A306005 -id:A306005 - cp's OEIS frontend

A000612 Number of P-equivalence classes of switching functions of n or fewer variables, divided by 2.

1, 2, 6, 40, 1992, 18666624, 12813206169137152, 33758171486592987164087845043830784, 1435913805026242504952006868879460423834904914948818373264705576411070464

Offset: 0

N. J. A. Sloane

Also number of nonisomorphic sets of nonempty subsets of an n-set.
Equivalently, number of nonisomorphic fillings of a Venn diagram of n sets. - Joerg Arndt, Mar 24 2020
Number of hypergraphs on n unlabeled nodes. - Charles R Greathouse IV, Apr 06 2021

			Non-isomorphic representatives of the a(2) = 6 set-systems are 0, {1}, {12}, {1}{2}, {1}{12}, {1}{2}{12}. - _Gus Wiseman_, Aug 07 2018

M. A. Harrison, Introduction to Switching and Automata Theory. McGraw Hill, NY, 1965, p. 153.
S. Muroga, Threshold Logic and Its Applications. Wiley, NY, 1971, p. 38 Table 2.3.2. - Row 5.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..12
M. A. Harrison, The number of equivalence classes of Boolean functions under groups containing negation, IEEE Trans. Electron. Comput. 12 (1963), 559-561.
M. A. Harrison, The number of equivalence classes of Boolean functions under groups containing negation, IEEE Trans. Electron. Comput. 12 (1963), 559-561. [Annotated scanned copy]
Geon Lee, Seokbum Yoon, Jihoon Ko, Hyunju Kim, and Kijung Shin, Hypergraph Motifs and Their Extensions Beyond Binary, arXiv:2310.15668 [cs.SI], 2023.
Peter H. van der Kamp, Hypergraphs and homogeneous Lotka-Volterra systems with linear Darboux polynomials, arXiv:2411.18264 [nlin.SI], 2024. See p. 4.
Wikipedia, Venn diagram
Index entries for sequences related to Boolean functions

a(n) = A003180(n)/2.

Cf. A007716, A055621, A058891, A283877, A300913, A306005, A317533, A317757.

a:= n-> add(1/(p-> mul((c-> j^c*c!)(coeff(p, x, j)), j=1..degree(p)))(
        add(x^i, i=l))*2^((w-> add(mul(2^igcd(t, l[i]), i=1..nops(l)),
        t=1..w)/w)(ilcm(l[]))), l=combinat[partition](n))/2:
seq(a(n), n=0..9);  # Alois P. Heinz, Aug 12 2019

sysnorm[{}] := {};sysnorm[m_]:=If[Union@@m!=Range[Max@@Flatten[m]],sysnorm[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[Sort[sysnorm[m,1]]]];sysnorm[m_,aft_]:=If[Length[Union@@m]<=aft,{m},With[{mx=Table[Count[m,i,{2}],{i,Select[Union@@m,#>=aft&]}]},Union@@(sysnorm[#,aft+1]&/@Union[Table[Map[Sort,m/.{par+aft-1->aft,aft->par+aft-1},{0,1}],{par,First/@Position[mx,Max[mx]]}]])]];
Table[Length[Union[sysnorm/@Subsets[Rest[Subsets[Range[n]]]]]],{n,4}] (* Gus Wiseman, Aug 07 2018 *)
a[n_] := Sum[1/Function[p, Product[Function[c, j^c*c!][Coefficient[p, x, j]], {j, 1, Exponent[p, x]}]][Total[x^l]]*2^(Function[w, Sum[Product[2^GCD[t, l[[i]]], {i, 1, Length[l]}], {t, 1, w}]/w][If[l=={}, 1, LCM @@ l]]), {l, IntegerPartitions[n]}]/2;
a /@ Range[0, 9] (* Jean-François Alcover, Feb 04 2020, after Alois P. Heinz *)

def partition(n, I=1):
  yield () if n==0 else (n,)
  for i in range(I, n//2 + 1):
    for p in partition(n-i, i):
      yield (i,) + p
def a(n):
  import math, operator, functools
  fracs = [(1<<(sum(functools.reduce(operator.mul, (1<Gregory Morse, Dec 23 2024

a(n) = A003180(n)/2.

