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* Copyright (C) 2019 The FindStatCrew <info@findstat.org> *
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Statistic identifier: St001721
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Collection: Binary words
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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.
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References: [1] Tasoulas, I., Manes, K., Sapounakis, A., Tsikouras, P. Chains with small intervals in the lattice of binary paths [[MathSciNet:4054761]]
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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
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Statistic 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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Created: May 20, 2021 at 12:23 by Martin Rubey
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Last Updated: May 20, 2021 at 12:23 by Martin Rubey