Abstract
Combinatorial power series are formal power series of the form \(\sum c_{n,H}X^n/H \) where, for each n, H runs through subgroups of the symmetric group \(S_n\) and the coefficients \(c_{n,H}\) are complex numbers (or ordinary power series involving some “weight variables”). Such series conveniently encode species of combinatorial (possibly weighted) structures according to their stabilizers (up to conjugacy). We give general lines for expressing these kinds of series – as well as the main operations \((+,\cdot ,\times ,\circ ,d/dX)\) between them – by making use of the GroupTheory package and give suggestions for possible extensions of that package and some other specific procedures such as collect, expand, series, etc. An analysis of multivariable combinatorial power series is also presented.
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Notes
- 1.
Two subgroups G of \(S_n\) and H of \(S_m\), with \(n \ne m\), are always considered to be be different, even if they consist of the "same" permutations.
- 2.
Technically, such classes are species in the sense of Joyal [2]. A species is an endofunctor F of the category of finite sets with bijections as morphisms. For each finite set U, each \(s \in F[U]\) is called an F-structure on U and for each bijection \(\beta : U \rightarrow V\), the bijection \(F[\beta ]:F[U]\rightarrow F[V]\) is said to “relabel” (or “transport”) each F-structure s on U to an isomorphic F-structure \(t=F[\beta ](s)\) on V..
- 3.
This kind of series was introduced by Yeh [5] to deal with species.
References
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Labelle, G. (2020). Classifying Discrete Structures by Their Stabilizers. In: Gerhard, J., Kotsireas, I. (eds) Maple in Mathematics Education and Research. MC 2019. Communications in Computer and Information Science, vol 1125. Springer, Cham. https://doi.org/10.1007/978-3-030-41258-6_30
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DOI: https://doi.org/10.1007/978-3-030-41258-6_30
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