A086162 Number of monomial ideals in two variables x, y that are Artinian, integrally closed, of colength n and contain x^3.

1, 1, 2, 3, 3, 5, 5, 5, 7, 8, 8, 11, 11, 11, 14, 15, 15, 19, 19, 19, 23, 24, 24, 29, 29, 29, 34, 35, 35, 41, 41, 41, 47, 48, 48, 55, 55, 55, 62, 63, 63, 71, 71, 71, 79, 80, 80, 89, 89, 89, 98, 99, 99, 109, 109, 109, 119, 120, 120, 131, 131, 131, 142, 143, 143, 155

Offset: 0

Author

Jan Snellman, Aug 25 2003

Keywords

• nonn
• easy

Comments

Alternatively, "concave partitions" of n with at most 3 parts, where a concave partition is defined by demanding that the monomial ideal, generated by the monomials whose exponents do not lie in the Ferrers diagram of the partition, is integrally closed.

References

• G. E. Andrews, The Theory of Partitions, Addison-Wesley Publishing Company, 1976.
• M. Paulsen and J. Snellman, Enumerativa egenskaper hos konkava partitioner (in Swedish), Department of Mathematics, Stockholm University.

Links

• Robert Israel, Table of n, a(n) for n = 0..10000
• V. Crispin Quinonez, Integrally closed monomial ideals and powers of ideals, Research Reports in Mathematics Number 7 2002, Department of Mathematics, Stockholm University.
• Jan Snellman and Michael Paulsen, Enumeration of Concave Integer Partitions, J. Integer Seq., Vol. 7 (2004), Article 04.1.3.
• Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1,0,1,-1,0,-1,1).

Crossrefs

Cf. A084913, A086161, A086163.

Programs

Maple

f:= gfun:-rectoproc({a(i+10)=a(i)-a(i+1)-a(i+3)+a(i+4)-a(i+6)+a(i+7)+a(i+9), seq(a(i)=[1, 1, 2, 3, 3, 5, 5, 5, 7, 8][i+1],i=0..9)},a(i),remember):
map(f, [$0..100]); # Robert Israel, May 22 2015

Mathematica

LinearRecurrence[{1, 0, 1, -1, 0, 1, -1, 0, -1, 1}, {1, 1, 2, 3, 3, 5, 5, 5, 7, 8}, 60] (* Jean-François Alcover, Aug 16 2022 *)

PARI

Vec((1+t^2+t^5-2*t^6-t^8+t^9)/((1-t)*(1-t^3)*(1-t^6)) + O(t^80)) \\ Michel Marcus, May 22 2015

Formula

G.f.: (1+t^2+t^5-2*t^6-t^8+t^9)/((1-t)*(1-t^3)*(1-t^6)).