A367902 -id:A367902 - cp's OEIS frontend

A368410 Number of non-isomorphic connected set-systems of weight n satisfying a strict version of the axiom of choice.

0, 1, 1, 2, 3, 7, 15, 32, 80, 198, 528

Offset: 0

Gus Wiseman, Dec 25 2023

A set-system is a finite set of finite nonempty sets. The weight of a set-system is the sum of cardinalities of its elements. Weight is generally not the same as number of vertices.

The axiom of choice says that, given any set of nonempty sets Y, it is possible to choose a set containing an element from each. The strict version requires this set to have the same cardinality as Y, meaning no element is chosen more than once.

			Non-isomorphic representatives of the a(1) = 1 through a(6) = 15 set-systems:
  {1}  {12}  {123}    {1234}    {12345}      {123456}
             {2}{12}  {13}{23}  {14}{234}    {125}{345}
                      {3}{123}  {23}{123}    {134}{234}
                                {4}{1234}    {15}{2345}
                                {2}{13}{23}  {34}{1234}
                                {2}{3}{123}  {5}{12345}
                                {3}{13}{23}  {1}{14}{234}
                                             {12}{13}{23}
                                             {1}{23}{123}
                                             {13}{24}{34}
                                             {14}{24}{34}
                                             {3}{14}{234}
                                             {3}{23}{123}
                                             {3}{4}{1234}
                                             {4}{14}{234}

Wikipedia, Axiom of choice.

For unlabeled graphs we have A005703, connected case of A134964.
For labeled graphs we have A129271, connected case of A133686.
The complement for labeled graphs is A140638, connected case of A367867.
The complement without connectedness is A367903, ranks A367907.
Without connectedness we have A368095, ranks A367906,
Complement with repeats: A368097, connected case of A368411, ranks A355529.
The complement is counted by A368409, connected case of A368094.
With repeats allowed: A368412, connected case of A368098, ranks A368100.
A000110 counts set-partitions, non-isomorphic A000041.
A003465 counts covering set-systems, unlabeled A055621.
A007716 counts non-isomorphic multiset partitions, connected A007718.
A058891 counts set-systems, unlabeled A000612, connected A323818.
A283877 counts non-isomorphic set-systems, connected A300913.
Cf. A001055, A140637, A302545, A306005, A321194, A321405, A367902, A368414, A368422.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mpm[n_]:=Join@@Table[Union[Sort[Sort/@(#/.x_Integer:>s[[x]])]& /@ sps[Range[n]]],{s,Flatten[MapIndexed[Table[#2,{#1}]&,#]]& /@ IntegerPartitions[n]}];
brute[m_]:=First[Sort[Table[Sort[Sort/@(m/.Rule@@@Table[{i,p[[i]]}, {i,Length[p]}])],{p,Permutations[Union@@m]}]]];
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]]]]]]]]];
Table[Length[Union[brute/@Select[mpm[n], UnsameQ@@#&&And@@UnsameQ@@@#&&Length[csm[#]]==1&&Select[Tuples[#], UnsameQ@@#&]!={}&]]],{n,0,6}]

A370642 Number of minimal subsets of {1..n} such that it is not possible to choose a different binary index of each element.

Original entry on oeis.org

0, 0, 0, 1, 1, 3, 9, 26, 26, 40, 82, 175, 338, 636, 1114

Offset: 0

Gus Wiseman, Mar 10 2024

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.

			The a(0) = 0 through a(6) = 9 subsets:
  .  .  .  {1,2,3}  {1,2,3}  {1,2,3}    {1,2,3}
                             {1,4,5}    {1,4,5}
                             {2,3,4,5}  {2,4,6}
                                        {1,2,5,6}
                                        {1,3,4,6}
                                        {1,3,5,6}
                                        {2,3,4,5}
                                        {2,3,5,6}
                                        {3,4,5,6}

For prime indices we have A370591, minima of A370583, complement A370582.
This is the minimal case of A370637, complement A370636.
The version for a unique choice is A370638, maxima A370640, diffs A370641.
The case without ones is A370644.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
A326031 gives weight of the set-system with BII-number n.
A367902 counts choosable set-systems, ranks A367906, unlabeled A368095.
A367903 counts non-choosable set-systems, ranks A367907, unlabeled A368094.
A368100 ranks choosable multisets, complement A355529.
A370585 counts maximal choosable sets.
Cf. A072639, A140637, A367905, A368109, A370589, A370593, A370639.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
fasmin[y_]:=Complement[y,Union@@Table[Union[s,#]& /@ Rest[Subsets[Complement[Union@@y,s]]],{s,y}]];
Table[Length[fasmin[Select[Subsets[Range[n]], Select[Tuples[bpe/@#],UnsameQ@@#&]=={}&]]],{n,0,10}]