More terms from Vladeta Jovovic, Feb 23 2000

A302545 Number of non-isomorphic multiset partitions of weight n with no singletons.

Original entry on oeis.org

1, 0, 2, 3, 12, 23, 84, 204, 682, 1977, 6546, 21003, 72038, 248055, 888771, 3240578, 12152775, 46527471, 182339441, 729405164, 2979121279, 12407308136, 52670355242, 227725915268, 1002285274515, 4487915293698, 20434064295155, 94559526596293, 444527730210294, 2122005930659752

Offset: 0

Gus Wiseman, Jun 20 2018

nonn

A multiset partition is a finite multiset of finite nonempty multisets of positive integers. A singleton is a multiset of size 1. The weight of a multiset partition is the sum of sizes of its elements. Weight is generally not the same as number of vertices.
Also non-isomorphic multiset partitions of weight n with no endpoints, where an endpoint is a vertex appearing only once (degree 1). For example, non-isomorphic representations of the a(4) = 12 multiset partitions are:
  {{1,1,1,1}}
  {{1,1,2,2}}
  {{1},{1,1,1}}
  {{1},{1,2,2}}
  {{1,1},{1,1}}
  {{1,1},{2,2}}
  {{1,2},{1,2}}
  {{1},{1},{1,1}}
  {{1},{1},{2,2}}
  {{1},{2},{1,2}}
  {{1},{1},{1},{1}}
  {{1},{1},{2},{2}}

			The a(4) = 12 multiset partitions:
  {{1,1,1,1}}
  {{1,1,2,2}}
  {{1,2,2,2}}
  {{1,2,3,3}}
  {{1,2,3,4}}
  {{1,1},{1,1}}
  {{1,1},{2,2}}
  {{1,2},{1,2}}
  {{1,2},{2,2}}
  {{1,2},{3,3}}
  {{1,2},{3,4}}
  {{1,3},{2,3}}

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

The set-system version is A330054 (no endpoints) or A306005 (no singletons).
Non-isomorphic multiset partitions are A007716.
Set-systems with no singletons are A016031.
Cf. A049311, A283877, A293606, A293607, A306008, A317533, A317794, A317795, A320665, A330053, A330055, A330058.

\\ compare with similar program for A007716.
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
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, k)={EulerT(Vec(sum(j=1, #q, gcd(t, q[j])*x^lcm(t, q[j])) + O(x*x^k), -k)) - Vec(sum(j=1, #q, if(t%q[j]==0, q[j]*x^t)) + O(x*x^k), -k)}
a(n)={my(s=0); forpart(q=n, s+=permcount(q)*polcoef(exp(x*Ser(sum(t=1, n, K(q, t, n)/t))), n)); s/n!} \\ Andrew Howroyd, Jan 15 2023

Extended by Gus Wiseman, Dec 09 2019

Terms a(11) and beyond from Andrew Howroyd, Jan 15 2023

A320924 Heinz numbers of multigraphical partitions.

Original entry on oeis.org

1, 4, 9, 12, 16, 25, 27, 30, 36, 40, 48, 49, 63, 64, 70, 75, 81, 84, 90, 100, 108, 112, 120, 121, 144, 147, 154, 160, 165, 169, 175, 189, 192, 196, 198, 210, 220, 225, 243, 250, 252, 256, 264, 270, 273, 280, 286, 289, 300, 324, 325, 336, 343, 351, 352, 360

Offset: 1

Gus Wiseman, Oct 24 2018

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
An integer partition is multigraphical if it comprises the multiset of vertex-degrees of some multigraph.
Also Heinz numbers of integer partitions of even numbers whose greatest part is less than or equal to half the sum of parts, i.e., numbers n whose sum of prime indices A056239(n) is even and at least twice the greatest prime index A061395(n). - Gus Wiseman, May 23 2021

