The MonotonicFactors( w ) command produces the monotonic factorization of the string w. It returns a list of strings whose catenation is the string w, such that each string in the list is maximal monotonic (either increasing or decreasing).

All of the StringTools package commands treat strings as (null-terminated) sequences of $8$-bit (ASCII) characters.  Thus, there is no support for multibyte character encodings, such as unicode encodings.

$\mathrm{with}\left(\mathrm{StringTools}\right)\:$

$\mathrm{MonotonicFactors}\left(''abc''\right)$

$\left[''abc''\right]$

$\mathrm{MonotonicFactors}\left(''abcdedcccbabab''\right)$

$\left[''abcde''\,''dc''\,''ccba''\,''ba''\,''b''\right]$

$\mathrm{MonotonicFactors}\left(\mathrm{ThueMorse}\left(60\right)\right)$

$\left[''01''\,''10''\,''10''\,''01''\,''10''\,''01''\,''01''\,''10''\,''10''\,''01''\,''01''\,''10''\,''01''\,''10''\,''10''\,''01''\,''10''\,''01''\,''01''\,''10''\,''01''\,''10''\,''10''\,''01''\,''01''\,''10''\,''10''\,''01''\,''10''\,''01''\right]$

$\mathrm{MonotonicFactors}\left(\mathrm{Fibonacci}\left(8\right)\right)$

$\left[''01''\,''001''\,''01''\,''001''\,''001''\,''01''\,''001''\,''01''\,''001''\,''001''\,''01''\,''001''\,''001''\right]$