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
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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.
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
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