			The sequence of all multigraphical partitions begins: (), (11), (22), (211), (1111), (33), (222), (321), (2211), (3111), (21111), (44), (422), (111111), (431), (332), (2222), (4211), (3221), (3311), (22211), (41111), (32111), (55), (221111).
From _Gus Wiseman_, May 23 2021: (Start)
The sequence of terms together with their prime indices and a multigraph realizing each begins:
    1:      () | {}
    4:    (11) | {{1,2}}
    9:    (22) | {{1,2},{1,2}}
   12:   (112) | {{1,3},{2,3}}
   16:  (1111) | {{1,2},{3,4}}
   25:    (33) | {{1,2},{1,2},{1,2}}
   27:   (222) | {{1,2},{1,3},{2,3}}
   30:   (123) | {{1,3},{2,3},{2,3}}
   36:  (1122) | {{1,2},{3,4},{3,4}}
   40:  (1113) | {{1,4},{2,4},{3,4}}
   48: (11112) | {{1,2},{3,5},{4,5}}
   49:    (44) | {{1,2},{1,2},{1,2},{1,2}}
   63:   (224) | {{1,3},{1,3},{2,3},{2,3}}
(End)

These partitions are counted by A209816.
The case with odd weights is A322109.
The conjugate case of equality is A340387.
The conjugate version with odd weights allowed is A344291.
The conjugate opposite version is A344292.
The opposite version with odd weights allowed is A344296.
The conjugate version is A344413.
The conjugate opposite version with odd weights allowed is A344414.
The case of equality is A344415.
The opposite version is A344416.
A000070 counts non-multigraphical partitions.
A025065 counts palindromic partitions.
A035363 counts partitions into even parts.
A056239 adds up prime indices, row sums of A112798.
A110618 counts partitions that are the vertex-degrees of some set multipartition with no singletons.
A334201 adds up all prime indices except the greatest.
Cf. A000041, A000569, A007717, A096373, A265640, A283877, A306005, A318361, A320459, A320911, A320922, A320923, A320925.

prptns[m_]:=Union[Sort/@If[Length[m]==0,{{}},Join@@Table[Prepend[#,m[[ipr]]]&/@prptns[Delete[m,List/@ipr]],{ipr,Select[Prepend[{#},1]&/@Select[Range[2,Length[m]],m[[#]]>m[[#-1]]&],UnsameQ@@m[[#]]&]}]]];
Select[Range[1000],prptns[Flatten[MapIndexed[Table[#2,{#1}]&,If[#==1,{},Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]]]]]!={}&]

Members m of A300061 such that A061395(m) <= A056239(m)/2. - Gus Wiseman, May 23 2021

A368094 Number of non-isomorphic set-systems of weight n contradicting a strict version of the axiom of choice.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 5, 12, 36, 97, 291

Offset: 0

Gus Wiseman, Dec 23 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(5) = 1 through a(7) = 12 set-systems:
  {{1},{2},{3},{2,3}}  {{1},{2},{1,3},{2,3}}    {{1},{2},{1,2},{3,4,5}}
                       {{1},{2},{3},{1,2,3}}    {{1},{3},{2,3},{1,2,3}}
                       {{2},{3},{1,3},{2,3}}    {{1},{4},{1,4},{2,3,4}}
                       {{3},{4},{1,2},{3,4}}    {{2},{3},{2,3},{1,2,3}}
                       {{1},{2},{3},{4},{3,4}}  {{3},{1,2},{1,3},{2,3}}
                                                {{1},{2},{3},{1,3},{2,3}}
                                                {{1},{2},{3},{2,4},{3,4}}
                                                {{1},{2},{3},{4},{2,3,4}}
                                                {{1},{3},{4},{2,4},{3,4}}
                                                {{1},{4},{5},{2,3},{4,5}}
                                                {{2},{3},{4},{1,2},{3,4}}
                                                {{1},{2},{3},{4},{5},{4,5}}

Wikipedia, Axiom of choice.

The case of unlabeled graphs is A140637, complement A134964.
The case of labeled graphs is A367867, complement A133686.
The labeled version is A367903, ranks A367907.
The complement is counted by A368095, connected A368410.
Repeats allowed: A368097, ranks A355529, complement A368098, ranks A368100.
Minimal multiset partitions of this type are ranked by A368187.
The connected case is A368409.
Factorizations of this type are counted by A368413, complement A368414.
Allowing repeated edges gives A368421, complement A368422.
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. A302545, A306005, A316983, A317533, A319616, A321155, A321405, A326031, A330223, A330227, A367905.

