// track characters we've seen an odd number of times
Set<Character> unpairedCharacters = new HashSet<>();

for (int i = 0; i < theString.length(); i++) {
    char c = theString.charAt(i);
    if (unpairedCharacters.contains(c)) {
        unpairedCharacters.remove(c);
    } else {
        unpairedCharacters.add(c);
    }
}

// the string has a palindrome permutation if it
// has one or zero characters without a pair
return unpairedCharacters.size() <= 1;