A306844 -id:A306844 - cp's OEIS frontend

A324765 Number of recursively anti-transitive rooted trees with n nodes.

1, 1, 2, 3, 6, 11, 26, 52, 119, 266, 618, 1432, 3402, 8093, 19505, 47228, 115244, 282529, 696388, 1723400

Offset: 1

Gus Wiseman, Mar 17 2019

An unlabeled rooted tree is recursively anti-transitive if no branch of a branch of a terminal subtree is a branch of the same subtree.

			The a(1) = 1 through a(6) = 11 recursively anti-transitive rooted trees:
  o  (o)  (oo)   (ooo)    (oooo)     (ooooo)
          ((o))  ((oo))   ((ooo))    ((oooo))
                 (((o)))  (((oo)))   (((ooo)))
                          ((o)(o))   ((o)(oo))
                          (o((o)))   (o((oo)))
                          ((((o))))  (oo((o)))
                                     ((((oo))))
                                     (((o)(o)))
                                     ((o((o))))
                                     (o(((o))))
                                     (((((o)))))

Cf. A000081, A290689, A301700, A304360, A306844, A317787, A318185.
Cf. A324695, A324751, A324756, A324758, A324764, A324766, A324767, A324768.
Cf. A324838, A324840, A324844.

nallt[n_]:=Select[Union[Sort/@Join@@(Tuples[nallt/@#]&/@IntegerPartitions[n-1])],Intersection[Union@@#,#]=={}&];
Table[Length[nallt[n]],{n,10}]

A324743 Number of maximal subsets of {1...n} containing no prime indices of the elements.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 8, 8, 8, 8, 12, 12, 18, 18, 19, 19, 30, 30, 54, 54, 54, 54, 96, 96, 96, 96, 96, 96, 156, 156, 244, 244, 248, 248, 248, 248, 440, 440, 440, 440, 688, 688, 1120, 1120, 1120, 1120, 1864, 1864, 1864, 1864, 1864, 1864, 3664, 3664, 3664, 3664, 3664

Offset: 0

Gus Wiseman, Mar 15 2019

nonn

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 a(0) = 1 through a(8) = 8 maximal subsets:
  {}  {1}  {1}  {2}    {1,3}  {1,3}    {1,3}    {1,3,7}  {1,3,7}
           {2}  {1,3}  {2,4}  {1,5}    {1,5}    {1,5,7}  {1,5,7}
                       {3,4}  {3,4}    {2,4,5}  {2,4,5}  {2,4,5,8}
                              {2,4,5}  {3,4,6}  {2,5,7}  {2,5,7,8}
                                       {4,5,6}  {3,4,6}  {3,4,6,8}
                                                {3,6,7}  {3,6,7,8}
                                                {4,5,6}  {4,5,6,8}
                                                {5,6,7}  {5,6,7,8}
An example for n = 15 is {1,5,7,9,13,15}, with prime indices:
  1: {}
  5: {3}
  7: {4}
  9: {2,2}
  13: {6}
  15: {2,3}
None of these prime indices {2,3,4,6} belong to the subset, as required.

Andrew Howroyd, Table of n, a(n) for n = 0..100

The non-maximal case is A324741. The case for subsets of {2...n} is A324763.
Cf. A000720, A001462, A007097, A084422, A085945, A112798, A276625, A290689, A290822, A304360, A306844, A320426, A324764.
Cf. A324695, A324736, A324742, A324744, A324751, A324756, A324758.

maxim[s_]:=Complement[s,Last/@Select[Tuples[s,2],UnsameQ@@#&&SubsetQ@@#&]];
Table[Length[maxim[Select[Subsets[Range[n]],Intersection[#,PrimePi/@First/@Join@@FactorInteger/@#]=={}&]]],{n,0,10}]

pset(n)={my(b=0, f=factor(n)[, 1]); sum(i=1, #f, 1<<(primepi(f[i])))}
a(n)={my(p=vector(n, k, pset(k)), d=0); for(i=1, #p, d=bitor(d, p[i]));
my(ismax(b)=my(e=0); forstep(k=#p, 1, -1, if(bittest(b,k), e=bitor(e,p[k]), if(!bittest(e,k) && !bitand(p[k], b), return(0)) )); 1);
((k, b)->if(k>#p, ismax(b), my(f=!bitand(p[k], b)); if(!f || bittest(d, k), self()(k+1, b)) + if(f, self()(k+1, b+(1<Andrew Howroyd, Aug 26 2019

Terms a(16) and beyond from Andrew Howroyd, Aug 26 2019

A324844 Number of unlabeled rooted trees with n nodes where the branches of no non-leaf branch of any terminal subtree form a submultiset of the branches of the same subtree.