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]}]]];
Table[Length[Union[brute/@Select[mpm[n], UnsameQ@@#&&And@@UnsameQ@@@# && Select[Tuples[#], UnsameQ@@#&]=={}&]]],{n,0,8}]

A368095 Number of non-isomorphic set-systems of weight n satisfying a strict version of the axiom of choice.

Original entry on oeis.org

1, 1, 2, 4, 8, 17, 39, 86, 208, 508, 1304

Offset: 0

Gus Wiseman, Dec 24 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.

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(5) = 17 set-systems:
  {1}  {12}    {123}      {1234}        {12345}
       {1}{2}  {1}{23}    {1}{234}      {1}{2345}
               {2}{12}    {12}{34}      {12}{345}
               {1}{2}{3}  {13}{23}      {14}{234}
                          {3}{123}      {23}{123}
                          {1}{2}{34}    {4}{1234}
                          {1}{3}{23}    {1}{2}{345}
                          {1}{2}{3}{4}  {1}{23}{45}
                                        {1}{24}{34}
                                        {1}{4}{234}
                                        {2}{13}{23}
                                        {2}{3}{123}
                                        {3}{13}{23}
                                        {4}{12}{34}
                                        {1}{2}{3}{45}
                                        {1}{2}{4}{34}
                                        {1}{2}{3}{4}{5}

For labeled graphs we have A133686, complement A367867.
For unlabeled graphs we have A134964, complement A140637.
For set-systems we have A367902, complement A367903.
These set-systems have BII-numbers A367906, complement A367907.
The complement is A368094, connected A368409.
Repeats allowed: A368098, ranks A368100, complement A368097, ranks A355529.
Minimal multiset partitions not of this type are counted by A368187.
The connected case is A368410.
Factorizations of this type are counted by A368414, complement A368413.
Allowing repeated edges gives A368422, complement A368421.
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, A007717, A302545, A306005, A316983, A319560, A321194, A321405, A330223, A367769, A367901.

Table[Length[Select[bmp[n], UnsameQ@@#&&And@@UnsameQ@@@#&&Select[Tuples[#], UnsameQ@@#&]!={}&]], {n,0,10}]

A368098 Number of non-isomorphic multiset partitions of weight n satisfying a strict version of the axiom of choice.

Original entry on oeis.org

1, 1, 3, 7, 21, 54, 165, 477, 1501, 4736, 15652

Offset: 0

Gus Wiseman, Dec 25 2023

A multiset partition is a finite multiset of finite nonempty multisets. The weight of a multiset partition 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) = 21 multiset partitions:
  {{1}}  {{1,1}}    {{1,1,1}}      {{1,1,1,1}}
         {{1,2}}    {{1,2,2}}      {{1,1,2,2}}
         {{1},{2}}  {{1,2,3}}      {{1,2,2,2}}
                    {{1},{2,2}}    {{1,2,3,3}}
                    {{1},{2,3}}    {{1,2,3,4}}
                    {{2},{1,2}}    {{1},{1,2,2}}
                    {{1},{2},{3}}  {{1,1},{2,2}}
                                   {{1,2},{1,2}}
                                   {{1},{2,2,2}}
                                   {{1,2},{2,2}}
                                   {{1},{2,3,3}}
                                   {{1,2},{3,3}}
                                   {{1},{2,3,4}}
                                   {{1,2},{3,4}}
                                   {{1,3},{2,3}}
                                   {{2},{1,2,2}}
                                   {{3},{1,2,3}}
                                   {{1},{2},{3,3}}
                                   {{1},{2},{3,4}}
                                   {{1},{3},{2,3}}
                                   {{1},{2},{3},{4}}

Wikipedia, Axiom of choice.

