,1,1,1] => 1111 => 4
0011 => 0101 => [1,1,1,1] => 1111 => 4
0101 => 1010 => [1,1,1,1] => 1111 => 4
0110 => 1011 => [1,1,2] => 1110 => 3
0111 => 1011 => [1,1,2] => 1110 => 3
1001 => 1010 => [1,1,1,1] => 1111 => 4
1101 => 1110 => [3,1] => 1001 => 2
1110 => 1111 => [4] => 1000 => 1
1111 => 1111 => [4] => 1000 => 1
00000 => 00001 => [4,1] => 10001 => 2
00101 => 01010 => [1,1,1,1,1] => 11111 => 5
00110 => 01011 => [1,1,1,2] => 11110 => 4
00111 => 01011 => [1,1,1,2] => 11110 => 4
01010 => 10101 => [1,1,1,1,1] => 11111 => 5
01011 => 10101 => [1,1,1,1,1] => 11111 => 5
01100 => 10101 => [1,1,1,1,1] => 11111 => 5
01110 => 10111 => [1,1,3] => 11100 => 3
01111 => 10111 => [1,1,3] => 11100 => 3
10010 => 10101 => [1,1,1,1,1] => 11111 => 5
10011 => 10101 => [1,1,1,1,1] => 11111 => 5
11101 => 11110 => [4,1] => 10001 => 2
11110 => 11111 => [5] => 10000 => 1
11111 => 11111 => [5] => 10000 => 1
000000 => 000001 => [5,1] => 100001 => 2
001010 => 010101 => [1,1,1,1,1,1] => 111111 => 6
001011 => 010101 => [1,1,1,1,1,1] => 111111 => 6
001100 => 010101 => [1,1,1,1,1,1] => 111111 => 6
001110 => 010111 => [1,1,1,3] => 111100 => 4
001111 => 010111 => [1,1,1,3] => 111100 => 4
010101 => 101010 => [1,1,1,1,1,1] => 111111 => 6
010110 => 101011 => [1,1,1,1,2] => 111110 => 5
010111 => 101011 => [1,1,1,1,2] => 111110 => 5
011001 => 101010 => [1,1,1,1,1,1] => 111111 => 6
011110 => 101111 => [1,1,4] => 111000 => 3
011111 => 101111 => [1,1,4] => 111000 => 3
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000381
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 1 => [1] => [1] => 1
1 => 1 => [1] => [1] => 1
00 => 01 => [1,1] => [2] => 2
01 => 10 => [1,1] => [2] => 2
10 => 11 => [2] => [1,1] => 1
11 => 11 => [2] => [1,1] => 1
000 => 001 => [2,1] => [2,1] => 2
001 => 010 => [1,1,1] => [3] => 3
010 => 101 => [1,1,1] => [3] => 3
011 => 101 => [1,1,1] => [3] => 3
100 => 101 => [1,1,1] => [3] => 3
101 => 110 => [2,1] => [2,1] => 2
110 => 111 => [3] => [1,1,1] => 1
111 => 111 => [3] => [1,1,1] => 1
0000 => 0001 => [3,1] => [2,1,1] => 2
0010 => 0101 => [1,1,1,1] => [4] => 4
0011 => 0101 => [1,1,1,1] => [4] => 4
0101 => 1010 => [1,1,1,1] => [4] => 4
0110 => 1011 => [1,1,2] => [1,3] => 3
0111 => 1011 => [1,1,2] => [1,3] => 3
1001 => 1010 => [1,1,1,1] => [4] => 4
1101 => 1110 => [3,1] => [2,1,1] => 2
1110 => 1111 => [4] => [1,1,1,1] => 1
1111 => 1111 => [4] => [1,1,1,1] => 1
00000 => 00001 => [4,1] => [2,1,1,1] => 2
00101 => 01010 => [1,1,1,1,1] => [5] => 5
00110 => 01011 => [1,1,1,2] => [1,4] => 4
00111 => 01011 => [1,1,1,2] => [1,4] => 4
01010 => 10101 => [1,1,1,1,1] => [5] => 5
01011 => 10101 => [1,1,1,1,1] => [5] => 5
01100 => 10101 => [1,1,1,1,1] => [5] => 5
01110 => 10111 => [1,1,3] => [1,1,3] => 3
01111 => 10111 => [1,1,3] => [1,1,3] => 3
10010 => 10101 => [1,1,1,1,1] => [5] => 5
10011 => 10101 => [1,1,1,1,1] => [5] => 5
11101 => 11110 => [4,1] => [2,1,1,1] => 2
11110 => 11111 => [5] => [1,1,1,1,1] => 1
11111 => 11111 => [5] => [1,1,1,1,1] => 1
000000 => 000001 => [5,1] => [2,1,1,1,1] => 2
001010 => 010101 => [1,1,1,1,1,1] => [6] => 6
001011 => 010101 => [1,1,1,1,1,1] => [6] => 6
001100 => 010101 => [1,1,1,1,1,1] => [6] => 6
001110 => 010111 => [1,1,1,3] => [1,1,4] => 4
001111 => 010111 => [1,1,1,3] => [1,1,4] => 4
010101 => 101010 => [1,1,1,1,1,1] => [6] => 6
010110 => 101011 => [1,1,1,1,2] => [1,5] => 5
010111 => 101011 => [1,1,1,1,2] => [1,5] => 5
011001 => 101010 => [1,1,1,1,1,1] => [6] => 6
011110 => 101111 => [1,1,4] => [1,1,1,3] => 3
011111 => 101111 => [1,1,4] => [1,1,1,3] => 3
Description
The largest part of an integer composition.
