## Example

\(
\def\sa#1{\tt{#1}}
\)

The words $\sa{abba}$ and $\sa{baab}$ are both palindromes
and conjugate of each other. On the contrary, the word $\sa{aabaa}$ has no
other conjugate palindrome, that is to say, its conjugacy class contains only
one palindrome.

The conjugacy class of $\sa{abba}$, set
$\{\sa{abba},\sa{bbaa},\sa{baab},\sa{aabb}\}$, contains only two palindromes.
This is also the case for the word $(\sa{abba})^3$ whose conjugacy class
contains $\sa{abbaab}\,\sa{baabba}$ and $\sa{baabba}\,\sa{abbaab}$, two
palindromes.
But the conjugacy class of $(\sa{abba})^2$ has only one
palindrome among its four conjugates.

#
Problem 12: Conjugate Palindromes

The problem is related to the two operations on words consisting of reversing
a word and taking one of its conjugate.
The operations are essentially incompatible in the sense that only
a few conjugates of a word are also its reverse.

To examine the situation, we consider palindromes that are conjugates of each
other.

What is the maximal number of palindromes in the conjugacy class of a word?

Consider the primitive root of two conjugate palindromes.

## References

C. Guo, J. Shallit, and A. M. Shur. On the combinatorics of palindromes
and antipalindromes. CoRR, abs/1503.09112, 2015.