- FindStat

Identifier

St000630: Binary words ⟶ ℤ

Values

0 => 1

1 => 1

00 => 1

01 => 2

10 => 2

11 => 1

000 => 1

001 => 2

010 => 1

011 => 2

100 => 2

101 => 1

110 => 2

111 => 1

0000 => 1

0001 => 2

0010 => 2

0011 => 2

0100 => 2

0101 => 2

0110 => 1

0111 => 2

1000 => 2

1001 => 1

1010 => 2

1011 => 2

1100 => 2

1101 => 2

1110 => 2

1111 => 1

00000 => 1

00001 => 2

00010 => 2

00011 => 2

00100 => 1

00101 => 2

00110 => 2

00111 => 2

01000 => 2

01001 => 2

01010 => 1

01011 => 2

01100 => 2

01101 => 2

01110 => 1

01111 => 2

10000 => 2

10001 => 1

10010 => 2

10011 => 2

10100 => 2

10101 => 1

10110 => 2

10111 => 2

11000 => 2

11001 => 2

11010 => 2

11011 => 1

11100 => 2

11101 => 2

11110 => 2

11111 => 1

000000 => 1

000001 => 2

000010 => 2

000011 => 2

000100 => 2

000101 => 2

000110 => 2

000111 => 2

001000 => 2

001001 => 2

001010 => 2

001011 => 3

001100 => 1

001101 => 3

001110 => 2

001111 => 2

010000 => 2

010001 => 2

010010 => 1

010011 => 3

010100 => 2

010101 => 2

010110 => 3

010111 => 2

011000 => 2

011001 => 3

011010 => 3

011011 => 2

011100 => 2

011101 => 2

011110 => 1

011111 => 2

100000 => 2

100001 => 1

100010 => 2

100011 => 2

100100 => 2

100101 => 3

100110 => 3

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Description

The length of the shortest palindromic decomposition of a binary word.
A palindromic decomposition (paldec for short) of a word $w=a_1,\dots,a_n$ is any list of factors $p_1,\dots,p_k$ such that $w=p_1\dots p_k$ and each $p_i$ is a palindrome, i.e. coincides with itself read backwards.

References

[1] მამუკა ჯიბლაძე Combinatorics of palindromic decompositions MathOverflow:201205
[2] Ravsky, A. On the palindromic decomposition of binary words arXiv:1004.1278

Code

def statistic(w):
    return min(len(p) for p in paldecs(w))

def paldecs(w):
    """
    Return all distinct palindromic decompositions of w.

    sage: w = Word([1,1])
    sage: paldecs(w)
    [[word: 1, word: 1], [word: 11]]

    sage: w = Word([1,0,1])
    sage: paldecs(w)
    [[word: 1, word: 0, word: 1], [word: 101]]

    sage: w = Word([1,0,1,0,0,1])
    sage: paldecs(w)
    [[word: 1, word: 0, word: 1, word: 0, word: 0, word: 1],
     [word: 1, word: 0, word: 1, word: 00, word: 1],
     [word: 1, word: 0, word: 1001],
     [word: 1, word: 010, word: 0, word: 1],
     [word: 101, word: 0, word: 0, word: 1],
     [word: 101, word: 00, word: 1]]

    sage: w = Word("referee")
    sage: paldecs(w)
    [[word: r, word: e, word: f, word: e, word: r, word: e, word: e],
     [word: r, word: e, word: f, word: e, word: r, word: ee],
     [word: r, word: e, word: f, word: ere, word: e],
     [word: r, word: efe, word: r, word: e, word: e],
     [word: r, word: efe, word: r, word: ee],
     [word: refer, word: e, word: e],
     [word: refer, word: ee]]

    """
    if len(w) == 0:
        return [[]]
    P1 = w.palindrome_prefixes()
    result = []
    for p1 in P1:
        l = len(p1)
        if l > 0:
            P2 = paldecs(w[l:])
            for p2 in P2:
                result.append([p1] + p2)
    return result

Created

Oct 16, 2016 at 21:19 by Martin Rubey

Updated

Dec 30, 2017 at 22:39 by Martin Rubey