def _cis(self, series_ring, base_ring): r""" The cycle index series for the species of partitions is given by .. math:: exp \sum_{n \ge 1} \frac{1}{n} \left( exp \left( \sum_{k \ge 1} \frac{x_{kn}}{k} \right) -1 \right). EXAMPLES:: sage: P = species.PartitionSpecies() sage: g = P.cycle_index_series() sage: g.coefficients(5) [p[], p[1], p[1, 1] + p[2], 5/6*p[1, 1, 1] + 3/2*p[2, 1] + 2/3*p[3], 5/8*p[1, 1, 1, 1] + 7/4*p[2, 1, 1] + 7/8*p[2, 2] + p[3, 1] + 3/4*p[4]] """ ciset = SetSpecies().cycle_index_series(base_ring) res = ciset.composition(ciset - 1) if self.is_weighted(): res *= self._weight return res