A086161 Number of monomial ideals in two variables x, y that are Artinian, integrally closed, of colength n and contain x^2.

1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 10, 11, 11, 11, 12, 12, 12, 13, 13, 13, 14, 14, 14, 15, 15, 15, 16, 16, 16, 17, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 20, 21, 21, 21, 22, 22, 22, 23, 23, 23, 24, 24, 24, 25, 25, 25

Offset: 0

Author

Jan Snellman, Aug 25 2003

Keywords

• nonn
• easy

Comments

Alternatively, "concave partitions" of n with at most 2 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 Seqs., Vol. 7, 2004.
• Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Crossrefs

Cf. A008620, A084913, A086162, A086163.

Programs

PARI

Vec((1+x^2-x^3)/((1-x)*(1-x^3)) + O(x^80)) \\ Michel Marcus, May 22 2015

Formula

G.f.: (1 + x^2 - x^3)/((1 - x)*(1 - x^3)).

a(n) = A008620(n+1). - R. J. Mathar, Sep 12 2008

E.g.f.: (3*exp(x)*(3 + x) - 2*sqrt(3)*exp(-x/2)*sin(sqrt(3)*x/2))/9. - Stefano Spezia, Feb 11 2023