A319608 Irregular triangle read by rows: T(n,k) is the number of irreducible numerical semigroups with Frobenius number n and k minimal generators less than n/2.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 4, 1, 1, 1, 1, 5, 2, 1, 4, 1, 1, 4, 2, 1, 4, 2, 1, 7, 6, 1, 1, 4, 2, 1, 8, 9, 2, 1, 5, 4, 1, 1, 7, 8, 2, 1, 8, 9, 2, 1, 10, 17, 7, 1, 1, 5, 6, 2, 1, 10, 19, 12, 2, 1, 10, 16, 7, 1, 1, 10, 21, 11, 2, 1, 9, 16, 9, 2, 1, 13, 34, 26, 8, 1, 1, 8, 15, 10, 2, 1, 14, 41, 37, 14, 2

Offset: 1

Author

Christopher O'Neill, Sep 24 2018

Keywords

• nonn
• tabf

Comments

The length of each row is floor((n+1)/2) - floor(n/3).

Summing rows yields A158206.

The expected number of minimal generators of a randomly selected numerical semigroup S(M,p) equals Sum_{n=1..M} ( p * (1 - p)^(floor(n/2)) * Product_{k>=0} T(n,k)*p^k ).

Examples

			T(13,2) = 2, since {5,6,9} and {7,8,9,10,11,12} minimally generate irreducible numerical semigroups with Frobenius number 13.
When written in rows:
  1
  1
  1
  1
  1,  1
  1
  1,  2
  1,  1
  1,  2
  1,  2
  1,  4,  1
  1,  1
  1,  5,  2
  1,  4,  1
  1,  4,  2
  1,  4,  2
  1,  7,  6,  1
  1,  4,  2
  1,  8,  9,  2
  1,  5,  4,  1
  1,  7,  8,  2
  1,  8,  9,  2
  1, 10, 17,  7,  1
  1,  5,  6,  2
  1, 10, 19, 12,  2
  1, 10, 16,  7,  1
  1, 10, 21, 11,  2
  1,  9, 16,  9,  2
  1, 13, 34, 26,  8,  1
  1,  8, 15, 10,  2
		

Links

• J. De Loera, C. O'Neill, and D. Wilbourne, Random numerical semigroups and a simplicial complex of irreducible semigroups, arXiv:1710.00979 [math.AC], 2017.
• C. Leng and C. O'Neill, A sequence of quasipolynomials arising from random numerical semigroups, arXiv:1809.09915 [math.CO], 2018.
• C. O'Neill, The first 90 rows formatted as a triangle

Crossrefs

Cf. A008615, A158206.