The case of labeled graphs is A133686, complement A367867.
The case of unlabeled graphs is A134964, complement A140637 (apparently).
Set-systems of this type are A367902, ranks A367906, connected A368410.
The complimentary set-systems are A367903, ranks A367907, connected A368409.
For set-systems we have A368095, complement A368094.
The complement is A368097, ranks A355529.
These multiset partitions have ranks A368100.
The connected case is A368412, complement A368411.
Factorizations of this type are counted by A368414, complement A368413.
For set multipartitions we have A368422, complement A368421.
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. A302545, A306005, A316983, A317533, A319616, A330223, A330227, 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]}]]];
Table[Length[Union[brute/@Select[mpm[n], Select[Tuples[#],UnsameQ@@#&]!={}&]]], {n,0,6}]

A306006 Number of non-isomorphic intersecting set-systems of weight n.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 10, 16, 30, 57, 109, 209, 431, 873, 1850, 3979, 8819, 19863

Offset: 0

Gus Wiseman, Jun 16 2018

An intersecting set-system S is a finite set of finite nonempty sets (edges), any two of which have a nonempty intersection. The weight of S is the sum of cardinalities of its elements. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(6) = 10 set-systems:
{{1,2,3,4,5,6}}
{{5},{1,2,3,4,5}}
{{1,5},{2,3,4,5}}
{{3,4},{1,2,3,4}}
{{1,2,5},{3,4,5}}
{{1,3,4},{2,3,4}}
{{3},{2,3},{1,2,3}}
{{4},{1,4},{2,3,4}}
{{1,2},{1,3},{2,3}}
{{1,4},{2,4},{3,4}}

Cf. A007716, A034691, A048143, A049311, A116540, A283877, A293606, A293607, A304867, A305999, A305854-A305857, A306005-A306008.

a(10)-a(17) from Bert Dobbelaere, May 04 2025

A110618 Number of partitions of n with no part larger than n/2. Also partitions of n into n/2 or fewer parts.

Original entry on oeis.org

1, 0, 1, 1, 3, 3, 7, 8, 15, 18, 30, 37, 58, 71, 105, 131, 186, 230, 318, 393, 530, 653, 863, 1060, 1380, 1686, 2164, 2637, 3345, 4057, 5096, 6158, 7665, 9228, 11395, 13671, 16765, 20040, 24418, 29098, 35251, 41869, 50460, 59755, 71669, 84626, 101050

Offset: 0

Henry Bottomley, Aug 01 2005

nonn

Also the number of integer partitions of n that are the vertex-degrees of some set multipartition (multiset of nonempty sets) with no singletons. - Gus Wiseman, Oct 30 2018

			a(5) = 3 since 5 can be partitioned as 1+1+1+1+1, 2+1+1+1, or 2+2+1; not counted are 5, 4+1, or 3+2.
a(6) = 7 since 6 can be partitioned as 1+1+1+1+1+1, 1+1+1+1+2, 1+1+2+2, 2+2+2, 1+1+1+3, 1+2+3, 3+3; not counted are 1+1+4, 2+4, 1+5, 6.
From _Gus Wiseman_, Oct 30 2018: (Start)
The a(2) = 1 through a(8) = 15 partitions with no part larger than n/2:
  (11)  (111)  (22)    (221)    (33)      (322)      (44)
               (211)   (2111)   (222)     (331)      (332)
               (1111)  (11111)  (321)     (2221)     (422)
                                (2211)    (3211)     (431)
                                (3111)    (22111)    (2222)
                                (21111)   (31111)    (3221)
                                (111111)  (211111)   (3311)
                                          (1111111)  (4211)
                                                     (22211)
                                                     (32111)
                                                     (41111)
                                                     (221111)
                                                     (311111)
                                                     (2111111)
                                                     (11111111)
The a(2) = 1 through a(8) = 15 partitions into n/2 or fewer parts:
  (2)  (3)  (4)   (5)   (6)    (7)    (8)
            (22)  (32)  (33)   (43)   (44)
            (31)  (41)  (42)   (52)   (53)
                        (51)   (61)   (62)
                        (222)  (322)  (71)
                        (321)  (331)  (332)
                        (411)  (421)  (422)
                               (511)  (431)
                                      (521)
                                      (611)
                                      (2222)
                                      (3221)
                                      (3311)
                                      (4211)
                                      (5111)
The a(6) = 7 integer partitions of 6 with no part larger than n/2 together with a realizing set multipartition of each (the parts of the partition count the appearances of each vertex in the set multipartition):
      (33): {{1,2},{1,2},{1,2}}
     (321): {{1,2},{1,2},{1,3}}
    (3111): {{1,2},{1,3},{1,4}}
     (222): {{1,2,3},{1,2,3}}
    (2211): {{1,2},{1,2,3,4}}
   (21111): {{1,2},{1,3,4,5}}
  (111111): {{1,2,3,4,5,6}}
(End)

Cf. A000070, A000569, A025065, A049311, A096373, A116540, A147878, A209816, A283877, A306005, A320921.

