Taking M to be the matrix [2] -----> [2] given by
M=[[ab+ba]], the cokernel decomposes into simples as follows:
[coker(M*)] = 1*[1, 1] + 1*[1, 1, 1].
In particular, the dimension is given as a function of n > 0:
(1/2) * (n - 1) * n.
Total computation time: 0.0955851078033 seconds

Taking M to be the matrix [3] -----> [3, 3] given by
M=[[abc-bac, abc+bca+cab]], the cokernel decomposes into simples as follows:
[coker(M*)] = 1*[1, 1] + 1*[2] + 1*[1, 1, 1] + 1*[2, 1].
In particular, the dimension is given as a function of n > 0:
(1/3) * (n - 1) * n * (n + 1).
Total computation time: 0.023894071579 seconds

Taking M to be the matrix [5] -----> [5, 5, 5] given by
M=[[abcde-bacde,abcde-bcdea,aadde-bbcce]], the cokernel decomposes into simples as follows:
[coker(M*)] = 1*[1] + 1*[1, 1] + 1*[2, 1] + 1*[3] + 1*[3, 1] + 1*[4] + 1*[5].
In particular, the dimension is given as a function of n > 0:
(1/120) * n * (n + 2) * (n^3 + 8*n^2 - 41*n + 72).
Total computation time: 0.449846029282 seconds

Taking M to be the matrix [5] -----> [4, 4, 4, 4, 4] given by
M=[[aabcd,abbcd,abccd,abcdd,abcda]], the cokernel decomposes into simples as follows:
[coker(M*)] = 5*[1, 1, 1] + 10*[2, 1] + 5*[3] + 10*[1, 1, 1, 1] + 15*[2, 1, 1] + 10*[2, 2] + 15*[3, 1] + 5*[4] + 6*[1, 1, 1, 1, 1] + 4*[2, 1, 1, 1] + 5*[2, 2, 1] + 6*[3, 1, 1] + 5*[3, 2] + 4*[4, 1] + 1*[5] + 1*[1, 1, 1, 1, 1, 1].
In particular, the dimension is given as a function of n > 0:
(n - 2) * (n - 1) * n * (n^2 - 2*n + 2).
Total computation time: 0.235414981842 seconds

Taking M to be the matrix [5] -----> [4, 4, 4, 4, 4, 5, 5] given by
M=[[aabcd,abbcd,abccd,abcdd,abcda,abcde-bcdea,abcde-edcba]], the cokernel decomposes into simples as follows:
[coker(M*)] = 1*[2, 1] + 1*[3] + 1*[2, 1, 1] + 1*[2, 2] + 2*[3, 1] + 1*[4] + 1*[1, 1, 1, 1, 1] + 1*[2, 2, 1] + 1*[3, 2] + 1*[5] + 1*[1, 1, 1, 1, 1, 1].
In particular, the dimension is given as a function of n > 0:
(1/10) * (n - 2) * (n - 1) * n * (n^2 - 2*n + 2).
Total computation time: 0.601917982101 seconds

Taking M to be the matrix [4] -----> [3, 4, 4, 3] given by
M=[[aabc,abcd-bacd,abcd+cdab,abbc+bcca+caab]], the cokernel decomposes into simples as follows:
[coker(M*)] = 1*[2, 1] + 1*[3, 1].
In particular, the dimension is given as a function of n > 0:
(1/8) * (n - 2) * (n - 1) * (n - 1/3) * n.
Total computation time: 0.17354798317 seconds

Taking M to be the matrix [4] -----> [4, 4, 5] given by
M=[[abcd+bacd,abcd+bcda,abcd+bcde+cdea+deab+eabc]], the cokernel decomposes into simples as follows:
[coker(M*)] = 1*[1, 1, 1, 1].
In particular, the dimension is given as a function of n > 0:
(1/6) * (n - 3) * (n - 2) * (n - 1).
Total computation time: 1.2256231308 seconds