A370645 Number of integer factorizations of n into unordered factors > 1 such that only one set can be obtained by choosing a different prime factor of each factor.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 1, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 5, 1, 1, 2, 2, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1

Offset: 1

Gus Wiseman, Mar 01 2024

nonn

All of these factorizations are co-balanced (A340596).

			The factorization f = (3*6*10) has prime factor choices (3,2,2), (3,3,2), (3,2,5), and (3,3,5), of which only (3,2,5) has all different parts, so f is counted under a(180).
The a(n) factorizations for n = 2, 12, 24, 36, 72, 120, 144, 180, 288:
  (2)  (2*6)  (3*8)   (4*9)   (8*9)   (3*5*8)   (2*72)   (4*5*9)   (3*96)
       (3*4)  (4*6)   (6*6)   (2*36)  (4*5*6)   (3*48)   (5*6*6)   (4*72)
              (2*12)  (2*18)  (3*24)  (2*3*20)  (4*36)   (2*3*30)  (6*48)
                      (3*12)  (4*18)  (2*5*12)  (6*24)   (2*5*18)  (8*36)
                              (6*12)  (2*6*10)  (8*18)   (2*6*15)  (9*32)
                                      (3*4*10)  (9*16)   (2*9*10)  (12*24)
                                                (12*12)  (3*4*15)  (16*18)
                                                         (3*5*12)  (2*144)
                                                         (3*6*10)

Multisets of this type are ranked by A368101, see also A368100, A355529.
For nonexistence we have A368413, complement A368414.
Subsets of this type are counted by A370584, see also A370582, A370583.
Maximal sets of this type are counted by A370585.
The version for partitions is A370594, see also A370592, A370593.
Subsets of this type are counted by A370638, see also A370636, A370637.
For unlabeled multiset partitions we have A370646, also A368098, A368097.
A001055 counts factorizations, strict A045778.
A006530 gives greatest prime factor, least A020639.
A007716 counts non-isomorphic multiset partitions, connected A007718.
A027746 lists prime factors, A112798 indices, length A001222.
A058891 counts set-systems, unlabeled A000612, connected A323818.
A283877 counts non-isomorphic set-systems, connected A300913.
A355741 counts ways to choose a prime factor of each prime index.
For set-systems see A367902-A367908.
Cf. A000040, A000720, A003963, A340596, A340653, A355744, A355745, A368110.

facs[n_]:=If[n<=1,{{}},Join @@ Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]], {d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n], Length[Union[Sort/@Select[Tuples[First /@ FactorInteger[#]&/@#], UnsameQ@@#&]]]==1&]],{n,100}]

A368186 Number of n-covers of an unlabeled n-set.

Original entry on oeis.org

1, 1, 2, 9, 87, 1973, 118827, 20576251, 10810818595, 17821875542809, 94589477627232498, 1651805220868992729874, 96651473179540769701281003, 19238331716776641088273777321428, 13192673305726630096303157068241728202, 31503323006770789288222386469635474844616195

Offset: 0

Gus Wiseman, Dec 19 2023

nonn

			Non-isomorphic representatives of the a(1) = 1 through a(3) = 9 set-systems:
  {{1}}  {{1},{2}}    {{1},{2},{3}}
         {{1},{1,2}}  {{1},{2},{1,3}}
                      {{1},{1,2},{1,3}}
                      {{1},{1,2},{2,3}}
                      {{1},{2},{1,2,3}}
                      {{1},{1,2},{1,2,3}}
                      {{1,2},{1,3},{2,3}}
                      {{1},{2,3},{1,2,3}}
                      {{1,2},{1,3},{1,2,3}}

Andrew Howroyd, Table of n, a(n) for n = 0..50

The labeled version is A054780, ranks A367917, non-covering A367916.
The case of graphs is A006649, labeled A367863, cf. A116508, A367862.
The case of connected graphs is A001429, labeled A057500.
Covers with any number of edges are counted by A003465, unlabeled A055621.
A046165 counts minimal covers, ranks A309326.
A058891 counts set-systems, unlabeled A000612, without singletons A016031.
A059201 counts covering T_0 set-systems, unlabeled A319637, ranks A326947.
Cf. A000088, A002494, A006126, A055130, A133686, A140638, A305000, A317795, A326754, A367901, A367902, A367903.