A000070 := proc(n) add( combinat[numbpart](i),i=0..n) ; end proc:
A110618 := proc(n) combinat[numbpart](n) - A000070(floor((n-1)/2)) ; end proc: # R. J. Mathar, Jan 24 2011

f[n_, 1] := 1; f[1, k_] := 1; f[n_, k_] := f[n, k] = If[k > n, f[n, k - 1], f[n, k - 1] + f[n - k, k]]; g[n_] := f[n, Floor[n/2]]; g[0] = 1; g[1] = 0; Array[g, 47, 0] (* Robert G. Wilson v, Jan 23 2011 *)
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
multhyp[m_]:=Select[mps[m],And[And@@UnsameQ@@@#,Min@@Length/@#>1]&];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
Table[Length[Select[strnorm[n],multhyp[#]!={}&]],{n,8}] (* Gus Wiseman, Oct 30 2018 *)

a(n) = numbpart(n) - sum(i=0, if (n%2, n\2, n/2-1), numbpart(i)); \\ Michel Marcus, Oct 31 2018

a(n) = A000041(n) - Sum_{i=0..floor((n-1)/2)} A000041(i) = A000041(n) - A000070(floor((n-1)/2)) = A110619(n, 2).

a(2*n) = A209816(n). - Gus Wiseman, Oct 30 2018

A317794 Number of non-isomorphic set-systems on n vertices with no singletons.

Original entry on oeis.org

1, 1, 2, 8, 180, 612032, 200253854316544, 263735716028826427534807159537664, 5609038300883759793482640992086670066760184863720423808367168537493504

Offset: 0

Gus Wiseman, Aug 07 2018

nonn

			Non-isomorphic representatives of the a(3) = 8 set-systems:
  0,
  {12}, {123},
  {12}{13}, {12}{123},
  {12}{13}{23}, {12}{13}{123},
  {12}{13}{23}{123}.

Loïc Foissy, Hopf algebraic structures on hypergraphs and multi-complexes, arXiv:2304.00810 [math.CO], 2023.
Peter H. van der Kamp, Hypergraphs and homogeneous Lotka-Volterra systems with linear Darboux polynomials, arXiv:2411.18264 [nlin.SI], 2024. See p. 4.

The spanning case is A317795.

Cf. A000088, A000612, A003180, A007716, A055621, A283877, A300913, A306005, A317533, A317757, A319876, A000616, A000370.

sysnorm[{}] := {};sysnorm[m_]:=If[Union@@m!=Range[Max@@Flatten[m]],sysnorm[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[Sort[sysnorm[m,1]]]];sysnorm[m_,aft_]:=If[Length[Union@@m]<=aft,{m},With[{mx=Table[Count[m,i,{2}],{i,Select[Union@@m,#>=aft&]}]},Union@@(sysnorm[#,aft+1]&/@Union[Table[Map[Sort,m/.{par+aft-1->aft,aft->par+aft-1},{0,1}],{par,First/@Position[mx,Max[mx]]}]])]];
Table[Length[Union[sysnorm/@Select[Subsets[Select[Subsets[Range[n]],Length[#]>1&]],Or[Length[#]==0,Union@@#==Range[Max@@Union@@#]]&]]],{n,4}]
(* second program *)
Table[Sum[2^PermutationCycles[Ordering[Map[Sort,Subsets[Range[n],{2,n}]/.Rule@@@Table[{i,prm[[i]]},{i,n}],{1}]],Length]/n!,{prm,Permutations[Range[n]]}],{n,6}] (* Gus Wiseman, Dec 12 2018 *)

a(n) = A000616(n) - A000370(n). - Tilman Piesk, Apr 14 2025

More terms from Gus Wiseman, Dec 12 2018

A320665 Number of non-isomorphic multiset partitions of weight n with no singletons or vertices that appear only once.

