@article{LABELLE1989269,
    Abstract = {We describe the close relationship between permutation groups and combinatorial species (introduced by A. Joyal, Adv. in Math. 42, 1981, 1--82). There is a bijection Φ between the set of transitive actions (up to isomorphism) of Sn on finite sets and the set of ``molecular'' species of degree n (up to isomorphism). This bijection extends to a ring isomorphism between B(Sn) (the Burnside ring of the symmetric group) and the ring V Sn (of virtual species of degree n).Since permutation groups are well known (and often studied using computers) this helps in finding examples and counterexamples in species. The cycle index series of a molecular species, which is hard to compute directly, is proved to be simply the (P{\'o}lya) cycle polynomial of the corresponding permutation group. Conversely, several operations which are hard to define in ΠnB(Sn) have a natural description in terms of species. Both situations are extended to coefficients in {$\lambda$}-rings and binomial rings in the last section.},
    Author = {Labelle, J and Yeh, Y.N},
    File = {The relation between burnside rings and combinatorial species - 1-s2.0-0097316589900198-main - a.pdf},
    ISSN = {0097-3165},
    Journal = {Journal of Combinatorial Theory, Series A},
    Number = {2},
    Pages = {269-284},
    Title = {The relation between burnside rings and combinatorial species},
    URL = {https://www.sciencedirect.com/science/article/pii/0097316589900198},
    Volume = {50},
    Year = {1989},
    bdsk-url-1 = {https://www.sciencedirect.com/science/article/pii/0097316589900198},
    bdsk-url-2 = {https://doi.org/10.1016/0097-3165(89)90019-8},
    date-added = {2023-02-20 12:04:39 +0100},
    date-modified = {2023-02-20 12:04:39 +0100},
    doi = {10.1016/0097-3165(89)90019-8}
}

@article{LABELLE1989269, Abstract = {We describe the close relationship between permutation groups and combinatorial species (introduced by A. Joyal, Adv. in Math. 42, 1981, 1--82). There is a bijection Φ between the set of transitive actions (up to isomorphism) of Sn on finite sets and the set of ``molecular'' species of degree n (up to isomorphism). This bijection extends to a ring isomorphism between B(Sn) (the Burnside ring of the symmetric group) and the ring V Sn (of virtual species of degree n).Since permutation groups are well known (and often studied using computers) this helps in finding examples and counterexamples in species. The cycle index series of a molecular species, which is hard to compute directly, is proved to be simply the (P{\'o}lya) cycle polynomial of the corresponding permutation group. Conversely, several operations which are hard to define in ΠnB(Sn) have a natural description in terms of species. Both situations are extended to coefficients in {$\lambda$}-rings and binomial rings in the last section.}, Author = {Labelle, J and Yeh, Y.N}, File = {The relation between burnside rings and combinatorial species - 1-s2.0-0097316589900198-main - a.pdf}, ISSN = {0097-3165}, Journal = {Journal of Combinatorial Theory, Series A}, Number = {2}, Pages = {269-284}, Title = {The relation between burnside rings and combinatorial species}, URL = {https://www.sciencedirect.com/science/article/pii/0097316589900198}, Volume = {50}, Year = {1989}, bdsk-url-1 = {https://www.sciencedirect.com/science/article/pii/0097316589900198}, bdsk-url-2 = {https://doi.org/10.1016/0097-3165(89)90019-8}, date-added = {2023-02-20 12:04:39 +0100}, date-modified = {2023-02-20 12:04:39 +0100}, doi = {10.1016/0097-3165(89)90019-8} }