A382018 Number of orbits under the action of the permutation group S(n) on the nonsingular n X n matrices over GF(2).

1, 1, 4, 33, 908, 85411, 28227922, 32597166327

Offset: 0

Author

Søren Fuglede Jørgensen, Mar 12 2025

Keywords

• nonn
• hard
• more

Comments

The action is defined by f.M(i,j)=M(f(i),f(j)).

Equivalently, the number of digraphs on n unlabeled nodes with loops allowed but no more than one arc with the same start and end node with adjacency matrices invertible over GF(2).

Examples

			For n = 2, representatives of the four different orbits are [[1, 0], [0, 1]], [[1, 1], [0, 1]], [[0, 1], [1, 1]], and [[0, 1], [1, 0]].
		

Links

• Jens Emil Christensen, Søren Fuglede Jørgensen, Andreas Pavlogiannis, and Jaco van de Pol, On Exact Sizes of Minimal CNOT Circuits, RC 2025, LNCS, vol 15716, pp. 71-88; arXiv:2503.01467 [quant-ph], 2025, Table 1.

Crossrefs

Cf. A000595, A002884, A224879.