A059201 -id:A059201 - cp's OEIS frontend

A059523 Number of n-element unlabeled ordered T_0-antichains without isolated vertices; number of T_1-hypergraphs (without empty edge and without multiple edges) on n labeled vertices.

1, 2, 2, 36, 19020, 2010231696, 9219217412568364176, 170141181796805105960861096082778425120, 57896044618658097536026644159052312977171804852352892309392604715987334365792

Offset: 0

Vladeta Jovovic and Goran Kilibarda, Jan 20 2001; revised Jun 03 2004

nonn

			Number of k-element T_1-hipergraphs (without empty edge and without multiple edges) on 3 labeled vertices is
C(7,k)-6*C(5,k)+6*C(4,k)+3*C(3,k)-6*C(2,k)+2*C(1,k),k=0..7; so a(3)=2+11+15+7+1=36=2^7-6*2^5+6*2^4+3*2^3-6*2^2+2*2.

V. Jovovic, T_1-hyper graphs on a labeled 3-set
Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, J. Integer Seqs., Vol. 7, 2004.
Goran Kilibarda and Vladeta Jovovic, Enumeration of some classes of T_0-hypergraphs, arXiv:1411.4187 [math.CO], 2014.

Cf. A003465, A059201, A059052, A326961.

a(n) = A059052(n)/2.

A326940 Number of T_0 set-systems on n vertices.

Original entry on oeis.org

1, 2, 7, 112, 32105, 2147161102, 9223372004645756887, 170141183460469231537996491362807709908, 57896044618658097711785492504343953921871039195927143534469727707459805807105

Offset: 0

Gus Wiseman, Aug 07 2019

nonn

The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. The T_0 condition means that the dual is strict (no repeated edges).

			The a(0) = 1 through a(2) = 7 set-systems:
  {}  {}     {}
      {{1}}  {{1}}
             {{2}}
             {{1},{2}}
             {{1},{1,2}}
             {{2},{1,2}}
             {{1},{2},{1,2}}

The non-T_0 version is A058891 shifted to the left.
The covering case is A059201.
The version with empty edges is A326941.
The unlabeled version is A326946.
Cf. A003180, A316978, A319559, A319564, A319637, A326939, A326947, A326949.

dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],UnsameQ@@dual[#]&]],{n,0,3}]

Binomial transform of A059201.

A368600 Number of ways to choose a set of n nonempty subsets of {1..n} such that it is not possible to choose a different element from each.

Original entry on oeis.org

0, 0, 0, 3, 164, 18625, 5491851, 4649088885, 12219849683346

Offset: 0

Gus Wiseman, Jan 01 2024

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.

			The a(3) = 3 set-systems:
  {{1},{2},{1,2}}
  {{1},{3},{1,3}}
  {{2},{3},{2,3}}

Wikipedia, Axiom of choice.

For a unique choice we have A003024, any length A367904 (ranks A367908).
Sets of n nonempty subsets of {1..n} are counted by A136556.
For any length we have A367903, ranks A367907, no singletons A367769.
The complement is A368601, any length A367902 (see also A367770, A367906).
A000372 counts antichains, covering A006126, nonempty A014466.
A003465 counts covering set-systems, unlabeled A055621.
A058891 counts set-systems, unlabeled A000612.
A059201 counts covering T_0 set-systems.
A323818 counts covering connected set-systems, unlabeled A323819.
Cf. A003025, A088957, A133686, A334282, A355529, A355740, A367862, A367867, A367868, A367901, A368094, A368097.

Table[Length[Select[Subsets[Rest[Subsets[Range[n]]], {n}],Length[Select[Tuples[#], UnsameQ@@#&]]==0&]],{n,0,3}]

from itertools import combinations, product, chain
from scipy.special import comb
def v(c):
    for elements in product(*c):
        if len(set(elements)) == len(elements):
            return True
    return False
def a(n):
    if n == 0:
        return 1
    subsets = list(chain.from_iterable(combinations(range(1, n + 1), r) for r in range(1, n + 1)))
    cs = combinations(subsets, n)
    c = sum(1 for c in cs if v(c))
    return c
[print(int(comb(2**n-1,n) - a(n))) for n in range(7)] # Robert P. P. McKone, Jan 02 2024

a(n) = A136556(n) - A368601(n).

a(6) from Robert P. P. McKone, Jan 02 2024

a(7)-a(8) from Christian Sievers, Jul 25 2024

A368601 Number of ways to choose a set of n nonempty subsets of {1..n} such that it is possible to choose a different element from each.