Original entry on oeis.org

1, 1, 2, 3, 7, 13, 32, 71, 170, 406, 1002, 2469, 6204, 15644, 39871, 102116, 263325, 682079, 1775600, 4640220

Offset: 1

Gus Wiseman, Mar 18 2019

nonn

			The a(1) = 1 through a(6) = 13 rooted trees:
  o  (o)  (oo)   (ooo)    (oooo)     (ooooo)
          ((o))  ((oo))   ((ooo))    ((oooo))
                 (((o)))  (o(oo))    (o(ooo))
                          (((oo)))   (((ooo)))
                          ((o)(o))   ((o)(oo))
                          (o((o)))   ((o(oo)))
                          ((((o))))  (o((oo)))
                                     (oo((o)))
                                     ((((oo))))
                                     (((o)(o)))
                                     ((o((o))))
                                     (o(((o))))
                                     (((((o)))))

The Matula-Goebel numbers of these trees are given by A324845.
Cf. A000081, A290689, A306844, A318185.
Cf. A324694, A324738, A324744, A324749, A324754, A324759, A324765, A324768, A324838, A324843, A324846, A324847, A324848, A324849.

submultQ[cap_,fat_]:=And@@Function[i,Count[fat,i]>=Count[cap,i]]/@Union[List@@cap];
rallt[n_]:=Select[Union[Sort/@Join@@(Tuples[rallt/@#]&/@IntegerPartitions[n-1])],And@@Table[!submultQ[b,#],{b,DeleteCases[#,{}]}]&];
Table[Length[rallt[n]],{n,10}]

A324840 Number of fully recursively anti-transitive rooted trees with n nodes.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 14, 23, 46, 85, 165, 313, 625, 1225, 2459, 4919, 9928, 20078, 40926, 83592

Offset: 1

Gus Wiseman, Mar 17 2019

An unlabeled rooted tree is fully recursively anti-transitive if no proper terminal subtree of any terminal subtree is a branch of the larger subtree.

			The a(1) = 1 through a(7) = 14 fully recursively anti-transitive rooted trees:
  o  (o)  (oo)   (ooo)    (oooo)     (ooooo)      (oooooo)
          ((o))  ((oo))   ((ooo))    ((oooo))     ((ooooo))
                 (((o)))  (((oo)))   (((ooo)))    (((oooo)))
                          ((o)(o))   ((o)(oo))    ((o)(ooo))
                          ((((o))))  ((((oo))))   ((oo)(oo))
                                     (((o)(o)))   ((((ooo))))
                                     (((((o)))))  (((o))(oo))
                                                  (((o)(oo)))
                                                  ((o)((oo)))
                                                  ((o)(o)(o))
                                                  (((((oo)))))
                                                  ((((o)(o))))
                                                  (((o))((o)))
                                                  ((((((o))))))

Gus Wiseman, The a(8) = 23 fully recursively anti-transitive rooted trees.
Gus Wiseman, The a(9) = 46 fully recursively anti-transitive rooted trees.
Gus Wiseman, The a(10) = 85 fully recursively anti-transitive rooted trees.

Cf. A000081, A279861, A290689, A304360, A306844, A318185.
Cf. A324695, A324751, A324756, A324758, A324763, A324765, A324768, A324769, A324770.
Cf. A324838, A324841, A324844, A324846.

dallt[n_]:=Select[Union[Sort/@Join@@(Tuples[dallt/@#]&/@IntegerPartitions[n-1])],Intersection[Union@@Rest[FixedPointList[Union@@#&,#]],#]=={}&];
Table[Length[dallt[n]],{n,10}]

A324744 Number of maximal subsets of {1...n} containing no element whose prime indices all belong to the subset.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 4, 5, 6, 8, 8, 11, 11, 22, 22, 22, 22, 28, 28, 44, 44, 52, 52, 76, 76, 88, 88, 96, 96, 184, 184, 240, 240, 264, 264, 296, 296, 592, 592, 592, 592, 728, 728, 1456, 1456, 1456, 1456, 2912, 2912, 3168, 3168, 3168, 3168, 5568, 5568, 5568, 5568

