A367901 -id:A367901 - cp's OEIS frontend

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

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.

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

A367772 Number of sets of nonempty subsets of {1..n} satisfying a strict version of the axiom of choice in more than one way.

Original entry on oeis.org

0, 0, 1, 23, 1105, 154941, 66072394, 88945612865, 396990456067403

Offset: 0

Gus Wiseman, Dec 12 2023

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(3) = 23 set-systems:
  {{1,2}}
  {{1,2,3}}
  {{1},{2,3}}
  {{1},{1,2,3}}
  {{1,2},{1,3}}
  {{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 at least one choice we have A367902.
For no choices we have A367903, no singletons A367769, ranks A367907.
For a unique choice we have A367904, ranks A367908.
These set-systems have ranks A367909.
A000372 counts antichains, covering A006126, nonempty A014466.
A003465 counts covering set-systems, unlabeled A055621.
A058891 counts set-systems, unlabeled A000612.
Cf. A059201, A102896, A133686, A283877, A306445, A323818, A355741, A367770, A367862, A367869, A367901, A367905.

Table[Length[Select[Subsets[Subsets[Range[n]]], Length[Select[Tuples[#], UnsameQ@@#&]]>1&]], {n,0,3}]

A367903(n) + A367904(n) + a(n) = A058891(n).

a(5)-a(8) from Christian Sievers, Jul 26 2024

A368730 Number of n-element sets of singletons or pairs of distinct elements of {1..n} with union {1..n}, or loop-graphs covering n vertices with n edges, such that it is not possible to choose a different element from each.

Original entry on oeis.org

0, 0, 0, 0, 6, 180, 4560, 117600, 3234588, 96119982, 3092585310, 107542211535, 4029055302855, 162040513972623, 6970457656110039, 319598974394563500, 15568332397812799920, 803271954062642638830, 43778508937914677872788, 2513783434620146896920843

Offset: 0

Gus Wiseman, Jan 04 2024

nonn

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(4) = 6 set-systems:
  {{1},{2},{1,2},{3,4}}
  {{1},{3},{1,3},{2,4}}
  {{1},{4},{1,4},{2,3}}
  {{2},{3},{1,4},{2,3}}
  {{2},{4},{1,3},{2,4}}
  {{3},{4},{1,2},{3,4}}

Andrew Howroyd, Table of n, a(n) for n = 0..200
Eric Weisstein's World of Mathematics, Graph Loop.
ProofWiki, Definition:Loop-Graph

The case of a unique choice appears to be A000272.
The version without the choice condition is A368597, non-covering A014068.
The complement appears to be A333331.
The non-covering case is A368596, allowing edges of any size A368600.
Allowing any number of edges of any size gives A367903, ranks A367907.
Allowing any number of non-singletons gives A367868, non-covering A367867.
A000085 counts set partitions into singletons or pairs.
A006125 counts graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A100861 counts set partitions into singletons or pairs by number of pairs.
A111924 counts set partitions into singletons or pairs by length.
A322661 counts labeled covering half-loop-graphs, connected A062740.
Cf. A000666, A116508, A133686, A367863, A367869, A367901, A368532.

Table[Length[Select[Subsets[Subsets[Range[n],{1,2}], {n}],Union@@#==Range[n] && Length[Select[Tuples[#],UnsameQ@@#&]]==0&]],{n,0,5}]

a(n) = A368596(n) + A368597(n) - A014068(n). - Andrew Howroyd, Jan 10 2024

Terms a(7) and beyond from Andrew Howroyd, Jan 10 2024

A387110 Number of ways to choose a sequence of distinct integer partitions, one of each prime index of n.

Original entry on oeis.org

1, 1, 2, 0, 3, 2, 5, 0, 2, 3, 7, 0, 11, 5, 6, 0, 15, 2, 22, 0, 10, 7, 30, 0, 6, 11, 0, 0, 42, 6, 56, 0, 14, 15, 15, 0, 77, 22, 22, 0, 101, 10, 135, 0, 6, 30, 176, 0, 20, 6, 30, 0, 231, 0, 21, 0, 44, 42, 297, 0, 385, 56, 10, 0, 33, 14, 490, 0, 60, 15, 627, 0

Offset: 1

Gus Wiseman, Aug 18 2025

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 axiom of choice says that, given any sequence of nonempty sets, it is possible to choose a sequence containing an element from each. In the strict version, the elements of this sequence must be distinct, meaning none is chosen more than once.