Original entry on oeis.org

1, 1, 3, 32, 1201, 151286, 62453670, 84707326890, 384641855115279

Offset: 0

Gus Wiseman, Jan 01 2024

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.

			The a(2) = 3 set-systems:
  {{1},{2}}
  {{1},{1,2}}
  {{2},{1,2}}
Non-isomorphic representatives of the a(3) = 32 set-systems:
  {{1},{2},{3}}
  {{1},{2},{1,3}}
  {{1},{2},{1,2,3}}
  {{1},{1,2},{1,3}}
  {{1},{1,2},{2,3}}
  {{1},{1,2},{1,2,3}}
  {{1},{2,3},{1,2,3}}
  {{1,2},{1,3},{2,3}}
  {{1,2},{1,3},{1,2,3}}

Wikipedia, Axiom of choice.

For a unique choice we have A003024, any length A367904 (ranks A367908).
Sets of n nonempty subsets of {1..n} are counted by A136556.
For any length we have A367902, ranks A367906, no singletons A367770.
The complement is A368600, any length A367903 (see also A367907, A367769).
A000372 counts antichains, covering A006126, nonempty A014466.
A003465 counts covering set-systems, unlabeled A055621.
A058891 counts set-systems, unlabeled A000612.
A059201 counts covering T_0 set-systems.
A323818 counts covering connected set-systems, unlabeled A323819.
Cf. A003025, A088957, A133686, A334282, A355529, A355740, A367862, A367867, A367901, A367905, A368094, A368097.

Table[Length[Select[Subsets[Rest[Subsets[Range[n]]], {n}],Length[Select[Tuples[#], UnsameQ@@#&]]>0&]],{n,0,3}]

from itertools import combinations, product, chain
def v(c):
    for elements in product(*c):
        if len(set(elements)) == len(elements):
            return True
    return False
def a(n):
    if n == 0:
        return 1
    subsets = list(chain.from_iterable(combinations(range(1, n + 1), r) for r in
range(1, n + 1)))
    cs = combinations(subsets, n)
    c = sum(1 for c in cs if v(c))
    return c
[print(a(n)) for n in range(7)] # Robert P. P. McKone, Jan 02 2024

a(n) + A368600(n) = A136556(n).

a(6) from Robert P. P. McKone, Jan 02 2024

a(7)-a(8) from Christian Sievers, Jul 25 2024

A326939 Number of T_0 sets of subsets of {1..n} that cover all n vertices.

Original entry on oeis.org

2, 2, 8, 192, 63384, 4294003272, 18446743983526539408, 340282366920938462946865774750753349904, 115792089237316195423570985008687907841019819456486779364848020385134373080448

Offset: 0

Gus Wiseman, Aug 07 2019

nonn

The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. The T_0 condition means that the dual is strict (no repeated edges).

			The a(0) = 2 through a(2) = 8 sets of subsets:
  {}    {{1}}     {{1},{2}}
  {{}}  {{},{1}}  {{1},{1,2}}
                  {{2},{1,2}}
                  {{},{1},{2}}
                  {{},{1},{1,2}}
                  {{},{2},{1,2}}
                  {{1},{2},{1,2}}
                  {{},{1},{2},{1,2}}

The non-T_0 version is A000371.
The case without empty edges is A059201.
The non-covering version is A326941.
The unlabeled version is A326942.
The case closed under intersection is A326943.
Cf. A003180, A003181, A003465, A316978, A319564, A319637, A326940, A326947.

dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
Table[Length[Select[Subsets[Subsets[Range[n]]],Union@@#==Range[n]&&UnsameQ@@dual[#]&]],{n,0,3}]

a(n) = 2 * A059201(n).

Inverse binomial transform of A326941.