Offset: 0

Gus Wiseman, Mar 15 2019

nonn

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 a(1) = 1 through a(8) = 6 maximal subsets:
  {1}  {1}  {2}    {1,3}  {1,3}    {1,3,6}    {3,4,6}    {1,3,6,7}
       {2}  {1,3}  {2,4}  {1,5}    {1,5,6}    {1,3,6,7}  {1,5,6,7}
                   {3,4}  {3,4}    {3,4,6}    {1,5,6,7}  {3,4,6,8}
                          {2,4,5}  {2,4,5,6}  {2,4,5,6}  {3,6,7,8}
                                              {2,5,6,7}  {2,4,5,6,8}
                                                         {2,5,6,7,8}

Andrew Howroyd, Table of n, a(n) for n = 0..100

The non-maximal case is A324738. The case for subsets of {2...n} is A324762.
Cf. A000720, A001462, A007097, A076078, A084422, A085945, A112798, A276625, A290822, A304360, A306844, A320426, A324764.
Cf. A324694, A324736, A324743, A324749, A324754, A324759.

maxim[s_]:=Complement[s,Last/@Select[Tuples[s,2],UnsameQ@@#&&SubsetQ@@#&]];
Table[Length[maxim[Select[Subsets[Range[n]],!MemberQ[#,k_/;SubsetQ[#,PrimePi/@First/@FactorInteger[k]]]&]]],{n,0,10}]

pset(n)={my(b=0, f=factor(n)[, 1]); sum(i=1, #f, 1<<(primepi(f[i])))}
a(n)={my(p=vector(n, k, if(k==1, 1, pset(k))), d=0); for(i=1, #p, d=bitor(d, p[i]));
my(ismax(b)=for(k=1, #p, if(!bittest(b,k) && bitnegimply(p[k], b), my(e=bitor(b, 1<#p, ismax(b), my(f=bitnegimply(p[k], b)); if(!f || bittest(d, k), self()(k+1, b)) + if(f, self()(k+1, b+(1<Andrew Howroyd, Aug 27 2019

Terms a(16) and beyond from Andrew Howroyd, Aug 27 2019

A324768 Number of fully anti-transitive rooted trees with n nodes.

Original entry on oeis.org

1, 1, 2, 3, 6, 11, 27, 60, 152, 376, 968, 2492, 6549, 17259, 46000, 123214, 332304, 900406, 2451999, 6703925

Offset: 1

Gus Wiseman, Mar 17 2019

An unlabeled rooted tree is fully anti-transitive if no proper terminal subtree of any branch of the root is a branch of the root.

			The a(1) = 1 through a(6) = 11 rooted trees:
  o  (o)  (oo)   (ooo)    (oooo)     (ooooo)
          ((o))  ((oo))   ((ooo))    ((oooo))
                 (((o)))  (((oo)))   (((ooo)))
                          ((o)(o))   ((o)(oo))
                          ((o(o)))   ((o(oo)))
                          ((((o))))  ((oo(o)))
                                     ((((oo))))
                                     (((o)(o)))
                                     (((o(o))))
                                     ((o((o))))
                                     (((((o)))))

Cf. A000081, A279861, A290689, A304360, A306844, A318185.
Cf. A324695, A324751, A324756, A324758, A324763, A324765, A324769, A324770.
Cf. A324838, A324840, A324844, A324846.

rtall[n_]:=Union[Sort/@Join@@(Tuples[rtall/@#]&/@IntegerPartitions[n-1])];
Table[Length[Select[rtall[n],Intersection[Union@@Rest[FixedPointList[Union@@#&,#]],#]=={}&]],{n,10}]

a(17)-a(20) from Jinyuan Wang, Jun 20 2020

A324738 Number of subsets of {1...n} containing no element > 1 whose prime indices all belong to the subset.

Original entry on oeis.org

1, 2, 3, 5, 8, 13, 26, 42, 72, 120, 232, 376, 752, 1128, 2256, 4512, 8256, 13632, 27264, 42048, 82944, 158976, 313344, 497664, 995328, 1700352, 3350016, 5815296, 11630592, 17491968, 34983936, 56954880, 108933120, 210788352, 418258944, 804667392, 1609334784

Offset: 0

Gus Wiseman, Mar 13 2019

nonn

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 a(0) = 1 through a(6) = 26 subsets:
  {}  {}   {}   {}     {}     {}       {}
      {1}  {1}  {1}    {1}    {1}      {1}
           {2}  {2}    {2}    {2}      {2}
                {3}    {3}    {3}      {3}
                {1,3}  {4}    {4}      {4}
                       {1,3}  {5}      {5}
                       {2,4}  {1,3}    {6}
                       {3,4}  {1,5}    {1,3}
                              {2,4}    {1,5}
                              {2,5}    {1,6}
                              {3,4}    {2,4}
                              {4,5}    {2,5}
                              {2,4,5}  {2,6}
                                       {3,4}
                                       {3,6}
                                       {4,5}
                                       {4,6}
                                       {5,6}
                                       {1,3,6}
                                       {1,5,6}
                                       {2,4,5}
                                       {2,4,6}
                                       {2,5,6}
                                       {3,4,6}
                                       {4,5,6}
                                       {2,4,5,6}

