- FindStat

Identifier

St001721: Binary words ⟶ ℤ

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
10100=>5
10101=>4
10110=>4
10111=>4
11000=>5
11001=>4
11010=>3
11011=>3
11100=>3
11101=>2
11110=>1
11111=>0
000000=>11
000001=>10
000010=>9
000011=>9
000100=>9
000101=>8
000110=>8
000111=>8
001000=>9
001001=>8
001010=>7
001011=>7
001100=>7
001101=>7
001110=>7
001111=>7
010000=>9
010001=>8
010010=>7
010011=>7
010100=>7
010101=>6
010110=>6
010111=>6
011000=>7
011001=>6
011010=>6
011011=>6
011100=>6
011101=>6
011110=>6
011111=>6
100000=>9
100001=>8
100010=>7
100011=>7
100100=>7
100101=>6
100110=>6
100111=>6
101000=>7
101001=>6
101010=>5
101011=>5
101100=>5
101101=>5
101110=>5
101111=>5
110000=>7
110001=>6
110010=>5
110011=>5
110100=>5
110101=>4
110110=>4
110111=>4
111000=>5
111001=>4
111010=>3
111011=>3
111100=>3
111101=>2
111110=>1
111111=>0
0000000=>13
0000001=>12
0000010=>11
0000011=>11
0000100=>11
0000101=>10
0000110=>10
0000111=>10
0001000=>11
0001001=>10
0001010=>9
0001011=>9
0001100=>9
0001101=>9
0001110=>9
0001111=>9
0010000=>11
0010001=>10
0010010=>9
0010011=>9
0010100=>9
0010101=>8
0010110=>8
0010111=>8
0011000=>9
0011001=>8
0011010=>8
0011011=>8
0011100=>8
0011101=>8
0011110=>8
0011111=>8
0100000=>11
0100001=>10
0100010=>9
0100011=>9
0100100=>9
0100101=>8
0100110=>8
0100111=>8
0101000=>9
0101001=>8
0101010=>7
0101011=>7
0101100=>7
0101101=>7
0101110=>7
0101111=>7
0110000=>9
0110001=>8
0110010=>7
0110011=>7
0110100=>7
0110101=>7
0110110=>7
0110111=>7
0111000=>7
0111001=>7
0111010=>7
0111011=>7
0111100=>7
0111101=>7
0111110=>7
0111111=>7
1000000=>11
1000001=>10
1000010=>9
1000011=>9
1000100=>9
1000101=>8
1000110=>8
1000111=>8
1001000=>9
1001001=>8
1001010=>7
1001011=>7
1001100=>7
1001101=>7
1001110=>7
1001111=>7
1010000=>9
1010001=>8
1010010=>7
1010011=>7
1010100=>7
1010101=>6
1010110=>6
1010111=>6
1011000=>7
1011001=>6
1011010=>6
1011011=>6
1011100=>6
1011101=>6
1011110=>6
1011111=>6
1100000=>9
1100001=>8
1100010=>7
1100011=>7
1100100=>7
1100101=>6
1100110=>6
1100111=>6
1101000=>7
1101001=>6
1101010=>5
1101011=>5
1101100=>5
1101101=>5
1101110=>5
1101111=>5
1110000=>7
1110001=>6
1110010=>5
1110011=>5
1110100=>5
1110101=>4
1110110=>4
1110111=>4
1111000=>5
1111001=>4
1111010=>3
1111011=>3
1111100=>3
1111101=>2
1111110=>1
1111111=>0
00000000=>15
00000001=>14
00000010=>13
00000011=>13
00000100=>13
00000101=>12
00000110=>12
00000111=>12
00001000=>13
00001001=>12
00001010=>11
00001011=>11
00001100=>11
00001101=>11
00001110=>11
00001111=>11
00010000=>13
00010001=>12
00010010=>11
00010011=>11
00010100=>11
00010101=>10
00010110=>10
00010111=>10
00011000=>11
00011001=>10
00011010=>10
00011011=>10
00011100=>10
00011101=>10
00011110=>10
00011111=>10
00100000=>13
00100001=>12
00100010=>11
00100011=>11
00100100=>11
00100101=>10
00100110=>10
00100111=>10
00101000=>11
00101001=>10
00101010=>9
00101011=>9
00101100=>9
00101101=>9
00101110=>9
00101111=>9
00110000=>11
00110001=>10
00110010=>9
00110011=>9
00110100=>9
00110101=>9
00110110=>9
00110111=>9
00111000=>9
00111001=>9
00111010=>9
00111011=>9
00111100=>9
00111101=>9
00111110=>9
00111111=>9
01000000=>13
01000001=>12
01000010=>11
01000011=>11
01000100=>11
01000101=>10
01000110=>10
01000111=>10
01001000=>11
01001001=>10
01001010=>9
01001011=>9
01001100=>9
