Experimental symbolics is very successful in discrete math, especially enumerative combinatorics. The natural question arises, how far can Sage help with this branch of symbolics? To this end I present a table of respective mathematical objects and algorithms, and the support Sage has for them. I'm leaning heavily on the recent summary of computer algebra relevant for enumerative combinatorics by Manuel Kauers (published in Bona's new Handbook of Combinatorics).

Computation in/with | Status | Comments |
---|---|---|

Finite fields | ✓ | Documentation |

Lattice reduction | ✓ | Documentation |

Multivariate polynomials | ✓ | Documentation |

Gröbner bases | ✓ | Documentation |

Algebraic number arithmetic | ✓ | Documentation |

Cylindrical Algebraic Decomposition |
✓ | Documentation (from Sage version 6.10.p2 up) |

Formal power series | ✓ | Two implementations, a fast one missing most symbolic function expansions, and a slower one with function expansions, but neglected having many bugs. Both not interoperating.---Documentation1, Documentation2 |

Lazy power series | ✓ | rudimentary---Documentation |

Laurent series | ✗ | Only univariate available |

Puiseux series | ✗ | |

Ore algebras | ✓ | optional package ore-algebra |

C-finite sequences | ✓ | Documentation |

D-finite sequences | ✗ | |

Combinatorial species | ✓ | Documentation |

Omega analysis (partitions) | ✗ | |

Ehrhart theory | ✗ | incomplete, in progress |

Computational group theory | ✓ | available via GAP |

Symbolic summation: Gosper's algorithm | ✓ | part of sum(), available via Maxima |

Zeilberger's algorithm | ✓ | part of sum(), available via Maxima |

Petkovšek's algorithm | ✗ | |

Karr's algorithm | ✗ | |

Creative telescoping | ✗ | |

ΠΣ-theory | ✗ | |

Holonomic functions | ✗ |