Original entry on oeis.org

1, 0, 1, 1, 5, 6, 27, 47, 169, 406, 1327, 3790, 12560, 39919, 136821, 470589, 1687981, 6162696, 23173374, 88981796, 349969596, 1405386733, 5764142220, 24111709328, 102825231702, 446665313598, 1975339030948, 8888051121242, 40667889052853, 189126710033882, 893526261542899

Offset: 0

Gus Wiseman, Oct 18 2018

nonn

The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}. This sequence counts non-isomorphic multiset partitions with no singletons whose dual also has no singletons.

			Non-isomorphic representatives of the a(2) = 1 through a(6) = 27 multiset partitions:
  {{1,1}}  {{1,1,1}}  {{1,1,1,1}}    {{1,1,1,1,1}}    {{1,1,1,1,1,1}}
                      {{1,1,2,2}}    {{1,1,2,2,2}}    {{1,1,1,2,2,2}}
                      {{1,1},{1,1}}  {{1,1},{1,1,1}}  {{1,1,2,2,2,2}}
                      {{1,1},{2,2}}  {{1,1},{1,2,2}}  {{1,1,2,2,3,3}}
                      {{1,2},{1,2}}  {{1,1},{2,2,2}}  {{1,1},{1,1,1,1}}
                                     {{1,2},{1,2,2}}  {{1,1,1},{1,1,1}}
                                                      {{1,1},{1,2,2,2}}
                                                      {{1,1,1},{2,2,2}}
                                                      {{1,1,2},{1,2,2}}
                                                      {{1,1},{2,2,2,2}}
                                                      {{1,1,2},{2,2,2}}
                                                      {{1,1},{2,2,3,3}}
                                                      {{1,1,2},{2,3,3}}
                                                      {{1,2},{1,1,2,2}}
                                                      {{1,2},{1,2,2,2}}
                                                      {{1,2},{1,2,3,3}}
                                                      {{1,2,2},{1,2,2}}
                                                      {{1,2,3},{1,2,3}}
                                                      {{2,2},{1,1,2,2}}
                                                      {{1,1},{1,1},{1,1}}
                                                      {{1,1},{1,2},{2,2}}
                                                      {{1,1},{2,2},{2,2}}
                                                      {{1,1},{2,2},{3,3}}
                                                      {{1,1},{2,3},{2,3}}
                                                      {{1,2},{1,2},{1,2}}
                                                      {{1,2},{1,2},{2,2}}
                                                      {{1,2},{1,3},{2,3}}

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

Row sums of A369287.

Cf. A001055, A007716, A306005, A317794, A318871, A320663, A320664, A321760.

\\ See links in A339645 for combinatorial species functions.
seq(n)={my(A=symGroupSeries(n)); NumUnlabeledObjsSeq(sCartProd(sExp(A-x*sv(1)), sExp(A-x*sv(1))))} \\ Andrew Howroyd, Jan 17 2023

Vec(G(20,1)) \\ G defined in A369287. - Andrew Howroyd, Jan 28 2024

Terms a(11) and beyond from Andrew Howroyd, Jan 17 2023

cp's OEIS Frontend

A000612 Number of P-equivalence classes of switching functions of n or fewer variables, divided by 2.

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A302545 Number of non-isomorphic multiset partitions of weight n with no singletons.

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A320924 Heinz numbers of multigraphical partitions.

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A368094 Number of non-isomorphic set-systems of weight n contradicting a strict version of the axiom of choice.

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A368095 Number of non-isomorphic set-systems of weight n satisfying a strict version of the axiom of choice.

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A368098 Number of non-isomorphic multiset partitions of weight n satisfying a strict version of the axiom of choice.

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A306006 Number of non-isomorphic intersecting set-systems of weight n.

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A110618 Number of partitions of n with no part larger than n/2. Also partitions of n into n/2 or fewer parts.

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