brute[m_]:=Table[Sort[Sort/@(m/.Rule@@@Table[{i, p[[i]]},{i,Length[p]}])], {p,Permutations[Union@@m]}];
Table[Length[Union[First[Sort[brute[#]]]& /@ Select[Subsets[Rest[Subsets[Range[n]]],{n}], Union@@#==Range[n]&]]], {n,0,3}]

permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t)={2^sum(j=1, #q, gcd(t, q[j])) - 1}
G(n,m)={if(n==0, 1, my(s=0); forpart(q=n, my(g=sum(t=1, m, K(q,t)*x^t/t, O(x*x^m))); s+=permcount(q)*exp(g - subst(g,x,x^2))); s/n!)}
a(n)=if(n ==0, 1, polcoef(G(n,n) - G(n-1,n), n)) \\ Andrew Howroyd, Jan 03 2024

a(n) = A055130(n, n) for n > 0. - Andrew Howroyd, Jan 03 2024

Terms a(6) and beyond from Andrew Howroyd, Jan 03 2024

A368412 Number of non-isomorphic connected multiset partitions of weight n satisfying a strict version of the axiom of choice.

Original entry on oeis.org

0, 1, 2, 4, 11, 25, 75, 206, 650, 2049, 6895

Offset: 0

Gus Wiseman, Dec 26 2023

A set-system is a finite set of finite nonempty sets. The weight of a set-system is the sum of cardinalities of its elements. Weight is generally not the same as number of vertices.

The axiom of choice says that, given any set of nonempty sets Y, it is possible to choose a set containing an element from each. The strict version requires this set to have the same cardinality as Y, meaning no element is chosen more than once.

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 11 multiset partitions:
  {{1}}  {{1,1}}  {{1,1,1}}    {{1,1,1,1}}
         {{1,2}}  {{1,2,2}}    {{1,1,2,2}}
                  {{1,2,3}}    {{1,2,2,2}}
                  {{2},{1,2}}  {{1,2,3,3}}
                               {{1,2,3,4}}
                               {{1},{1,2,2}}
                               {{1,2},{1,2}}
                               {{1,2},{2,2}}
                               {{1,3},{2,3}}
                               {{2},{1,2,2}}
                               {{3},{1,2,3}}

Wikipedia, Axiom of choice.

The case of labeled graphs is A129271, connected case of A133686.
The complement for labeled graphs is A140638, connected case of A367867.
This is the connected case of A368098, ranks A368100.
Complement set-systems: A368409, connected case of A368094, ranks A367907.
For set-systems we have A368410, connected case of A368095, ranks A367906.
The complement is A368411, connected case of A368097, ranks A355529.
A000110 counts set partitions, non-isomorphic A000041.
A003465 counts covering set-systems, unlabeled A055621.
A007716 counts non-isomorphic multiset partitions, connected A007718.
A058891 counts set-systems, unlabeled A000612, connected A323818.
A283877 counts non-isomorphic set-systems, connected A300913.
Cf. A140637, A302545, A316983, A317533, A319616, A367902, A367905, A368187.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mpm[n_]:=Join@@Table[Union[Sort[Sort /@ (#/.x_Integer:>s[[x]])]&/@sps[Range[n]]], {s,Flatten[MapIndexed[Table[#2,{#1}]&,#]]& /@ IntegerPartitions[n]}];
brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])], {p,Permutations[Union@@m]}]]];
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]]]]]]]]];
Table[Length[Union[brute /@ Select[mpm[n],Length[csm[#]]==1&&Select[Tuples[#], UnsameQ@@#&]!={}&]]],{n,0,6}]

A370641 Number of maximal subsets of {1..n} containing n such that it is possible to choose a different binary index of each element.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 9, 15, 32, 45, 67, 98, 141, 197, 263, 358, 1201, 1493, 1920, 2482, 3123, 3967, 4884, 6137, 7584, 9369, 11169, 13664, 15818, 19152, 22418, 26905, 151286, 173409, 202171, 237572, 273651, 320040, 367792, 428747, 485697, 562620, 637043, 734738, 815492

Offset: 0

Gus Wiseman, Mar 11 2024

nonn

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.

