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

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

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

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

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

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

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

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