01001101=>9
01001110=>9
01001111=>9
01010000=>11
01010001=>10
01010010=>9
01010011=>9
01010100=>9
01010101=>8
01010110=>8
01010111=>8
01011000=>9
01011001=>8
01011010=>8
01011011=>8
01011100=>8
01011101=>8
01011110=>8
01011111=>8
01100000=>11
01100001=>10
01100010=>9
01100011=>9
01100100=>9
01100101=>8
01100110=>8
01100111=>8
01101000=>9
01101001=>8
01101010=>8
01101011=>8
01101100=>8
01101101=>8
01101110=>8
01101111=>8
01110000=>9
01110001=>8
01110010=>8
01110011=>8
01110100=>8
01110101=>8
01110110=>8
01110111=>8
01111000=>8
01111001=>8
01111010=>8
01111011=>8
01111100=>8
01111101=>8
01111110=>8
01111111=>8
10000000=>13
10000001=>12
10000010=>11
10000011=>11
10000100=>11
10000101=>10
10000110=>10
10000111=>10
10001000=>11
10001001=>10
10001010=>9
10001011=>9
10001100=>9
10001101=>9
10001110=>9
10001111=>9
10010000=>11
10010001=>10
10010010=>9
10010011=>9
10010100=>9
10010101=>8
10010110=>8
10010111=>8
10011000=>9
10011001=>8
10011010=>8
10011011=>8
10011100=>8
10011101=>8
10011110=>8
10011111=>8
10100000=>11
10100001=>10
10100010=>9
10100011=>9
10100100=>9
10100101=>8
10100110=>8
10100111=>8
10101000=>9
10101001=>8
10101010=>7
10101011=>7
10101100=>7
10101101=>7
10101110=>7
10101111=>7
10110000=>9
10110001=>8
10110010=>7
10110011=>7
10110100=>7
10110101=>7
10110110=>7
10110111=>7
10111000=>7
10111001=>7
10111010=>7
10111011=>7
10111100=>7
10111101=>7
10111110=>7
10111111=>7
11000000=>11
11000001=>10
11000010=>9
11000011=>9
11000100=>9
11000101=>8
11000110=>8
11000111=>8
11001000=>9
11001001=>8
11001010=>7
11001011=>7
11001100=>7
11001101=>7
11001110=>7
11001111=>7
11010000=>9
11010001=>8
11010010=>7
11010011=>7
11010100=>7
11010101=>6
11010110=>6
11010111=>6
11011000=>7
11011001=>6
11011010=>6
11011011=>6
11011100=>6
11011101=>6
11011110=>6
11011111=>6
11100000=>9
11100001=>8
11100010=>7
11100011=>7
11100100=>7
11100101=>6
11100110=>6
11100111=>6
11101000=>7
11101001=>6
11101010=>5
11101011=>5
11101100=>5
11101101=>5
11101110=>5
11101111=>5
11110000=>7
11110001=>6
11110010=>5
11110011=>5
11110100=>5
11110101=>4
11110110=>4
11110111=>4
11111000=>5
11111001=>4
11111010=>3
11111011=>3
11111100=>3
11111101=>2
11111110=>1
11111111=>0
000000000=>17
000000001=>16
000000010=>15
000000011=>15
000000100=>15
000000101=>14
000000110=>14
000000111=>14
000001000=>15
000001001=>14
000001010=>13
000001011=>13
000001100=>13
000001101=>13
000001110=>13
000001111=>13
000010000=>15
000010001=>14
000010010=>13
000010011=>13
000010100=>13
000010101=>12
000010110=>12
000010111=>12
000011000=>13
000011001=>12
000011010=>12
000011011=>12
000011100=>12
000011101=>12
000011110=>12
000011111=>12
000100000=>15
000100001=>14
000100010=>13
000100011=>13
000100100=>13
000100101=>12
000100110=>12
000100111=>12
000101000=>13
000101001=>12
000101010=>11
000101011=>11
000101100=>11
000101101=>11
000101110=>11
000101111=>11
000110000=>13
000110001=>12
000110010=>11
000110011=>11
000110100=>11
000110101=>11
000110110=>11
000110111=>11
000111000=>11
000111001=>11
000111010=>11
000111011=>11
000111100=>11
000111101=>11
000111110=>11
000111111=>11
001000000=>15
001000001=>14
001000010=>13
001000011=>13
001000100=>13
001000101=>12
001000110=>12
001000111=>12
001001000=>13
001001001=>12
001001010=>11
001001011=>11
001001100=>11
001001101=>11