Also choices of A070939(n) elements of {1..n} containing n such that it is possible to choose a different binary index of each.

			The a(0) = 0 through a(7) = 15 subsets:
  .  {1}  {1,2}  {1,3}  {1,2,4}  {1,2,5}  {1,2,6}  {1,2,7}
                 {2,3}  {1,3,4}  {1,3,5}  {1,3,6}  {1,3,7}
                        {2,3,4}  {2,3,5}  {1,4,6}  {1,4,7}
                                 {2,4,5}  {1,5,6}  {1,5,7}
                                 {3,4,5}  {2,3,6}  {1,6,7}
                                          {2,5,6}  {2,3,7}
                                          {3,4,6}  {2,4,7}
                                          {3,5,6}  {2,5,7}
                                          {4,5,6}  {2,6,7}
                                                   {3,4,7}
                                                   {3,5,7}
                                                   {3,6,7}
                                                   {4,5,7}
                                                   {4,6,7}
                                                   {5,6,7}

A version for set-systems is A368601.
For prime indices we have A370590, without n A370585, see also A370591.
This is the maximal case of A370636 requiring n, complement A370637.
This is the maximal case of A370639, complement A370589.
Without requiring n we have A370640.
Dominated by A370819.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A058891 counts set-systems, A003465 covering, A323818 connected.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
A367902 counts choosable set-systems, ranks A367906, unlabeled A368095.
A367903 counts non-choosable set-systems, ranks A367907, unlabeled A368094.
Cf. A133686, A326031, A326702, A357812, A367905, A368100, A368109, A370586, A370638, A370642.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Table[Length[Select[Subsets[Range[n],{IntegerLength[n,2]}],MemberQ[#,n] && Length[Union[Sort/@Select[Tuples[bpe/@#], UnsameQ@@#&]]]>0&]],{n,0,25}]

More terms from Jinyuan Wang, Mar 28 2025

A370591 Number of minimal subsets of {1..n} such that it is not possible to choose a different prime factor of each element (non-choosable).

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 4, 4, 7, 11, 16, 16, 30, 30, 39, 73

Offset: 0

Gus Wiseman, Feb 28 2024

			The a(1) = 1 through a(10) = 16 subsets:
{1}  {1}  {1}  {1}    {1}    {1}      {1}      {1}      {1}      {1}
               {2,4}  {2,4}  {2,4}    {2,4}    {2,4}    {2,4}    {2,4}
                             {2,3,6}  {2,3,6}  {2,8}    {2,8}    {2,8}
                             {3,4,6}  {3,4,6}  {4,8}    {3,9}    {3,9}
                                               {2,3,6}  {4,8}    {4,8}
                                               {3,4,6}  {2,3,6}  {2,3,6}
                                               {3,6,8}  {2,6,9}  {2,6,9}
                                                        {3,4,6}  {3,4,6}
                                                        {3,6,8}  {3,6,8}
                                                        {4,6,9}  {4,6,9}
                                                        {6,8,9}  {6,8,9}
                                                                 {2,5,10}
                                                                 {4,5,10}
                                                                 {5,8,10}
                                                                 {3,5,6,10}
                                                                 {5,6,9,10}

Minimal case of A370583, complement A370582.
For binary indices instead of factors we have A370642, minima of A370637.
A006530 gives greatest prime factor, least A020639.
A027746 lists prime factors, indices A112798, length A001222.
A355741 counts choices of a prime factor of each prime index.
A367902 counts choosable set-systems, ranks A367906, unlabeled A368095.
A367903 counts non-choosable set-systems, ranks A367907, unlabeled A368094.
A368098 counts choosable unlabeled multiset partitions, complement A368097.
A368100 ranks choosable multisets, complement A355529.
A368414 counts choosable factorizations, complement A368413.
A370585 counts maximal choosable sets.
A370592 counts choosable partitions, complement A370593.
Cf. A000040, A000720, A045778, A133686, A355739, A355744, A355745, A367771, A370584, A370586, A370587, A370589.

Table[Length[fasmin[Select[Subsets[Range[n]], Length[Select[Tuples[prix/@#],UnsameQ@@#&]]==0&]]], {n,0,15}]

A370644 Number of minimal subsets of {2..n} such that it is not possible to choose a different binary index of each element.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 4, 13, 13, 26, 56, 126, 243, 471, 812, 1438

Offset: 0

Gus Wiseman, Mar 11 2024

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.