Andrew Howroyd, Table of n, a(n) for n = 0..100

The maximal case is A324744. The case of subsets of {2...n} is A324739. The strict integer partition version is A324749. The integer partition version is A324754. The Heinz number version is A324759. An infinite version is A324694.
Cf. A000720, A001221, A001462, A007097, A076078, A084422, A085945, A112798, A276625, A279861, A290689, A290822, A304360, A306844.
Cf. A324695, A324736, A324741, A324750, A324755, A324760.

Table[Length[Select[Subsets[Range[n]],!MemberQ[#,k_/;SubsetQ[#,PrimePi/@First/@FactorInteger[k]]]&]],{n,0,10}]

pset(n)={my(b=0,f=factor(n)[,1]); sum(i=1, #f, 1<<(primepi(f[i])))}
a(n)={my(p=vector(n,k,if(k==1, 1, pset(k))), d=0); for(i=1, #p, d=bitor(d, p[i]));
((k,b)->if(k>#p, 1, my(t=self()(k+1,b)); if(bitnegimply(p[k], b), t+=if(bittest(d,k), self()(k+1, b+(1<Andrew Howroyd, Aug 16 2019

Terms a(21) and beyond from Andrew Howroyd, Aug 16 2019

A324742 Number of subsets of {2...n} containing no prime indices of the elements.

Original entry on oeis.org

1, 2, 3, 6, 10, 16, 24, 48, 84, 144, 228, 420, 648, 1080, 1800, 3600, 5760, 11136, 16704, 31104, 53568, 90624, 136896, 269952, 515712, 862080, 1708800, 3171840, 4832640, 9325440, 14890752, 29781504, 52245504, 88418304, 166017024, 331628544, 497645568, 829409280

Offset: 1

Gus Wiseman, Mar 15 2019

nonn

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 a(1) = 1 through a(6) = 16 subsets:
  {}  {}   {}   {}     {}       {}
      {2}  {2}  {2}    {2}      {2}
           {3}  {3}    {3}      {3}
                {4}    {4}      {4}
                {2,4}  {5}      {5}
                {3,4}  {2,4}    {6}
                       {2,5}    {2,4}
                       {3,4}    {2,5}
                       {4,5}    {3,4}
                       {2,4,5}  {3,6}
                                {4,5}
                                {4,6}
                                {5,6}
                                {2,4,5}
                                {3,4,6}
                                {4,5,6}
An example for n = 20 is {4,5,6,12,17,18,19}, with prime indices:
   4: {1,1}
   5: {3}
   6: {1,2}
  12: {1,1,2}
  17: {7}
  18: {1,2,2}
  19: {8}
None of these prime indices {1,2,3,7,8} belong to the set, as required.

Andrew Howroyd, Table of n, a(n) for n = 1..100

The maximal case is A324763. The version for subsets of {1...n} is A324741. The strict integer partition version is A324752. The integer partition version is A324757. The Heinz number version is A324761. An infinite version is A304360.
Cf. A000720, A001462, A007097, A076078, A084422, A085945, A112798, A276625, A290689, A290822, A306844, A324764.
Cf. A324695, A324737, A324743, A324751, A324756, A324758.

Table[Length[Select[Subsets[Range[2,n]],Intersection[#,PrimePi/@First/@Join@@FactorInteger/@#]=={}&]],{n,10}]

pset(n)={my(b=0,f=factor(n)[,1]); sum(i=1, #f, 1<<(primepi(f[i])))}
a(n)={my(p=vector(n-1,k,pset(k+1)>>1), d=0); for(i=1, #p, d=bitor(d, p[i]));
((k,b)->if(k>#p, 1, my(t=self()(k+1,b)); if(!bitand(p[k], b), t+=if(bittest(d,k), self()(k+1, b+(1<Andrew Howroyd, Aug 16 2019

Terms a(21) and beyond from Andrew Howroyd, Aug 16 2019

A324838 Number of unlabeled rooted trees with n nodes where the branches of no branch of the root form a submultiset of the branches of the root.

