A325110 -id:A325110 - cp's OEIS frontend

A325108 Number of maximal subsets of {1...n} with no binary containments.

1, 1, 1, 2, 2, 4, 5, 6, 6, 11, 13, 16, 17, 22, 27, 28

Offset: 0

Gus Wiseman, Mar 28 2019

A pair of positive integers is a binary containment if the positions of 1's in the reversed binary expansion of the first are a subset of the positions of 1's in the reversed binary expansion of the second.

			The a(0) = 1 through a(7) = 6 maximal subsets:
  {}  {1}  {1,2}  {3}    {3,4}    {2,5}    {1,6}    {7}
                  {1,2}  {1,2,4}  {3,4}    {2,5}    {1,6}
                                  {3,5}    {3,4}    {2,5}
                                  {1,2,4}  {1,2,4}  {3,4}
                                           {3,5,6}  {1,2,4}
                                                    {3,5,6}

Cf. A006126, A014466, A019565, A267610.

Cf. A325095, A325096, A325101, A325106, A325107, A325109, A325110.

binpos[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
maxim[s_]:=Complement[s,Last/@Select[Tuples[s,2],UnsameQ@@#&&SubsetQ@@#&]];
Table[Length[maxim[Select[Subsets[Range[n]],stableQ[#,SubsetQ[binpos[#1],binpos[#2]]&]&]]],{n,0,10}]

A325099 Number of binary carry-connected strict integer partitions of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 1, 4, 5, 8, 6, 11, 11, 15, 13, 18, 20, 30, 29, 43, 49, 68, 66, 84, 94, 125, 131, 165, 184, 237, 251, 291, 315, 383, 408, 486, 536, 663, 714, 832, 912, 1104, 1195, 1405, 1554, 1877, 2046, 2348, 2559, 2998, 3256, 3730, 4084, 4793, 5230, 5938

Offset: 0

Gus Wiseman, Mar 28 2019

nonn

A binary carry of two positive integers is an overlap of the positions of 1's in their reversed binary expansion. An integer partition is binary carry-connected if the graph whose vertices are the parts and whose edges are binary carries is connected.

			The a(1) = 1 through a(11) = 6 strict partitions (A = 10, B = 11):
  (1)  (2)  (3)  (4)   (5)   (6)    (7)  (8)   (9)    (A)    (B)
                 (31)  (32)  (51)        (53)  (54)   (64)   (65)
                             (321)       (62)  (63)   (73)   (74)
                                         (71)  (72)   (91)   (632)
                                               (531)  (532)  (731)
                                                      (541)  (5321)
                                                      (631)
                                                      (721)

Cf. A050315, A080572, A247935, A267610, A267700.

Cf. A325095, A325096, A325098, A325104, A325106, A325110, A325119.

binpos[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Length[csm[binpos/@#]]<=1&]],{n,0,30}]

A325100 Heinz numbers of strict integer partitions with no binary carries.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 11, 13, 14, 17, 19, 21, 23, 26, 29, 31, 33, 35, 37, 38, 41, 42, 43, 47, 53, 57, 58, 59, 61, 67, 69, 71, 73, 74, 79, 83, 86, 89, 95, 97, 101, 103, 106, 107, 109, 111, 113, 114, 122, 123, 127, 131, 133, 137, 139, 142, 149, 151, 157, 158, 159

Offset: 1

Gus Wiseman, Mar 28 2019

nonn

A binary carry of two positive integers is an overlap of the positions of 1's in their reversed binary expansion.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1) * ... * prime(y_k), so these are squarefree numbers whose prime indices have no carries. A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   3: {2}
   5: {3}
   6: {1,2}
   7: {4}
  11: {5}
  13: {6}
  14: {1,4}
  17: {7}
  19: {8}
  21: {2,4}
  23: {9}
  26: {1,6}
  29: {10}
  31: {11}
  33: {2,5}
  35: {3,4}
  37: {12}
  38: {1,8}
  41: {13}
  42: {1,2,4}

Cf. A050315, A056239, A080572, A112798, A247935, A267610.

Cf. A325095, A325096, A325097, A325100, A325101, A325103, A325110, A325119.

binpos[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Select[Range[100],SquareFreeQ[#]&&stableQ[PrimePi/@First/@FactorInteger[#],Intersection[binpos[#1],binpos[#2]]!={}&]&]

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A325108 Number of maximal subsets of {1...n} with no binary containments.

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A325099 Number of binary carry-connected strict integer partitions of n.

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A325100 Heinz numbers of strict integer partitions with no binary carries.

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