Is this fact true about anagrammatical substrings?

Question

I'm trying to write an algorithm that finds the number of anagrammatical substrings of a string. For instance, the string "abba" has 4:

(1) "a", "a"

(2) "b", "b"

(3) "ab", "ba"

(4) "abb", "bba"

A fact I'm trying to use to optimize is

If a string has no anagrammatical pairs of substrings of length k, then it has no anagrammatical pairs of substrings of length k+1

Can you confirm whether that's true or not?

Because my algorithm

static int NumAnagrammaticalPairs(string str)
{
    int count = 0; // count of anagrammatical pairs found
    int n = str.Length / 2; // OPTIMIZATION: only need to look through the substrings of half the size or less
    for(int k = 1; k <= n; ++k) 
    {
        // get all substrings of length k
        var subsk = GetSubstrings(str,k).ToList();

        // count the number of anagrammatical pairs
        var indices = Enumerable.Range(0, subsk.Count);
        int anapairs = (from i in indices
                        from j in indices
                        where i < j && IsAnagrammaticalPair(subsk[i], subsk[j])
                        select 1).Count();

        // OPTIMIZATION: if didn't find any anagrammatical pairs in the substrings of length k, 
        // there are no anagrammatical pairs in the substrings of length k+1, so we can exit
        // the loop early
        if(anapairs == 0)
            break;
        else
            count += anapairs;
    }
    return count;       
}

is getting results sliggggtttthhhhly off (usually off by 1) the actual results in the test cases.

Answer 1

That's not the case - abcd and cdab are anagrams of length 4 but you can't find length 3 anagram substrings. Concretely, abcdab will not work, since it contains both abcd and cdab, but no 3 anagrams (from abc, bcd, cda, dab).