			The prime indices of 9 are (2,2), and there are a(9) = 2 choices:
  ((2),(1,1))
  ((1,1),(2))
The prime indices of 15 are (2,3), and there are a(15) = 5 choices:
  ((2),(3))
  ((2),(2,1))
  ((2),(1,1,1))
  ((1,1),(2,1))
  ((1,1),(1,1,1))

Positions of zeros are A276078 (choosable), complement A276079 (non-choosable).
Allowing repeated partitions gives A299200, A357977, A357982, A357978.
For multiset systems see A355529, A355744, A367771, set systems A367901-A367905.
For divisors see A355731, A355733, A355734, A355735, A355737, A355739, A355740.
For prime factors instead of partitions see A355741, A355742, A387136.
The disjoint case is A383706.
For initial intervals instead of partitions we have A387111.
The case of strict partitions is A387115.
The case of constant partitions is A387120.
Taking each prime factor (instead of index) gives A387133.
A000041 counts integer partitions, strict A000009.
A003963 multiplies together prime indices.
A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
A120383 lists numbers divisible by all of their prime indices.
A289509 lists numbers with relatively prime prime indices.
Cf. A008284, A063834, A261049, A296122, A299201, A335433, A335448, A355745, A387135.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Tuples[IntegerPartitions/@prix[n]],UnsameQ@@#&]],{n,100}]

A387111 Number of ways to choose a sequence of distinct positive integers, one in the initial interval of each prime index of n.

Original entry on oeis.org

1, 1, 2, 0, 3, 1, 4, 0, 2, 2, 5, 0, 6, 3, 4, 0, 7, 0, 8, 0, 6, 4, 9, 0, 6, 5, 0, 0, 10, 1, 11, 0, 8, 6, 9, 0, 12, 7, 10, 0, 13, 2, 14, 0, 2, 8, 15, 0, 12, 2, 12, 0, 16, 0, 12, 0, 14, 9, 17, 0, 18, 10, 4, 0, 15, 3, 19, 0, 16, 4, 20, 0, 21, 11, 4, 0, 16, 4, 22

Offset: 1

Gus Wiseman, Aug 18 2025

nonn

The initial interval of a nonnegative integer x is the set {1,...,x}.
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 axiom of choice says that, given any sequence of nonempty sets, it is possible to choose a sequence containing an element from each. In the strict version, the elements of this sequence must be distinct, meaning none is chosen more than once.

			The prime indices of 75 are (2,3,3), with initial intervals ({1,2},{1,2,3},{1,2,3}), with choices (1,2,3), (1,3,2), (2,1,3), (2,3,1), so a(75) = 4.

Allowing repeated partitions gives A003963.
For constant instead of distinct we have A055396.
For multiset systems see A355529, A355744, A367771, set systems A367901-A367905.
For divisors we have A355739, zeros A355740, strict case of A355731.
For prime factors we have A355741, prime powers A355742, weakly increasing A355745.
For integer partitions we have A387110.
Positions of nonzero terms are A387112 (choosable).
Positions of 0 are A387134 (non-choosable).
A001414 adds up distinct prime divisors, counted by A001221.
A061395 gives greatest prime index.
A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
A120383 lists numbers divisible by all of their prime indices.
A289509 lists numbers with relatively prime prime indices.
A324850 lists numbers divisible by the product of their prime indices.
Cf. A000720, A326841, A335433, A335448, A355733, A355737, A355747, A357980, A383706, A387120, A387133, A387135.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Tuples[Range/@prix[n]],UnsameQ@@#&]],{n,100}]

A369143 Number of labeled simple graphs with n edges and n vertices such that it is not possible to choose a different vertex from each edge (non-choosable).

Original entry on oeis.org

0, 0, 0, 0, 0, 30, 1335, 47460, 1651230, 59636640, 2284113762, 93498908580, 4099070635935, 192365988161490, 9646654985111430, 515736895712230192, 29321225548502776980, 1768139644819077541440, 112805126206185257070660, 7595507651522103787077270, 538504704005397535690160274

Offset: 0

Gus Wiseman, Jan 21 2024

nonn

			The term a(5) = 30 counts all permutations of the graph {{1,2},{1,3},{1,4},{2,3},{2,4}}.

Andrew Howroyd, Table of n, a(n) for n = 0..200

The version without the choice condition is A116508, covering A367863.
The complement is A137916.
Allowing any number of edges gives A367867, covering A367868.
The version with loops is A368596, covering A368730, unlabeled A368835.
For set-systems we have A368600, for any number of edges A367903.
The covering case is A369144.
A006125 counts simple graphs, unlabeled A000088.
A058891 counts set-systems (without singletons A016031), unlabeled A000612.
Cf. A000085, A057500, A062740, A129271, A133686, A333331, A367769, A367869, A367901, A367907.

