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# Many Words with Many Palindromes

This slide is about Section 13 in the book 125 problems in Text Algorithms. October 25, 2021

## Transcript

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
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
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
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
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
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
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
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