A086163 Number of monomial ideals in two variables x, y that are Artinian, integrally closed, of colength n and contain x^4.

1, 1, 2, 3, 4, 6, 7, 7, 10, 13, 13, 16, 18, 19, 23, 27, 28, 32, 36, 39, 43, 48, 50, 56, 61, 65, 71, 77, 81, 90, 95, 100, 108, 116, 121, 132, 139, 145, 156, 167, 172, 185, 194, 202, 215, 228, 235, 250, 262, 273, 287, 302, 311, 329, 343, 356, 373, 390, 402, 424, 439, 454

Offset: 0

Author

Jan Snellman, Aug 25 2003

Keywords

• nonn
• easy

Comments

Alternatively, "concave partitions" of n with at most 4 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

• 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.

Crossrefs

Cf. A084913, A086162, A086163.

Programs

Mathematica

CoefficientList[ Series[ (1 + t^2 + t^4 + t^5 - t^6 - t^7 + 2*t^9 - 2*t^10 - t^11 - 2*t^12 + 2*t^13 - t^14 - t^15 + t^16 + t^17 + t^18 - t^19) / ((1 - t)*(1 - t^3)*(1 - t^6)*(1 - t^10)), {t, 0, 65}], t]

Formula

G.f.: (1 + t^2 + t^4 + t^5 - t^6 - t^7 + 2*t^9 - 2*t^10 - t^11 - 2*t^12 + 2*t^13 - t^14 - t^15 + t^16 + t^17 + t^18 - t^19)/((1-t)*(1-t^3)*(1-t^6)*(1-t^10)).

Extensions

More terms from Robert G. Wilson v, Aug 27 2003