- FindStat

Identifier

St000519: Binary words ⟶ ℤ

Values

0 => 0

1 => 0

00 => 1

01 => 1

10 => 1

11 => 1

000 => 2

001 => 2

010 => 2

011 => 2

100 => 2

101 => 2

110 => 2

111 => 2

0000 => 3

0001 => 3

0010 => 2

0011 => 2

0100 => 2

0101 => 3

0110 => 2

0111 => 3

1000 => 3

1001 => 2

1010 => 3

1011 => 2

1100 => 2

1101 => 2

1110 => 3

1111 => 3

00000 => 4

00001 => 4

00010 => 3

00011 => 3

00100 => 3

00101 => 3

00110 => 2

00111 => 3

01000 => 3

01001 => 3

01010 => 4

01011 => 3

01100 => 2

01101 => 3

01110 => 3

01111 => 4

10000 => 4

10001 => 3

10010 => 3

10011 => 2

10100 => 3

10101 => 4

10110 => 3

10111 => 3

11000 => 3

11001 => 2

11010 => 3

11011 => 3

11100 => 3

11101 => 3

11110 => 4

11111 => 4

000000 => 5

000001 => 5

000010 => 4

000011 => 4

000100 => 3

000101 => 3

000110 => 3

000111 => 3

001000 => 3

001001 => 4

001010 => 4

001011 => 3

001100 => 3

001101 => 3

001110 => 3

001111 => 4

010000 => 4

010001 => 3

010010 => 4

010011 => 3

010100 => 4

010101 => 5

010110 => 3

010111 => 3

011000 => 3

011001 => 3

011010 => 3

011011 => 4

011100 => 3

011101 => 3

011110 => 4

011111 => 5

100000 => 5

100001 => 4

100010 => 3

100011 => 3

100100 => 4

100101 => 3

100110 => 3

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Description

The largest length of a factor maximising the subword complexity.
Let $p_w(n)$ be the number of distinct factors of length $n$. Then the statistic is the largest $n$ such that $p_w(n)$ is maximal:
$$ H_w = \max\{n: p_w(n)\text{ is maximal}\} $$
A related statistic is the number of distinct factors of arbitrary length, also known as subword complexity, St000294The number of distinct factors of a binary word..

References

[1] Berstel, J. 50ieme Seminaire Lotharingien de Combinatoire, www-igm.univ-mlv.fr/~berstel/Exposes/2003-03-23Lotharingien.pdf

Code

def statistic(w):
    m = max(len(w.factor_set(n=n)) for n in range(len(w)))
    return max(n for n in range(len(w)) if len(w.factor_set(n=n)) == m)

Created

Jun 01, 2016 at 09:54 by Martin Rubey

Updated

Jun 07, 2016 at 19:55 by Martin Rubey