001001110=>11
001001111=>11
001010000=>13
001010001=>12
001010010=>11
001010011=>11
001010100=>11
001010101=>10
001010110=>10
001010111=>10
001011000=>11
001011001=>10
001011010=>10
001011011=>10
001011100=>10
001011101=>10
001011110=>10
001011111=>10
001100000=>13
001100001=>12
001100010=>11
001100011=>11
001100100=>11
001100101=>10
001100110=>10
001100111=>10
001101000=>11
001101001=>10
001101010=>10
001101011=>10
001101100=>10
001101101=>10
001101110=>10
001101111=>10
001110000=>11
001110001=>10
001110010=>10
001110011=>10
001110100=>10
001110101=>10
001110110=>10
001110111=>10
001111000=>10
001111001=>10
001111010=>10
001111011=>10
001111100=>10
001111101=>10
001111110=>10
001111111=>10
010000000=>15
010000001=>14
010000010=>13
010000011=>13
010000100=>13
010000101=>12
010000110=>12
010000111=>12
010001000=>13
010001001=>12
010001010=>11
010001011=>11
010001100=>11
010001101=>11
010001110=>11
010001111=>11
010010000=>13
010010001=>12
010010010=>11
010010011=>11
010010100=>11
010010101=>10
010010110=>10
010010111=>10
010011000=>11
010011001=>10
010011010=>10
010011011=>10
010011100=>10
010011101=>10
010011110=>10
010011111=>10
010100000=>13
010100001=>12
010100010=>11
010100011=>11
010100100=>11
010100101=>10
010100110=>10
010100111=>10
010101000=>11
010101001=>10
010101010=>9
010101011=>9
010101100=>9
010101101=>9
010101110=>9
010101111=>9
010110000=>11
010110001=>10
010110010=>9
010110011=>9
010110100=>9
010110101=>9
010110110=>9
010110111=>9
010111000=>9
010111001=>9
010111010=>9
010111011=>9
010111100=>9
010111101=>9
010111110=>9
010111111=>9
011000000=>13
011000001=>12
011000010=>11
011000011=>11
011000100=>11
011000101=>10
011000110=>10
011000111=>10
011001000=>11
011001001=>10
011001010=>9
011001011=>9
011001100=>9
011001101=>9
011001110=>9
011001111=>9
011010000=>11
011010001=>10
011010010=>9
011010011=>9
011010100=>9
011010101=>9
011010110=>9
011010111=>9
011011000=>9
011011001=>9
011011010=>9
011011011=>9
011011100=>9
011011101=>9
011011110=>9
011011111=>9
011100000=>11
011100001=>10
011100010=>9
011100011=>9
011100100=>9
011100101=>9
011100110=>9
011100111=>9
011101000=>9
011101001=>9
011101010=>9
011101011=>9
011101100=>9
011101101=>9
011101110=>9
011101111=>9
011110000=>9
011110001=>9
011110010=>9
011110011=>9
011110100=>9
011110101=>9
011110110=>9
011110111=>9
011111000=>9
011111001=>9
011111010=>9
011111011=>9
011111100=>9
011111101=>9
011111110=>9
011111111=>9
100000000=>15
100000001=>14
100000010=>13
100000011=>13
100000100=>13
100000101=>12
100000110=>12
100000111=>12
100001000=>13
100001001=>12
100001010=>11
100001011=>11
100001100=>11
100001101=>11
100001110=>11
100001111=>11
100010000=>13
100010001=>12
100010010=>11
100010011=>11
100010100=>11
100010101=>10
100010110=>10
100010111=>10
100011000=>11
100011001=>10
100011010=>10
100011011=>10
100011100=>10
100011101=>10
100011110=>10
100011111=>10
100100000=>13
100100001=>12
100100010=>11
100100011=>11
100100100=>11
100100101=>10
100100110=>10
100100111=>10
100101000=>11
100101001=>10
100101010=>9
100101011=>9
100101100=>9
100101101=>9
100101110=>9
100101111=>9
100110000=>11
100110001=>10
100110010=>9
100110011=>9
100110100=>9
100110101=>9
100110110=>9
100110111=>9
100111000=>9
100111001=>9
100111010=>9
100111011=>9
100111100=>9