Original entry on oeis.org

1, 0, 1, 2, 5, 10, 28, 64, 169, 422, 1108, 2872, 7627, 20202, 54216, 145867, 395288

Offset: 1

Gus Wiseman, Mar 18 2019

			The a(1) = 1 through a(6) = 10 rooted trees:
  o  ((o))  ((oo))   ((ooo))    ((oooo))
            (((o)))  (((oo)))   (((ooo)))
                     ((o)(o))   ((o)(oo))
                     ((o(o)))   ((o(oo)))
                     ((((o))))  ((oo(o)))
                                ((((oo))))
                                (((o)(o)))
                                (((o(o))))
                                ((o((o))))
                                (((((o)))))

Cf. A000081, A290689, A304360, A306844, A317787, A318185.

Cf. A324694, A324696, A324704, A324738, A324744, A324758, A324759, A324765, A324768, A324771, A324839, A324840, A324844, A324846.

submultQ[cap_,fat_]:=And@@Function[i,Count[fat,i]>=Count[cap,i]]/@Union[List@@cap];
rtall[n_]:=Union[Sort/@Join@@(Tuples[rtall/@#]&/@IntegerPartitions[n-1])];
Table[Length[Select[rtall[n],And@@Table[!submultQ[b,#],{b,#}]&]],{n,10}]

A324763 Number of maximal subsets of {2...n} containing no prime indices of the elements.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 6, 6, 6, 6, 10, 10, 16, 16, 16, 16, 24, 24, 48, 48, 48, 48, 84, 84, 84, 84, 84, 84, 144, 144, 228, 228, 228, 228, 228, 228, 420, 420, 420, 420, 648, 648, 1080, 1080, 1080, 1080, 1800, 1800, 1800, 1800, 1800, 1800, 3600, 3600, 3600, 3600, 3600

Offset: 1

Gus Wiseman, Mar 17 2019

nonn

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 a(1) = 1 through a(9) = 6 subsets:
  {}  {2}  {2}  {2,4}  {3,4}    {2,4,5}  {2,4,5}  {2,4,5,8}  {2,4,5,8}
           {3}  {3,4}  {2,4,5}  {3,4,6}  {2,5,7}  {2,5,7,8}  {2,5,7,8}
                                {4,5,6}  {3,4,6}  {3,4,6,8}  {3,4,6,8,9}
                                         {3,6,7}  {3,6,7,8}  {3,6,7,8,9}
                                         {4,5,6}  {4,5,6,8}  {4,5,6,8,9}
                                         {5,6,7}  {5,6,7,8}  {5,6,7,8,9}

Andrew Howroyd, Table of n, a(n) for n = 1..100

The non-maximal version is A324742.
The version for subsets of {1...n} is A324741.
An infinite version is A304360.
Cf. A076078, A084422, A085945, A112798, A276625, A306844, A324764.
Cf. A324695, A324743, A324751, A324756, A324758, A324762.

maxim[s_]:=Complement[s,Last/@Select[Tuples[s,2],UnsameQ@@#&&SubsetQ@@#&]];
Table[Length[maxim[Select[Subsets[Range[2,n]],Intersection[#,PrimePi/@First/@Join@@FactorInteger/@#]=={}&]]],{n,10}]

pset(n)={my(b=0, f=factor(n)[, 1]); sum(i=1, #f, 1<<(primepi(f[i])))}
a(n)={my(p=vector(n-1, k, pset(k+1)>>1), d=0); for(i=1, #p, d=bitor(d, p[i]));
my(ismax(b)=my(e=0); forstep(k=#p, 1, -1, if(bittest(b,k), e=bitor(e,p[k]), if(!bittest(e,k) && !bitand(p[k], b), return(0)) )); 1);
((k, b)->if(k>#p, ismax(b), my(f=!bitand(p[k], b)); if(!f || bittest(d, k), self()(k+1, b)) + if(f, self()(k+1, b+(1<Andrew Howroyd, Aug 26 2019

Terms a(16) and beyond from Andrew Howroyd, Aug 26 2019

cp's OEIS Frontend

A324765 Number of recursively anti-transitive rooted trees with n nodes.

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A324743 Number of maximal subsets of {1...n} containing no prime indices of the elements.

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A324844 Number of unlabeled rooted trees with n nodes where the branches of no non-leaf branch of any terminal subtree form a submultiset of the branches of the same subtree.

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A324840 Number of fully recursively anti-transitive rooted trees with n nodes.

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A324744 Number of maximal subsets of {1...n} containing no element whose prime indices all belong to the subset.

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A324768 Number of fully anti-transitive rooted trees with n nodes.

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A324738 Number of subsets of {1...n} containing no element > 1 whose prime indices all belong to the subset.

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A324742 Number of subsets of {2...n} containing no prime indices of the elements.

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