			The a(0) = 0 through a(7) = 13 subsets:
  .  .  .  .  .  {2,3,4,5}  {2,4,6}    {2,4,6}
                            {2,3,4,5}  {2,3,4,5}
                            {2,3,5,6}  {2,3,4,7}
                            {3,4,5,6}  {2,3,5,6}
                                       {2,3,5,7}
                                       {2,3,6,7}
                                       {2,4,5,7}
                                       {2,5,6,7}
                                       {3,4,5,6}
                                       {3,4,5,7}
                                       {3,4,6,7}
                                       {3,5,6,7}
                                       {4,5,6,7}
The a(0) = 0 through a(7) = 13 set-systems:
  .  .  .  .  .  {2}{12}{3}{13}  {2}{3}{23}       {2}{3}{23}
                                 {2}{12}{3}{13}   {2}{12}{3}{13}
                                 {12}{3}{13}{23}  {12}{3}{13}{23}
                                 {2}{12}{13}{23}  {2}{12}{13}{23}
                                                  {2}{12}{3}{123}
                                                  {2}{3}{13}{123}
                                                  {12}{3}{13}{123}
                                                  {12}{3}{23}{123}
                                                  {2}{12}{13}{123}
                                                  {2}{12}{23}{123}
                                                  {2}{13}{23}{123}
                                                  {3}{13}{23}{123}
                                                  {12}{13}{23}{123}

The version with ones allowed is A370642, minimal case of A370637.
This is the minimal case of A370643.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
A367902 counts choosable set-systems, ranks A367906, unlabeled A368095.
A367903 counts non-choosable set-systems, ranks A367907, unlabeled A368094.
A370585 counts maximal choosable sets.
Cf. A072639, A140637, A326031, A355529, A367905, A368109, A370589, A370591, A370636, A370639, A370640.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
fasmin[y_]:=Complement[y,Union@@Table[Union[s,#]& /@ Rest[Subsets[Complement[Union@@y,s]]],{s,y}]];
Table[Length[fasmin[Select[Subsets[Range[2,n]], Select[Tuples[bpe/@#],UnsameQ@@#&]=={}&]]],{n,0,10}]

A387118 Number of integer partitions of n without choosable initial intervals.

Original entry on oeis.org

0, 0, 1, 1, 2, 4, 6, 8, 13, 19, 28, 37, 52, 70, 97, 130, 172, 224, 293, 378, 492, 630, 806, 1018, 1286, 1609, 2019, 2514, 3131, 3874, 4784, 5872, 7198, 8786, 10712, 13013, 15794, 19100, 23063, 27752, 33341, 39939, 47781, 57013, 67955, 80816, 95992, 113773, 134668

Offset: 0

Gus Wiseman, Sep 02 2025

The initial interval of a nonnegative integer x is the set {1,...,x}.

We say that a sequence of nonempty sets is choosable iff it is possible to choose a different element from each set. For example, ({1,2},{1},{1,3}) is choosable because we have the choice (2,1,3), but ({1},{2},{1,3},{2,3}) is not.

			The partition y = (2,2,1) has initial intervals ({1,2},{1,2},{1}), which are not choosable, so y is counted under a(5).
The a(2) = 1 through a(8) = 13 partitions:
  (11)  (111)  (211)   (221)    (222)     (511)      (611)
               (1111)  (311)    (411)     (2221)     (2222)
                       (2111)   (2211)    (3211)     (3221)
                       (11111)  (3111)    (4111)     (3311)
                                (21111)   (22111)    (4211)
                                (111111)  (31111)    (5111)
                                          (211111)   (22211)
                                          (1111111)  (32111)
                                                     (41111)
                                                     (221111)
                                                     (311111)
                                                     (2111111)
                                                     (11111111)

The complement is counted by A238873, ranks A387112.
The complement for divisors is A239312, ranks A368110.
For divisors instead of initial intervals we have A370320, ranks A355740.
The complement for prime factors is A370592, ranks A368100.
For prime factors instead of initial intervals we have A370593, ranks A355529.
These partitions have ranks A387113.
For partitions instead of initial intervals we have A387134.
The complement for partitions is A387328.
For strict partitions instead of initial intervals we have A387137, ranks A387176.
The complement for strict partitions is A387178.
A000005 counts divisors.
A000041 counts integer partitions, strict A000009.
A367902 counts choosable set-systems, complement A367903.
A370582 counts sets with choosable prime factors, complement A370583.
Cf. A005703, A052335, A367867, A367901, A367905, A367907, A370585, A370594, A387111.

