A086162 -id:A086162 - cp's OEIS frontend

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

Jan Snellman, Aug 25 2003

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.

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.

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.

Cf. A084913, A086162, A086163.

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]

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)).

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

A084913 Number of monomial ideals in two variables that are Artinian, integrally closed and of colength n.

Original entry on oeis.org

1, 2, 3, 4, 7, 9, 11, 17, 23, 28, 39, 48, 59, 79, 100, 121, 152, 185, 225, 280, 338, 404, 492, 584, 696, 835, 983, 1162, 1385, 1612

Offset: 0

Jan Snellman and Michael Paulsen, Jul 03 2003

Alternatively, "concave partitions" of n, 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.

			a(4) = 4 because the Artinian monomial ideals in two variables that have colength 4 are (x^4,y), (x^3,y^2), (x^2, y^2), (x^2,xy,y^3), (x,y^4), corresponding to the partitions (1,1,1,1), (3,1), (2,2), (2,1,1), (4); the ideal (x^2,y^2) is not integrally closed, hence the partition (2,2) is not concave.

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

V. Crispin Quinonez, Integrally closed monomial ideals and powers of ideals, Research Reports in Mathematics Number 7 2002, Department of Mathematics, Stockholm University.

Cf. A086161, A086162, A086163.

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

Original entry on oeis.org

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

Jan Snellman, Aug 25 2003

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.

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.

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).

Cf. A008620, A084913, A086162, A086163.

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

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

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

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A084913 Number of monomial ideals in two variables that are Artinian, integrally closed and of colength n.

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

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