A217990 Size of largest semigroup generated by one Boolean n X n matrix.

1, 2, 5, 10, 17, 26, 37, 50, 65, 82, 101, 122, 145, 170, 197, 226, 257, 290, 420

Offset: 1

Author

Jeffrey Shallit, Oct 17 2012

Keywords

• nonn
• hard
• more

Examples

			a(4) = 10: the matrix [[0,0,0,1], [0,0,1,0], [1,0,0,0], [0,1,1,0]] generates a semigroup of order 10 under Boolean matrix multiplication.
		

References

• J. Denes, K. H. Kim, and F. W. Roush, Automata on one symbol, in Studies in Pure Mathematics: To the Memory of Paul Turan, Birkhäuser, 1983, pp. 127-134.

Links

• George Markowsky, Bounds on the index and period of a binary relation on a finite set, Semigroup Forum 13 (1977), 253-259.

Crossrefs

Cf. A000793, A202140.

Formula

For n < 19, a(n) = n^2 - 2n + 2. For large n, l(n) <= a(n) < n^2 - 2n + 2 + l(n), where l(n) is Landau's function (A000793). It seems the exact value is still unknown.