Table[Length[Select[IntegerPartitions[n],Select[Tuples[Range/@#],UnsameQ@@#&]=={}&]],{n,0,10}]

More terms from Jinyuan Wang, Sep 05 2025

A387137 Number of integer partitions of n whose parts do not have choosable sets of strict integer partitions.

Original entry on oeis.org

0, 0, 1, 1, 3, 4, 6, 9, 14, 20, 29, 39, 56, 74, 101, 134, 178, 232, 305, 392, 508, 646, 825, 1042, 1317, 1649, 2066, 2567, 3190, 3937, 4859, 5960, 7306, 8914, 10863, 13183, 15984, 19304, 23288, 28003, 33631, 40272, 48166, 57453, 68448, 81352, 96568, 114383

Offset: 0

Gus Wiseman, Sep 02 2025

We say that a sequence of nonempty sets is choosable iff it is possible to choose a different element from each set. For example, ({1,2},{1},{1,3}) is choosable because we have the choice (2,1,3), but ({1},{2},{1,3},{2,3}) is not.
a(n) is the number of integer partitions of n such that it is not possible to choose a sequence of distinct strict integer partitions, one of each part.
Also the number of integer partitions of n with at least one part k whose multiplicity exceeds A000009(k).

			The a(2) = 1 through a(8) = 14 partitions:
  (11)  (111)  (22)    (221)    (222)     (322)      (422)
               (211)   (311)    (411)     (511)      (611)
               (1111)  (2111)   (2211)    (2221)     (2222)
                       (11111)  (3111)    (3211)     (3221)
                                (21111)   (4111)     (3311)
                                (111111)  (22111)    (4211)
                                          (31111)    (5111)
                                          (211111)   (22211)
                                          (1111111)  (32111)
                                                     (41111)
                                                     (221111)
                                                     (311111)
                                                     (2111111)
                                                     (11111111)

The complement for initial intervals is A238873, ranks A387112.
The complement for divisors is A239312, ranks A368110.
Twice-partitions of this type (into distinct strict partitions) are counted by A358914.
For divisors instead of strict partitions we have A370320, ranks A355740.
The complement for prime factors is A370592, ranks A368100.
For prime factors instead of strict partitions we have A370593, ranks A355529.
For initial intervals instead of strict partitions we have A387118, ranks A387113.
For all partitions instead of strict partitions we have A387134, ranks A387577.
These partitions are ranked by A387176.
The complement is counted by A387178, ranks A387177.
The complement for partitions is A387328, ranks A387576.
The version for constant partitions is A387329, ranks A387180.
The complement for constant partitions is A387330, ranks A387181.
A000005 counts divisors.
A000041 counts integer partitions, strict A000009.
A367902 counts choosable set-systems, complement A367903.
Cf. A005703, A052335, A261049, A270995, A276078, A335448, A355535, A367867, A367901, A367905, A383706, A387115.

strptns[n_]:=Select[IntegerPartitions[n],UnsameQ@@#&];
Table[Length[Select[IntegerPartitions[n],Length[Select[Tuples[strptns/@#],UnsameQ@@#&]]==0&]],{n,0,15}]

cp's OEIS Frontend

A368410 Number of non-isomorphic connected set-systems of weight n satisfying a strict version of the axiom of choice.

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A370642 Number of minimal subsets of {1..n} such that it is not possible to choose a different binary index of each element.

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A370645 Number of integer factorizations of n into unordered factors > 1 such that only one set can be obtained by choosing a different prime factor of each factor.

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A368186 Number of n-covers of an unlabeled n-set.

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A368412 Number of non-isomorphic connected multiset partitions of weight n satisfying a strict version of the axiom of choice.

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A370641 Number of maximal subsets of {1..n} containing n such that it is possible to choose a different binary index of each element.

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A370591 Number of minimal subsets of {1..n} such that it is not possible to choose a different prime factor of each element (non-choosable).

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A370644 Number of minimal subsets of {2..n} such that it is not possible to choose a different binary index of each element.

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A387118 Number of integer partitions of n without choosable initial intervals.

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