Matching statistic: St000808
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
St000808: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
St000808: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 1 => [1] => [1] => 1
1 => 1 => [1] => [1] => 1
00 => 01 => [1,1] => [2] => 2
01 => 10 => [1,1] => [2] => 2
10 => 11 => [2] => [1,1] => 1
11 => 11 => [2] => [1,1] => 1
000 => 001 => [2,1] => [2,1] => 2
001 => 010 => [1,1,1] => [3] => 3
010 => 101 => [1,1,1] => [3] => 3
011 => 101 => [1,1,1] => [3] => 3
100 => 101 => [1,1,1] => [3] => 3
101 => 110 => [2,1] => [2,1] => 2
110 => 111 => [3] => [1,1,1] => 1
111 => 111 => [3] => [1,1,1] => 1
0000 => 0001 => [3,1] => [2,1,1] => 2
0010 => 0101 => [1,1,1,1] => [4] => 4
0011 => 0101 => [1,1,1,1] => [4] => 4
0101 => 1010 => [1,1,1,1] => [4] => 4
0110 => 1011 => [1,1,2] => [1,3] => 3
0111 => 1011 => [1,1,2] => [1,3] => 3
1001 => 1010 => [1,1,1,1] => [4] => 4
1101 => 1110 => [3,1] => [2,1,1] => 2
1110 => 1111 => [4] => [1,1,1,1] => 1
1111 => 1111 => [4] => [1,1,1,1] => 1
00000 => 00001 => [4,1] => [2,1,1,1] => 2
00101 => 01010 => [1,1,1,1,1] => [5] => 5
00110 => 01011 => [1,1,1,2] => [1,4] => 4
00111 => 01011 => [1,1,1,2] => [1,4] => 4
01010 => 10101 => [1,1,1,1,1] => [5] => 5
01011 => 10101 => [1,1,1,1,1] => [5] => 5
01100 => 10101 => [1,1,1,1,1] => [5] => 5
01110 => 10111 => [1,1,3] => [1,1,3] => 3
01111 => 10111 => [1,1,3] => [1,1,3] => 3
10010 => 10101 => [1,1,1,1,1] => [5] => 5
10011 => 10101 => [1,1,1,1,1] => [5] => 5
11101 => 11110 => [4,1] => [2,1,1,1] => 2
11110 => 11111 => [5] => [1,1,1,1,1] => 1
11111 => 11111 => [5] => [1,1,1,1,1] => 1
000000 => 000001 => [5,1] => [2,1,1,1,1] => 2
001010 => 010101 => [1,1,1,1,1,1] => [6] => 6
001011 => 010101 => [1,1,1,1,1,1] => [6] => 6
001100 => 010101 => [1,1,1,1,1,1] => [6] => 6
001110 => 010111 => [1,1,1,3] => [1,1,4] => 4
001111 => 010111 => [1,1,1,3] => [1,1,4] => 4
010101 => 101010 => [1,1,1,1,1,1] => [6] => 6
010110 => 101011 => [1,1,1,1,2] => [1,5] => 5
010111 => 101011 => [1,1,1,1,2] => [1,5] => 5
011001 => 101010 => [1,1,1,1,1,1] => [6] => 6
011110 => 101111 => [1,1,4] => [1,1,1,3] => 3
011111 => 101111 => [1,1,4] => [1,1,1,3] => 3
Description
The number of up steps of the associated bargraph.
Interpret the composition as the sequence of heights of the bars of a bargraph. This statistic is the number of up steps.
The following 29 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001330The hat guessing number of a graph. St001372The length of a longest cyclic run of ones of a binary word. St001176The size of a partition minus its first part. St001581The achromatic number of a graph. St000172The Grundy number of a graph. St001029The size of the core of a graph. St001116The game chromatic number of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St000806The semiperimeter of the associated bargraph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001963The tree-depth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001235The global dimension of the corresponding Comp-Nakayama algebra. St000388The number of orbits of vertices of a graph under automorphisms. St000822The Hadwiger number of the graph. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St001352The number of internal nodes in the modular decomposition of a graph. St001812The biclique partition number of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001488The number of corners of a skew partition. St000307The number of rowmotion orbits of a poset. St000632The jump number of the poset.
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