A199711 Triangular array: T(n,k) gives the number of numerical semigroups of genus n and multiplicity k, (n>=1, k>=2).

1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 4, 4, 1, 1, 3, 6, 7, 5, 1, 1, 3, 7, 10, 11, 6, 1, 1, 3, 9, 13, 17, 16, 7, 1, 1, 4, 11, 16, 27, 28, 22, 8, 1, 1, 4, 13, 22, 37, 44, 44, 29, 9, 1, 1, 4, 15, 24, 49, 64, 72, 66, 37, 10, 1, 1, 5, 18, 32, 66, 85, 116, 116, 95, 46, 11, 1

Offset: 1

Author

Peter Bala, Nov 09 2011

Keywords

• nonn
• tabl

Comments

A numerical semigroup is a subset S of N, the nonnegative integers, that is closed under addition, contains the element 0 and such that N-S is finite. The cardinality of N-S is called the genus of S. The least positive integer belonging to S is called the multiplicity of S. The number of numerical semigroups of genus n is A007323(n).

Examples

			Triangle begins
.n\k.|..2....3....4....5....6....7....8....9...10
= = = = = = = = = = = = = = = = = = = = = = = = =
..1..|..1
..2..|..1....1
..3..|..1....2....1
..4..|..1....2....3....1
..5..|..1....2....4....4....1
..6..|..1....3....6....7....5....1
..7..|..1....3....7...10...11....6....1
..8..|..1....3....9...13...17...16....7....1
..9..|..1....4...11...16...27...28...22....8....1
...
T(3,3) = 2: The two numerical semigroups of genus 3 and multiplicity 3 are S = N - {1,2,4} and S = N - {1,2,5}.
		

Links

• V. Blanco, P. A. Garcia-Sanchez and J. Puerto, Computing the number of numerical semigroups using generating functions, arXiv:0901.1228v3 [math.CO], 2009.
• Nathan Kaplan, Counting Numerical Semigroups, arXiv:1707.02551 [math.CO], 2017.

Crossrefs

Cf. A007323 (row sums).