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Identifier

St001885: Binary words ⟶ ℤ

Values

0 => 2

1 => 2

00 => 1

01 => 2

10 => 2

11 => 1

000 => 1

001 => 4

010 => 1

011 => 4

100 => 4

101 => 1

110 => 4

111 => 1

0000 => 1

0001 => 6

0010 => 3

0011 => 6

0100 => 3

0101 => 1

0110 => 3

0111 => 6

1000 => 6

1001 => 3

1010 => 1

1011 => 3

1100 => 6

1101 => 3

1110 => 6

1111 => 1

00000 => 1

00001 => 12

00010 => 5

00011 => 12

00100 => 1

00101 => 12

00110 => 5

00111 => 12

01000 => 5

01001 => 2

01010 => 1

01011 => 12

01100 => 5

01101 => 2

01110 => 5

01111 => 12

10000 => 12

10001 => 5

10010 => 2

10011 => 5

10100 => 12

10101 => 1

10110 => 2

10111 => 5

11000 => 12

11001 => 5

11010 => 12

11011 => 1

11100 => 12

11101 => 5

11110 => 12

11111 => 1

000000 => 1

000001 => 20

000010 => 11

000011 => 20

000100 => 3

000101 => 20

000110 => 11

000111 => 20

001000 => 3

001001 => 1

001010 => 11

001011 => 20

001100 => 3

001101 => 20

001110 => 11

001111 => 20

010000 => 11

010001 => 3

010010 => 1

010011 => 20

010100 => 11

010101 => 1

010110 => 11

010111 => 20

011000 => 11

011001 => 3

011010 => 11

011011 => 1

011100 => 11

011101 => 3

011110 => 11

011111 => 20

100000 => 20

100001 => 11

100010 => 3

100011 => 11

100100 => 1

100101 => 11

100110 => 3

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Description

The number of binary words with the same proper border set.
The proper border set of a binary word $w$ is the set of proper prefixes which are also suffixes of $w$.
For example, $0010000010$, $0010100010$ and $0010110010$ are the words with proper border set $\{0, 0010\}$, whereas $0010010010$ has proper border set $\{0, 0010, 0010010\}$.

Code

@cached_function
def proper_border_set(w):
    B = []
    for i in range(1, len(w)):
        if w[:i] == w[-i:]:
            B.append(w[:i])
    return B

def statistic(w):
    B = proper_border_set(w)
    return sum(1 for u in Words([0,1], len(w)) if B == proper_border_set(u))

Created

Dec 21, 2022 at 11:43 by Martin Rubey

Updated

Dec 21, 2022 at 11:43 by Martin Rubey