IntegerVectorsModPermutationGroup

Higher Edu - Research dev card
Development from the higher education and research community
  • Creation or important update: 08/09/13
  • Minor correction: 08/09/13

IntegerVectorsModPermutationGroup : enumeration up to the action of a permutation group

This software was developed (or is under development) within the higher education and research community. Its stability can vary (see fields below) and its working state is not guaranteed.
  • Web site
  • System:
  • License(s): GPL
  • Status: stable release
  • Support: maintained, ongoing development
  • Designer(s): Nicolas Borie
  • Contact designer(s): nicolas.borie@univ-mlv.fr
  • Laboratory, service:

 

General software features

IntegerVectorsModPermutationGroup is an enumeration engine of integer vectors up to the action of a permutation group.

Let n a positif integer and G a permutation group, subgroup of the symmetric group of order n. This Sage module IntegerVectorsModPermutationGroup allows to enumerate tuples of length n modulo the action by position of G. This problem generalizes the enumeration of unlabelled graphs up to an isomorphism. One can also add some constraints like the sum of the entries or their maximum size.

This module is completly integrated in Sage since the version 4.7.

Exemple

Exemple for the cyclic group over 4 elements:

sage: G = PermutationGroup([[(1,2,3,4)]]); G
Permutation Group with generators [(1,2,3,4)]
sage: G.cardinality()
4
sage: S = IntegerVectorsModPermutationGroup(G); S
Integer vectors of length 4 enumerated up to the action of Permutation Group with generators [(1,2,3,4)]
sage: S.cardinality()
+Infinity
sage: it = iter(S)
sage: for i in range(25): v = it.next(); print v, " : ", S.orbit(v)
....:
[0, 0, 0, 0]  :  set([[0, 0, 0, 0]])
[1, 0, 0, 0]  :  set([[1, 0, 0, 0], [0, 0, 1, 0], [0, 1, 0, 0], [0, 0, 0, 1]])
[2, 0, 0, 0]  :  set([[2, 0, 0, 0], [0, 0, 2, 0], [0, 0, 0, 2], [0, 2, 0, 0]])
[1, 1, 0, 0]  :  set([[1, 0, 0, 1], [0, 0, 1, 1], [1, 1, 0, 0], [0, 1, 1, 0]])
[1, 0, 1, 0]  :  set([[0, 1, 0, 1], [1, 0, 1, 0]])
[3, 0, 0, 0]  :  set([[0, 0, 3, 0], [0, 3, 0, 0], [3, 0, 0, 0], [0, 0, 0, 3]])
[2, 1, 0, 0]  :  set([[0, 2, 1, 0], [0, 0, 2, 1], [1, 0, 0, 2], [2, 1, 0, 0]])
[2, 0, 1, 0]  :  set([[0, 1, 0, 2], [0, 2, 0, 1], [1, 0, 2, 0], [2, 0, 1, 0]])
[2, 0, 0, 1]  :  set([[2, 0, 0, 1], [0, 1, 2, 0], [1, 2, 0, 0], [0, 0, 1, 2]])
[1, 1, 1, 0]  :  set([[1, 1, 1, 0], [1, 1, 0, 1], [1, 0, 1, 1], [0, 1, 1, 1]])
[4, 0, 0, 0]  :  set([[4, 0, 0, 0], [0, 4, 0, 0], [0, 0, 4, 0], [0, 0, 0, 4]])
[3, 1, 0, 0]  :  set([[0, 0, 3, 1], [1, 0, 0, 3], [0, 3, 1, 0], [3, 1, 0, 0]])
[3, 0, 1, 0]  :  set([[0, 3, 0, 1], [0, 1, 0, 3], [3, 0, 1, 0], [1, 0, 3, 0]])
[3, 0, 0, 1]  :  set([[0, 0, 1, 3], [3, 0, 0, 1], [0, 1, 3, 0], [1, 3, 0, 0]])
[2, 2, 0, 0]  :  set([[0, 2, 2, 0], [2, 2, 0, 0], [2, 0, 0, 2], [0, 0, 2, 2]])
[2, 1, 1, 0]  :  set([[2, 1, 1, 0], [1, 0, 2, 1], [1, 1, 0, 2], [0, 2, 1, 1]])
[2, 1, 0, 1]  :  set([[0, 1, 2, 1], [1, 0, 1, 2], [2, 1, 0, 1], [1, 2, 1, 0]])
[2, 0, 2, 0]  :  set([[2, 0, 2, 0], [0, 2, 0, 2]])
[2, 0, 1, 1]  :  set([[1, 2, 0, 1], [2, 0, 1, 1], [0, 1, 1, 2], [1, 1, 2, 0]])
[1, 1, 1, 1]  :  set([[1, 1, 1, 1]])
[5, 0, 0, 0]  :  set([[0, 0, 0, 5], [5, 0, 0, 0], [0, 5, 0, 0], [0, 0, 5, 0]])
[4, 1, 0, 0]  :  set([[0, 0, 4, 1], [1, 0, 0, 4], [0, 4, 1, 0], [4, 1, 0, 0]])
[4, 0, 1, 0]  :  set([[0, 4, 0, 1], [1, 0, 4, 0], [0, 1, 0, 4], [4, 0, 1, 0]])
[4, 0, 0, 1]  :  set([[4, 0, 0, 1], [1, 4, 0, 0], [0, 0, 1, 4], [0, 1, 4, 0]])
[3, 2, 0, 0]  :  set([[3, 2, 0, 0], [0, 0, 3, 2], [2, 0, 0, 3], [0, 3, 2, 0]])

Context in which the software is used

The development of a such engine was necessary for the thesis work of the author. The thesis is about effective invariant theory. This module is also usefull in the following fields:

  • Effective invariant theory,
  • Effective Galois theory,
  • Structure Species theory.
Publications related to the software