A319608 - OEIS

%I #36 Jun 14 2020 12:48:50

%S 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,

%T 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,

%U 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

%N 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.

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

%C Summing rows yields A158206.

%C 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 ).

%H J. De Loera, C. O'Neill, and D. Wilbourne, <a href="https://arxiv.org/abs/1710.00979">Random numerical semigroups and a simplicial complex of irreducible semigroups</a>, arXiv:1710.00979 [math.AC], 2017.

%H C. Leng and C. O'Neill, <a href="https://arxiv.org/abs/1809.09915">A sequence of quasipolynomials arising from random numerical semigroups</a>, arXiv:1809.09915 [math.CO], 2018.

%H C. O'Neill, <a href="https://gist.github.com/coneill-math/c2f12c94c7ee12ac7652096329417b7d">The first 90 rows formatted as a triangle</a>

%e T(13,2) = 2, since {5,6,9} and {7,8,9,10,11,12} minimally generate irreducible numerical semigroups with Frobenius number 13.

%e When written in rows:

%e   1

%e   1

%e   1

%e   1

%e   1,  1

%e   1

%e   1,  2

%e   1,  1

%e   1,  2

%e   1,  2

%e   1,  4,  1

%e   1,  1

%e   1,  5,  2

%e   1,  4,  1

%e   1,  4,  2

%e   1,  4,  2

%e   1,  7,  6,  1

%e   1,  4,  2

%e   1,  8,  9,  2

%e   1,  5,  4,  1

%e   1,  7,  8,  2

%e   1,  8,  9,  2

%e   1, 10, 17,  7,  1

%e   1,  5,  6,  2

%e   1, 10, 19, 12,  2

%e   1, 10, 16,  7,  1

%e   1, 10, 21, 11,  2

%e   1,  9, 16,  9,  2

%e   1, 13, 34, 26,  8,  1

%e   1,  8, 15, 10,  2

%Y Cf. A008615, A158206.

%K nonn,tabf

%O 1,9

%A _Christopher O'Neill_, Sep 24 2018