  Two strings x and y are said to be Abelian equivalent if x is a permutation
of y, or vice versa. If a string z satisfies z = xy with x and y being Abelian
equivalent, then z is said to be an Abelian square. If a string w can be
factorized into a sequence v_1,...,v_s of strings such that v_1 ,..., v_{s-1}
are all Abelian equivalent and vs is a substring of a permutation of v_1, then
w is said to have a regular Abelian period (p,t) where p = |v_1| and t = |v_s|.
If a substring w_1[i..i+l-1] of a string w_1 and a substring w_2[j..j+l-1] of
another string w_2 are Abelian equivalent, then the substrings are said to be a
common Abelian factor of w_1 and w_2 and if the length l is the maximum of such
then the substrings are said to be a longest common Abelian factor of w_1 and
w_2. We propose efficient algorithms which compute these Abelian regularities
using the run length encoding (RLE) of strings. For a given string w of length
n whose RLE is of size m, we propose algorithms which compute all Abelian
squares occurring in w in O(mn) time, and all regular Abelian periods of w in
O(mn) time. For two given strings w_1 and w_2 of total length n and of total
RLE size m, we propose an algorithm which computes all longest common Abelian
factors in O(m^2n) time.
