def statistic(pi):
    ct = pi.cycle_tuples()
    one = None # cycle containing 1
    two = 0    # other cycles of length at least 2
    fix = 0    # other fixed points
    for c in ct:
        if one is None and 1 in c:
            one = True
        elif len(c) > 1:
            two += 1
        else:
            fix += 1

    return len(pi) + two - fix - 1

# alternative code for checking the lemma
def statistic_alt1(pi):
    def gens(n):
        S = Permutations(n)
        return [S(Permutation((1, i))) for i in range(2, n+1)]

    G = PermutationGroup(gens(len(pi)))
    pi = G(pi)
    w = gap.Factorization(G._gap_(), pi._gap_())
    return w.Length()