Table[Length[Select[Subsets[Subsets[Range[n],{2}], {n}],Length[Select[Tuples[#],UnsameQ@@#&]]==0&]],{n,0,5}]

a(n) = A116508(n) - A137916(n). - Andrew Howroyd, Feb 02 2024

a(8) onwards from Andrew Howroyd, Feb 02 2024

A370804 Number of non-condensed integer partitions of n into parts > 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 1, 2, 3, 3, 6, 6, 12, 14, 21, 25, 37, 43, 62, 75, 101, 124, 167, 198, 261, 316, 401, 488, 618, 745, 930, 1119, 1379, 1664, 2032, 2433, 2960, 3537, 4259, 5076, 6094, 7227, 8629, 10205, 12126, 14302, 16932, 19893, 23471, 27502, 32315, 37775

Offset: 0

Gus Wiseman, Mar 03 2024

nonn

These are partitions without ones such that it is not possible to choose a different divisor of each part.

			The a(6) = 1 through a(14) = 12 partitions:
  (222)  .  (2222)  (333)   (3322)   (3332)   (3333)    (4333)    (4442)
                    (3222)  (4222)   (5222)   (4422)    (7222)    (5333)
                            (22222)  (32222)  (6222)    (33322)   (5522)
                                              (33222)   (43222)   (8222)
                                              (42222)   (52222)   (33332)
                                              (222222)  (322222)  (43322)
                                                                  (44222)
                                                                  (53222)
                                                                  (62222)
                                                                  (332222)
                                                                  (422222)
                                                                  (2222222)

These partitions have as ranks the odd terms of A355740.
The version with ones is A370320, complement A239312.
The complement without ones is A370805.
The version for prime factors is A370807, with ones A370593.
The version for factorizations is A370813, complement A370814.
A000005 counts divisors.
A000041 counts integer partitions, strict A000009.
A027746 lists prime factors, A112798 indices, length A001222.
A355731 counts choices of a divisor of each prime index, firsts A355732.
A355741, A355744, A355745 choose prime factors of prime indices.
Cf. A355529, A355739, A367867, A367901, A368110, A368413, A370595, A370806, A370808, A370810.

Table[Length[Select[IntegerPartitions[n],FreeQ[#,1] && Length[Select[Tuples[Divisors/@#],UnsameQ@@#&]]==0&]],{n,0,30}]

More terms from Jinyuan Wang, Feb 14 2025

A387120 Number of ways to choose a different constant integer partition of each prime index of n.

Original entry on oeis.org

1, 1, 2, 0, 2, 2, 3, 0, 2, 2, 2, 0, 4, 3, 4, 0, 2, 2, 4, 0, 6, 2, 3, 0, 2, 4, 0, 0, 4, 4, 2, 0, 4, 2, 6, 0, 6, 4, 8, 0, 2, 6, 4, 0, 4, 3, 4, 0, 6, 2, 4, 0, 5, 0, 4, 0, 8, 4, 2, 0, 6, 2, 6, 0, 8, 4, 2, 0, 6, 6, 6, 0, 4, 6, 4, 0, 6, 8, 4, 0, 0, 2, 2, 0, 4, 4, 8

Offset: 1

Gus Wiseman, Aug 26 2025

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 prime indices of 90 are {1,2,2,3}, with choices:
  ((1),(2),(1,1),(3))
  ((1),(1,1),(2),(3))
  ((1),(2),(1,1),(1,1,1))
  ((1),(1,1),(2),(1,1,1))
so a(90) = 4.

For multiset systems see A355529, set systems A367901.
For not necessarily different choices we have A355731, see A355740.
For divisors instead of constant partitions we have A355739 (also the disjoint case).
For prime factors instead of constant partitions we have A387136.
For all instead of just constant partitions we have A387110, disjoint case A383706.
For initial intervals instead of partitions we have A387111.
For strict instead of constant partitions we have A387115.
Twice partitions of this type are counted by A387179, constant-block case of A296122.
Positions of zero are A387180 (non-choosable), complement A387181 (choosable).
A000041 counts integer partitions, strict A000009.
A003963 multiplies together prime indices.
A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
A120383 lists numbers divisible by all of their prime indices.
A289509 lists numbers with relatively prime prime indices.
Cf. A000005, A063834, A261049, A299200, A299201, A335433, A335448, A355741, A357977, A387135.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Tuples[Select[IntegerPartitions[#],SameQ@@#&]&/@prix[n]],UnsameQ@@#&]],{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

cp's OEIS Frontend

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

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

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A367772 Number of sets of nonempty subsets of {1..n} satisfying a strict version of the axiom of choice in more than one way.

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A368730 Number of n-element sets of singletons or pairs of distinct elements of {1..n} with union {1..n}, or loop-graphs covering n vertices with n edges, such that it is not possible to choose a different element from each.

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A387110 Number of ways to choose a sequence of distinct integer partitions, one of each prime index of n.

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A387111 Number of ways to choose a sequence of distinct positive integers, one in the initial interval of each prime index of n.

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A369143 Number of labeled simple graphs with n edges and n vertices such that it is not possible to choose a different vertex from each edge (non-choosable).

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A370804 Number of non-condensed integer partitions of n into parts > 1.

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