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A086162
Number of monomial ideals in two variables x, y that are Artinian, integrally closed, of colength n and contain x^3.
4
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
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.
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
KEYWORD
nonn,easy
AUTHOR
Jan Snellman (Jan.Snellman(AT)math.su.se), Aug 25 2003
STATUS
approved