cp's OEIS Frontend

A371446 Number of carry-connected integer partitions whose distinct parts have no binary containments.

%I A371446 #17 Apr 03 2024 05:04:17
%S A371446 1,1,2,2,3,2,4,2,5,4,4,4,8,4,7,7,12,10,14,12,15,19,19,21,32,27,33,40,
%T A371446 46,47,61,52,75,89,95,104,129,129,149,176,188,208,249,257,296,341,373,
%U A371446 394,476,496,552
%N A371446 Number of carry-connected integer partitions whose distinct parts have no binary containments.
%C A371446 These partitions are ranked by A371445.
%C A371446 A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.
%C A371446 A binary carry of two positive integers is an overlap of binary indices. An integer partition is binary carry-connected iff the graph with one vertex for each part and edges corresponding to binary carries is connected.
%C A371446 A binary containment is a containment of binary indices. For example, the numbers {3,5} have binary indices {{1,2},{1,3}}, so there is a binary carry but not a binary containment.
%e A371446 The a(12) = 8 through a(14) = 7 partitions:
%e A371446   (12)             (13)                         (14)
%e A371446   (6,6)            (10,3)                       (7,7)
%e A371446   (9,3)            (5,5,3)                      (9,5)
%e A371446   (4,4,4)          (1,1,1,1,1,1,1,1,1,1,1,1,1)  (6,5,3)
%e A371446   (6,3,3)                                       (5,3,3,3)
%e A371446   (3,3,3,3)                                     (2,2,2,2,2,2,2)
%e A371446   (2,2,2,2,2,2)                                 (1,1,1,1,1,1,1,1,1,1,1,1,1,1)
%e A371446   (1,1,1,1,1,1,1,1,1,1,1,1)
%t A371446 stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
%t A371446 bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A371446 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A371446 csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}], Length[Intersection@@s[[#]]]>0&]},If[c=={},s, csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
%t A371446 Table[Length[Select[IntegerPartitions[n], stableQ[bix/@Union[#],SubsetQ]&&Length[csm[bix/@#]]<=1&]],{n,0,30}]
%Y A371446 The first condition (carry-connected) is A325098.
%Y A371446 The second condition (stable) is A325109.
%Y A371446 Ranks for binary indices of binary indices are A326750 = A326704 /\ A326749.
%Y A371446 Ranks for prime indices of prime indices are A329559 = A305078 /\ A316476.
%Y A371446 Ranks for prime indices of binary indices are A371294 = A087086 /\ A371291.
%Y A371446 Ranks for binary indices of prime indices are A371445 = A325118 /\ A371455.
%Y A371446 A001187 counts connected graphs.
%Y A371446 A007718 counts non-isomorphic connected multiset partitions.
%Y A371446 A048143 counts connected antichains of sets.
%Y A371446 A048793 lists binary indices, reverse A272020, length A000120, sum A029931.
%Y A371446 A070939 gives length of binary expansion.
%Y A371446 A326964 counts connected set-systems, covering A323818.
%Y A371446 Cf. A019565, A304713, A304716, A305079, A305148, A325097, A325105, A325107, A325119, A371452.
%K A371446 nonn,more
%O A371446 0,3
%A A371446 _Gus Wiseman_, Apr 02 2024