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%I #29 Feb 12 2023 11:06:37
%S 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,
%T 34,35,35,41,41,41,47,48,48,55,55,55,62,63,63,71,71,71,79,80,80,89,89,
%U 89,98,99,99,109,109,109,119,120,120,131,131,131,142,143,143,155
%N Number of monomial ideals in two variables x, y that are Artinian, integrally closed, of colength n and contain x^3.
%C 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.
%D G. E. Andrews, The Theory of Partitions, Addison-Wesley Publishing Company, 1976.
%D M. Paulsen and J. Snellman, Enumerativa egenskaper hos konkava partitioner (in Swedish), Department of Mathematics, Stockholm University.
%H Robert Israel, <a href="/A086162/b086162.txt">Table of n, a(n) for n = 0..10000</a>
%H V. Crispin Quinonez, <a href="https://www2.math.su.se/reports/2002/7/2002-7.pdf">Integrally closed monomial ideals and powers of ideals</a>, Research Reports in Mathematics Number 7 2002, Department of Mathematics, Stockholm University.
%H Jan Snellman and Michael Paulsen, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Snellman/snellman2.html">Enumeration of Concave Integer Partitions</a>, J. Integer Seq., Vol. 7 (2004), Article 04.1.3.
%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,-1,0,1,-1,0,-1,1).
%F G.f.: (1+t^2+t^5-2*t^6-t^8+t^9)/((1-t)*(1-t^3)*(1-t^6)).
%p 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):
%p map(f, [$0..100]); # _Robert Israel_, May 22 2015
%t 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 *)
%o (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
%Y Cf. A084913, A086161, A086163.
%K nonn,easy
%O 0,3
%A Jan Snellman (Jan.Snellman(AT)math.su.se), Aug 25 2003
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