thm33 [-t target] alphabet [p1 p2 ...]
where target is sympy (default) or sage and p1, p2, ...
are the patterns/factors to be avoided
Calling thm33 abc cba aa results in SymPy code for counting words over
the alphabet {a,b,c} avoiding cba and aa as factors:
$ thm33 abc cba aa
from sympy import *
x = Symbol('x')
A = Matrix([
[1-3*x,1,1],
[x**5,-x**3*(1+x),-x**2*(x**2)],
[x**5,-x**3*(0),-x**2*(1)]
])
b = Matrix([[1],[0],[0]])
F = A.solve(b)[0]
F = F.factor()
print(F)
print(series(F,n=10))
Executing this script, e.g. by piping through python3, we find the
rational generating function and the first few terms of its Taylor
series:
$ thm33 abc cba aa | python3 -
1/(x**2 - 3*x + 1)
1 + 3*x + 8*x**2 + 21*x**3 + 55*x**4 + 144*x**5 + 377*x**6 + 987*x**7 + 2584*x**8 + 6765*x**9 + O(x**10)
L. J. Guibas and A. M. Odlyzko, String overlaps, pattern matching, and nontransitive games, Journal of Combinatorial Theory, Series A, Volume 30, Issue 2, March 1981, Pages 183-208.