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

0,3

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).

Formula

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

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 *)

Prog

(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

Crossrefs

Cf. A084913, A086161, A086163.

Sequence in context: A362840 A302607 A098567 * A036703 A306253 A117629

Adjacent sequences: A086159 A086160 A086161 * A086163 A086164 A086165

Keyword

nonn,easy

Author

Jan Snellman (Jan.Snellman(AT)math.su.se), Aug 25 2003

Status

approved