kgoto
October 25, 2021
68

# 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

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

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