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%I #6 Feb 01 2020 07:09:30
%S 1,1,2,3,6,14,30,68,157,379,901,2229,5488,13846,34801,89368,228186,
%T 592943,1533511,4026833
%N Number of locally disjoint enriched identity p-trees of weight n.
%C A locally disjoint enriched identity p-tree of weight n is either the number n itself or a finite sequence of distinct non-overlapping locally disjoint enriched identity p-trees whose weights are weakly decreasing and sum to n.
%e The a(1) = 1 through a(6) = 14 enriched p-trees:
%e 1 2 3 4 5 6
%e (21) (31) (32) (42)
%e ((21)1) (41) (51)
%e ((21)2) (321)
%e ((31)1) ((21)3)
%e (((21)1)1) ((31)2)
%e ((32)1)
%e (3(21))
%e ((41)1)
%e ((21)21)
%e (((21)1)2)
%e (((21)2)1)
%e (((31)1)1)
%e ((((21)1)1)1)
%t disjointQ[u_]:=Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
%t ldeip[n_]:=Prepend[Select[Join@@Table[Tuples[ldeip/@p],{p,Rest[IntegerPartitions[n]]}],UnsameQ@@#&&disjointQ[DeleteCases[#,_Integer]]&],n];
%t Table[Length[ldeip[n]],{n,12}]
%Y The orderless version is A316694.
%Y The non-identity version is A331687.
%Y Identity trees are A004111.
%Y P-trees are A196545.
%Y Enriched p-trees are A289501.
%Y Locally disjoint identity trees are A316471.
%Y Enriched identity p-trees are A331875, with locally disjoint case A331687.
%Y Cf. A000669, A005804, A141268, A300660, A316696, A316697, A331678, A331679, A331680, A331683, A331686, A331783, A331874.
%K nonn,more
%O 1,3
%A _Gus Wiseman_, Jan 31 2020
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