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

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

, 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

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

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

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