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Many Words with Many Palindromes

kgoto
October 25, 2021

Many Words with Many Palindromes

This slide is about Section 13 in the book 125 problems in Text Algorithms.

kgoto

October 25, 2021
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  1. Many Words with Many Palindromes (Section 13)
    Section 13 in 125 problems book.
    The problem in the section is based on
    the following survery.
    A. Glen, J. Justin, S. Widmer and L. Q.
    Zamboni. Palindromic richness. Eur. J.
    Comb., 30(2):510–531, 2009.
    Speaker: @kgoto

    View full-size slide

  2. Example: "poor", "abac" are palindrome rich, and "abca" is not.
    Def. Palindrome Rich
    If a word contains distinct non-empty palindromes,

    is called palindrome rich.
    w ∣w∣
    w
    2
    2

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  3. For each palindrome, mark the beginning position of the right most
    occurrence.
    There are at most one mark for each position.
    (The proof will be given in the next slide)
    Observation

    contains at most distinct palindromes.
    w ∣w∣
    3
    3

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  4. Assume we have two marks for a position, and let , for
    be corresponding distinct palindromes.
    occurs also the suffix of , and it contradicts that is the right
    most occurrence.
    Observation

    contains at most distinct palindromes.
    w ∣w∣
    s p ∣s∣ < ∣p∣
    s p s
    4
    4

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  5. The answer is yes.
    We use the fact that the number of partitions of integer grows
    exponentially.
    Def. Rich Word

    is the number of rich words of length on alphabet of size
    .
    Rich

    (n)
    k
    n k
    Observation

    For binary words , we have the following,
    for
    for

    ( is this trivial?)
    k = 2
    Rich (n) =
    2
    2n n < 8
    Rich (n) <
    2
    2n n ≥ 8
    Question
    Does grow exponentially?
    (Same) is there positive constant for which ?
    Rich (2n)
    2
    c Rich

    (2n) ≥
    2
    2cn
    5
    5

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  6. Consider of length .
    Example: for

    These are right most occurrences of each palindrome in .

    Def. Par tition of Integer

    The partition of integer is a tuple of integers

    such that , and .
    n π = (n , … , n

    )
    1 k
    n

    <
    1
    n

    <
    2
    ⋯ < n

    k
    n =

    n

    ∑i i
    w

    =
    π a ba b ⋯ ba
    n

    1 n

    2 n

    k n + k − 1
    π = (1, 2, 3, 5, 6) n = 17
    w

    π
    6
    6

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  7. So, is palindrome rich.
    Consider of length .
    has aditional palindromes of the form and for .

    Their numbers are and .
    So the # of total distinct palindromes in is .
    is also rich.
    w

    π
    v

    =
    π w

    b
    π
    n−k+1 2n
    v

    π ba b
    n

    k bx x > 1
    1 n − k
    v

    π 2n
    v

    π
    Fact

    # of partitions an integer into distinct positive integers grows
    exponentially with .
    n
    n
    Answer of Question

    Q: Does grow exponentially?
    A: Yes, also grows exponentially.
    Rich (2n)
    2
    Rich

    (2n)
    2
    7
    7

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  8. Reference
    The problem is based on the survery by Glen et al.
    A. Glen, J. Justin, S. Widmer and L. Q. Zamboni. Palindromic richness.
    Eur. J. Comb., 30(2):510–531, 2009.
    8
    8

    View full-size slide