100111101=>9
100111110=>9
100111111=>9
101000000=>13
101000001=>12
101000010=>11
101000011=>11
101000100=>11
101000101=>10
101000110=>10
101000111=>10
101001000=>11
101001001=>10
101001010=>9
101001011=>9
101001100=>9
101001101=>9
101001110=>9
101001111=>9
101010000=>11
101010001=>10
101010010=>9
101010011=>9
101010100=>9
101010101=>8
101010110=>8
101010111=>8
101011000=>9
101011001=>8
101011010=>8
101011011=>8
101011100=>8
101011101=>8
101011110=>8
101011111=>8
101100000=>11
101100001=>10
101100010=>9
101100011=>9
101100100=>9
101100101=>8
101100110=>8
101100111=>8
101101000=>9
101101001=>8
101101010=>8
101101011=>8
101101100=>8
101101101=>8
101101110=>8
101101111=>8
101110000=>9
101110001=>8
101110010=>8
101110011=>8
101110100=>8
101110101=>8
101110110=>8
101110111=>8
101111000=>8
101111001=>8
101111010=>8
101111011=>8
101111100=>8
101111101=>8
101111110=>8
101111111=>8
110000000=>13
110000001=>12
110000010=>11
110000011=>11
110000100=>11
110000101=>10
110000110=>10
110000111=>10
110001000=>11
110001001=>10
110001010=>9
110001011=>9
110001100=>9
110001101=>9
110001110=>9
110001111=>9
110010000=>11
110010001=>10
110010010=>9
110010011=>9
110010100=>9
110010101=>8
110010110=>8
110010111=>8
110011000=>9
110011001=>8
110011010=>8
110011011=>8
110011100=>8
110011101=>8
110011110=>8
110011111=>8
110100000=>11
110100001=>10
110100010=>9
110100011=>9
110100100=>9
110100101=>8
110100110=>8
110100111=>8
110101000=>9
110101001=>8
110101010=>7
110101011=>7
110101100=>7
110101101=>7
110101110=>7
110101111=>7
110110000=>9
110110001=>8
110110010=>7
110110011=>7
110110100=>7
110110101=>7
110110110=>7
110110111=>7
110111000=>7
110111001=>7
110111010=>7
110111011=>7
110111100=>7
110111101=>7
110111110=>7
110111111=>7
111000000=>11
111000001=>10
111000010=>9
111000011=>9
111000100=>9
111000101=>8
111000110=>8
111000111=>8
111001000=>9
111001001=>8
111001010=>7
111001011=>7
111001100=>7
111001101=>7
111001110=>7
111001111=>7
111010000=>9
111010001=>8
111010010=>7
111010011=>7
111010100=>7
111010101=>6
111010110=>6
111010111=>6
111011000=>7
111011001=>6
111011010=>6
111011011=>6
111011100=>6
111011101=>6
111011110=>6
111011111=>6
111100000=>9
111100001=>8
111100010=>7
111100011=>7
111100100=>7
111100101=>6
111100110=>6
111100111=>6
111101000=>7
111101001=>6
111101010=>5
111101011=>5
111101100=>5
111101101=>5
111101110=>5
111101111=>5
111110000=>7
111110001=>6
111110010=>5
111110011=>5
111110100=>5
111110101=>4
111110110=>4
111110111=>4
111111000=>5
111111001=>4
111111010=>3
111111011=>3
111111100=>3
111111101=>2
111111110=>1
111111111=>0
                    

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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.

References

[1] Tasoulas, I., Manes, K., Sapounakis, A., Tsikouras, P. Chains with small intervals in the lattice of binary paths MathSciNet:4054761

Code

def filling(w):
    w = list(w)
    f = list(w)
    for i in range(len(w)-1):
        if w[i:i+2] == [0, 1]:
            f[i:i+2] = [1, 0]
    if w[len(w)-1] == 0:
        f[len(w)-1] = 1
    return Words([0,1])(f)

def degree(w):
    d = 0
    while w.count(0):
        w = filling(w)
        d += 1
    return d

Created

May 20, 2021 at 12:23 by Martin Rubey

Updated

May 20, 2021 at 12:23 by Martin Rubey