Your data matches 2 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001721
(load all 2 compositions to match this statistic)
Mapping
St001721: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 1
1 => 0
00 => 3
01 => 2
10 => 1
11 => 0
000 => 5
001 => 4
010 => 3
011 => 3
100 => 3
101 => 2
110 => 1
111 => 0
0000 => 7
0001 => 6
0010 => 5
0011 => 5
0100 => 5
0101 => 4
0110 => 4
0111 => 4
1000 => 5
1001 => 4
1010 => 3
1011 => 3
1100 => 3
1101 => 2
1110 => 1
1111 => 0
00000 => 9
00001 => 8
00010 => 7
00011 => 7
00100 => 7
00101 => 6
00110 => 6
00111 => 6
01000 => 7
01001 => 6
01010 => 5
01011 => 5
01100 => 5
01101 => 5
01110 => 5
01111 => 5
10000 => 7
10001 => 6
10010 => 5
10011 => 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: St001232
Mapping
Mp00105: Binary words —complement⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 39%
Values
0 => 1 => [1,1] => [1,0,1,0]
 => 1
1 => 0 => [2] => [1,1,0,0]
 => 0
00 => 11 => [1,1,1] => [1,0,1,0,1,0]
 => ? = 3
01 => 10 => [1,2] => [1,0,1,1,0,0]
 => 2
10 => 01 => [2,1] => [1,1,0,0,1,0]
 => 1
11 => 00 => [3] => [1,1,1,0,0,0]
 => 0
000 => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => ? = 5
001 => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => ? = 4
010 => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 3
011 => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 3
100 => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => ? = 3
101 => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
110 => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
111 => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
0000 => 1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => ? = 7
0001 => 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => ? = 6
0010 => 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => ? = 5
0011 => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => ? = 5
0100 => 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => ? = 5
0101 => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 4
0110 => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 4
0111 => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 4
1000 => 0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => ? = 5
1001 => 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => ? = 4
1010 => 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3
1011 => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 3
1100 => 0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => ? = 3
1101 => 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
1110 => 0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1
1111 => 0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 0
00000 => 11111 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 9
00001 => 11110 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 8
00010 => 11101 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 7
00011 => 11100 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 7
00100 => 11011 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 7
00101 => 11010 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 6
00110 => 11001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 6
00111 => 11000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 6
01000 => 10111 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 7
01001 => 10110 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 6
01010 => 10101 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 5
01011 => 10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 5
01100 => 10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 5
01101 => 10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 5
01110 => 10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 5
01111 => 10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 5
10000 => 01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 7
10001 => 01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 6
10010 => 01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => ? = 5
10011 => 01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => ? = 5
10100 => 01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => ? = 5
10101 => 01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 4
10110 => 01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 4
10111 => 01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 4
11000 => 00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => ? = 5
11001 => 00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => ? = 4
11010 => 00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 3
11011 => 00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 3
11100 => 00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 3
11101 => 00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 2
11110 => 00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
11111 => 00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
000000 => 111111 => [1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 11
000001 => 111110 => [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 10
000010 => 111101 => [1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 9
000011 => 111100 => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 9
000100 => 111011 => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 9
000101 => 111010 => [1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 8
000110 => 111001 => [1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 8
000111 => 111000 => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 8
001000 => 110111 => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 9
001001 => 110110 => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 8
001010 => 110101 => [1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 7
001011 => 110100 => [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 7
001100 => 110011 => [1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 7
001101 => 110010 => [1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 7
001110 => 110001 => [1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 7
001111 => 110000 => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 7
010000 => 101111 => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 9
010001 => 101110 => [1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 8
010010 => 101101 => [1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => ? = 7
010101 => 101010 => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => 6
010110 => 101001 => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => 6
010111 => 101000 => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => 6
011010 => 100101 => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => 6
011011 => 100100 => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => 6
011101 => 100010 => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => 6
011110 => 100001 => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => 6
011111 => 100000 => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => 6
101010 => 010101 => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => 5
101011 => 010100 => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => 5
101101 => 010010 => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => 5
101110 => 010001 => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => 5
101111 => 010000 => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => 5
110101 => 001010 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => 4
110110 => 001001 => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => 4
110111 => 001000 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => 4
111010 => 000101 => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => 3
111011 => 000100 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => 3
111101 => 000010 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => 2
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.