A326970 Number of set-systems covering n vertices whose dual is a weak antichain.

Original entry on oeis.org

1, 1, 3, 43, 19251

Offset: 0

Gus Wiseman, Aug 10 2019

A set-system is a finite set of finite nonempty sets. The dual of a set-system has, for each vertex, one edges consisting of the indices (or positions) of the edges containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. A weak antichain is a multiset of sets, none of which is a proper subset of any other.

			The a(3) = 43 set-systems:
  {123}  {1}{23}  {1}{2}{3}     {1}{2}{3}{12}
         {2}{13}  {12}{13}{23}  {1}{2}{3}{13}
         {3}{12}  {1}{23}{123}  {1}{2}{3}{23}
                  {2}{13}{123}  {1}{2}{13}{23}
                  {3}{12}{123}  {1}{2}{3}{123}
                                {1}{3}{12}{23}
                                {2}{3}{12}{13}
                                {1}{12}{13}{23}
                                {2}{12}{13}{23}
                                {3}{12}{13}{23}
                                {12}{13}{23}{123}
.
  {1}{2}{3}{12}{13}     {1}{2}{3}{12}{13}{23}    {1}{2}{3}{12}{13}{23}{123}
  {1}{2}{3}{12}{23}     {1}{2}{3}{12}{13}{123}
  {1}{2}{3}{13}{23}     {1}{2}{3}{12}{23}{123}
  {1}{2}{12}{13}{23}    {1}{2}{3}{13}{23}{123}
  {1}{2}{3}{12}{123}    {1}{2}{12}{13}{23}{123}
  {1}{2}{3}{13}{123}    {1}{3}{12}{13}{23}{123}
  {1}{2}{3}{23}{123}    {2}{3}{12}{13}{23}{123}
  {1}{3}{12}{13}{23}
  {2}{3}{12}{13}{23}
  {1}{2}{13}{23}{123}
  {1}{3}{12}{23}{123}
  {2}{3}{12}{13}{123}
  {1}{12}{13}{23}{123}
  {2}{12}{13}{23}{123}
  {3}{12}{13}{23}{123}

Covering set-systems are A003465.
Covering set-systems whose dual is strict are A059201.
The T_1 case is A326961.
The BII-numbers of these set-systems are A326966.
The non-covering case is A326968.
The unlabeled version is A326973.
Cf. A006126, A059052, A059523, A326950, A326960, A326965, A326969, A326971, A326974, A326975, A326978.

dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&stableQ[dual[#],SubsetQ]&]],{n,0,3}]

Inverse binomial transform of A326968.

A367916 Number of sets of nonempty subsets of {1..n} with the same number of edges as covered vertices.

Original entry on oeis.org

1, 2, 6, 45, 1376, 161587, 64552473, 85987037645, 386933032425826, 6005080379837219319, 328011924848834642962619, 64153024576968812343635391868, 45547297603829979923254392040011994, 118654043008142499115765307533395739785599

Offset: 0

Gus Wiseman, Dec 08 2023

nonn

			The a(0) = 1 through a(2) = 6 set-systems:
  {}  {}     {}
      {{1}}  {{1}}
             {{2}}
             {{1},{2}}
             {{1},{1,2}}
             {{2},{1,2}}

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

The covering case is A054780.
For graphs we have A367862, covering A367863, unlabeled A006649.
These set-systems have ranks A367917.
A000372 counts antichains, covering A006126, nonempty A014466.
A003465 counts set-systems covering {1..n}, unlabeled A055621.
A058891 counts set-systems, unlabeled A000612.
A059201 counts covering T_0 set-systems.
A136556 counts set-systems on {1..n} with n edges.
Cf. A092918, A102896, A133686, A306445, A323818, A355740, A367770, A367869, A367901, A367902, A367905.

Table[Length[Select[Subsets[Rest[Subsets[Range[n]]]], Length[Union@@#]==Length[#]&]],{n,0,3}]

\\ Here b(n) is A054780(n).
b(n) = sum(k=0, n, (-1)^(n-k) * binomial(n,k) * binomial(2^k-1, n))
a(n) = sum(k=0, n, binomial(n,k) * b(k)) \\ Andrew Howroyd, Dec 29 2023

Binomial transform of A054780.

A326941 Number of T_0 sets of subsets of {1..n}.

Original entry on oeis.org

2, 4, 14, 224, 64210, 4294322204, 18446744009291513774, 340282366920938463075992982725615419816, 115792089237316195423570985008687907843742078391854287068939455414919611614210

Offset: 0

Gus Wiseman, Aug 07 2019

nonn

The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. The T_0 condition means that the dual is strict (no repeated edges).

			The a(0) = 2 through a(2) = 14 sets of subsets:
  {}    {}        {}
  {{}}  {{}}      {{}}
        {{1}}     {{1}}
        {{},{1}}  {{2}}
                  {{},{1}}
                  {{},{2}}
                  {{1},{2}}
                  {{1},{1,2}}
                  {{2},{1,2}}
                  {{},{1},{2}}
                  {{},{1},{1,2}}
                  {{},{2},{1,2}}
                  {{1},{2},{1,2}}
                  {{},{1},{2},{1,2}}

The non-T_0 version is A001146.
The covering case is A326939.
The case without empty edges is A326940.
The unlabeled version is A326949.
Cf. A003180, A059201, A316978, A319559, A319564, A319637, A326946, A326947.

dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
Table[Length[Select[Subsets[Subsets[Range[n]]],UnsameQ@@dual[#]&]],{n,0,3}]

a(n) = 2 * A326940(n).

Binomial transform of A326939.

a(5)-a(8) from Andrew Howroyd, Aug 14 2019

A330229 Number of fully chiral set-systems covering n vertices.

Original entry on oeis.org

1, 1, 2, 42, 21336

Offset: 0

Gus Wiseman, Dec 08 2019

A set-system is a finite set of finite nonempty sets. It is fully chiral if every permutation of the vertices gives a different representative.

			The a(3) = 42 set-systems:
  {1}{2}{13}    {1}{2}{12}{13}    {1}{2}{12}{13}{123}
  {1}{2}{23}    {1}{2}{12}{23}    {1}{2}{12}{23}{123}
  {1}{3}{12}    {1}{3}{12}{13}    {1}{3}{12}{13}{123}
  {1}{3}{23}    {1}{3}{13}{23}    {1}{3}{13}{23}{123}
  {2}{3}{12}    {2}{3}{12}{23}    {2}{3}{12}{23}{123}
  {2}{3}{13}    {2}{3}{13}{23}    {2}{3}{13}{23}{123}
  {1}{12}{23}   {1}{2}{13}{123}
  {1}{13}{23}   {1}{2}{23}{123}
  {2}{12}{13}   {1}{3}{12}{123}
  {2}{13}{23}   {1}{3}{23}{123}
  {3}{12}{13}   {2}{3}{12}{123}
  {3}{12}{23}   {2}{3}{13}{123}
  {1}{12}{123}  {1}{12}{23}{123}
  {1}{13}{123}  {1}{13}{23}{123}
  {2}{12}{123}  {2}{12}{13}{123}
  {2}{23}{123}  {2}{13}{23}{123}
  {3}{13}{123}  {3}{12}{13}{123}
  {3}{23}{123}  {3}{12}{23}{123}

The non-covering version is A330282.
Costrict (or T_0) covering set-systems are A059201.
BII-numbers of fully chiral set-systems are A330226.
Non-isomorphic, fully chiral multiset partitions are A330227.
Fully chiral partitions are counted by A330228.
Fully chiral covering set-systems are A330229.
Fully chiral factorizations are A330235.
MM-numbers of fully chiral multisets of multisets are A330236.
Cf. A000612, A055621, A083323, A283877, A319559, A319564, A330224, A330234.

graprms[m_]:=Union[Table[Sort[Sort/@(m/.Rule@@@Table[{p[[i]],i},{i,Length[p]}])],{p,Permutations[Union@@m]}]];
Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&Length[graprms[#]]==n!&]],{n,0,3}]

Binomial transform is A330282.

A054780 Number of n-covers of a labeled n-set.

Original entry on oeis.org

1, 1, 3, 32, 1225, 155106, 63602770, 85538516963, 386246934638991, 6001601072676524540, 327951891446717800997416, 64149416776011080449232990868, 45546527789182522411309599498741023, 118653450898277491435912500458608964207578

Offset: 0

Vladeta Jovovic, May 21 2000

Also, number of n X n rational {0,1}-matrices with no zero rows or columns and with all rows distinct, up to permutation of rows.

			From _Gus Wiseman_, Dec 19 2023: (Start)
Number of ways to choose n nonempty sets with union {1..n}. For example, the a(3) = 32 covers are:
  {1}{2}{3}  {1}{2}{13}  {1}{2}{123}  {1}{12}{123}  {12}{13}{123}
             {1}{2}{23}  {1}{3}{123}  {1}{13}{123}  {12}{23}{123}
             {1}{3}{12}  {1}{12}{13}  {1}{23}{123}  {13}{23}{123}
             {1}{3}{23}  {1}{12}{23}  {2}{12}{123}
             {2}{3}{12}  {1}{13}{23}  {2}{13}{123}
             {2}{3}{13}  {2}{3}{123}  {2}{23}{123}
                         {2}{12}{13}  {3}{12}{123}
                         {2}{12}{23}  {3}{13}{123}
                         {2}{13}{23}  {3}{23}{123}
                         {3}{12}{13}  {12}{13}{23}
                         {3}{12}{23}
                         {3}{13}{23}
(End)

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

Main diagonal of A055154.
Cf. A048291, A088310, A181230, A259763.
Covers with any number of edges are counted by A003465, unlabeled A055621.
Connected graphs of this type are counted by A057500, unlabeled A001429.
This is the covering case of A136556.
The case of graphs is A367863, covering case of A116508, unlabeled A006649.
Binomial transform is A367916.
These set-systems have ranks A367917.
The unlabeled version is A368186.
A006129 counts covering graphs, connected A001187, unlabeled A002494.
A046165 counts minimal covers, ranks A309326.
Cf. A006126, A058891, A059201, A133686, A323816, A323817, A323818, A326754, A367862, A367901.

Join[{1}, Table[Sum[StirlingS1[n+1, k+1]*(2^k - 1)^n, {k, 0, n}]/n!, {n, 1, 15}]] (* Vaclav Kotesovec, Jun 04 2022 *)
Table[Length[Select[Subsets[Rest[Subsets[Range[n]]],{n}],Union@@#==Range[n]&]],{n,0,4}] (* Gus Wiseman, Dec 19 2023 *)

a(n) = sum(k=0, n, (-1)^k*binomial(n, k)*binomial(2^(n-k)-1, n)) \\ Andrew Howroyd, Jan 20 2024

a(n) = Sum_{k=0..n} (-1)^k*binomial(n, k)*binomial(2^(n-k)-1, n).
a(n) = (1/n!)*Sum_{k=0..n} Stirling1(n+1, k+1)*(2^k-1)^n.
G.f.: Sum_{n>=0} log(1+(2^n-1)*x)^n/((1+(2^n-1)*x)*n!). - Paul D. Hanna and Vladeta Jovovic, Jan 16 2008
a(n) ~ 2^(n^2) / n!. - Vaclav Kotesovec, Jun 04 2022
Inverse binomial transform of A367916. - Gus Wiseman, Dec 19 2023

cp's OEIS Frontend

A059523 Number of n-element unlabeled ordered T_0-antichains without isolated vertices; number of T_1-hypergraphs (without empty edge and without multiple edges) on n labeled vertices.

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A326940 Number of T_0 set-systems on n vertices.

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A368600 Number of ways to choose a set of n nonempty subsets of {1..n} such that it is not possible to choose a different element from each.

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A368601 Number of ways to choose a set of n nonempty subsets of {1..n} such that it is possible to choose a different element from each.

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A326939 Number of T_0 sets of subsets of {1..n} that cover all n vertices.

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A326970 Number of set-systems covering n vertices whose dual is a weak antichain.

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A367916 Number of sets of nonempty subsets of {1..n} with the same number of edges as covered vertices.

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A326941 Number of T_0 sets of subsets of {1..n}.

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