A326030 -id:A326030 - cp's OEIS frontend

A326950 Number of T_0 antichains of nonempty subsets of {1..n}.

1, 2, 4, 12, 107, 6439, 7726965, 2414519001532, 56130437161079183223017, 286386577668298409107773412840148848120595

Offset: 0

Gus Wiseman, Aug 08 2019

The dual of a set-system has, for each vertex, one edge 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}}. The T_0 condition means that the dual is strict (no repeated edges).

			The a(0) = 1 through a(3) = 12 antichains:
  {}  {}     {}         {}
      {{1}}  {{1}}      {{1}}
             {{2}}      {{2}}
             {{1},{2}}  {{3}}
                        {{1},{2}}
                        {{1},{3}}
                        {{2},{3}}
                        {{1,2},{1,3}}
                        {{1,2},{2,3}}
                        {{1},{2},{3}}
                        {{1,3},{2,3}}
                        {{1,2},{1,3},{2,3}}

Antichains of nonempty sets are A014466.
T_0 set-systems are A326940.
The covering case is A245567.
Cf. A006126, A059201, A059052, A245567, A319559, A319564, A326030, A326946, A326947.

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}]],stableQ[#,SubsetQ]&&UnsameQ@@dual[#]&]],{n,0,3}]

Binomial transform of A245567, if we assume A245567(0) = 1.

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

a(9), based on A245567, from Patrick De Causmaecker, Jun 01 2023

A326574 Number of antichains of subsets of {1..n} with equal edge-sums.

Original entry on oeis.org

2, 3, 5, 10, 22, 61, 247, 2096, 81896, 52260575

Offset: 0

Gus Wiseman, Jul 18 2019

An antichain is a finite set of finite sets, none of which is a subset of any other. The edge-sums are the sums of vertices in each edge, so for example the edge sums of {{1,3},{2,5},{3,4,5}} are {4,7,12}.

			The a(0) = 2 through a(4) = 22 antichains:
  {}    {}     {}       {}           {}
  {{}}  {{}}   {{}}     {{}}         {{}}
        {{1}}  {{1}}    {{1}}        {{1}}
               {{2}}    {{2}}        {{2}}
               {{1,2}}  {{3}}        {{3}}
                        {{1,2}}      {{4}}
                        {{1,3}}      {{1,2}}
                        {{2,3}}      {{1,3}}
                        {{1,2,3}}    {{1,4}}
                        {{3},{1,2}}  {{2,3}}
                                     {{2,4}}
                                     {{3,4}}
                                     {{1,2,3}}
                                     {{1,2,4}}
                                     {{1,3,4}}
                                     {{2,3,4}}
                                     {{1,2,3,4}}
                                     {{3},{1,2}}
                                     {{4},{1,3}}
                                     {{1,4},{2,3}}
                                     {{2,4},{1,2,3}}
                                     {{3,4},{1,2,4}}

Set partitions with equal block-sums are A035470.
Antichains with different edge-sums are A326030.
MM-numbers of multiset partitions with equal part-sums are A326534.
The covering case is A326566.
Cf. A000372, A003182, A006126, A307249, A321455, A321717, A321718, A326518, A326565, A326572.

stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
cleqset[set_]:=stableSets[Subsets[set],SubsetQ[#1,#2]||Total[#1]!=Total[#2]&];
Table[Length[cleqset[Range[n]]],{n,0,5}]

a(9) from Andrew Howroyd, Aug 13 2019

A327903 Number of set-systems covering n vertices where every edge has a different sum.

Original entry on oeis.org

1, 1, 5, 77, 7369, 10561753, 839653402893, 15924566366443524837, 315320784127456186118309342769, 29238175285109256786706269143580213236526609, 59347643832090275881798554403880633753161146711444051797893301

Offset: 0

Gus Wiseman, Sep 30 2019

nonn

A set-system is a set of nonempty sets. It is covering if there are no isolated (uncovered) vertices.

			The a(3) = 77 set-systems:
  123  1-23    1-2-3      1-2-3-13      1-2-3-13-23     1-2-3-13-23-123
       2-13    1-2-13     1-2-3-23      1-2-12-13-23    1-2-12-13-23-123
       1-123   1-2-23     1-2-12-13     1-2-3-13-123
       12-13   1-3-23     1-2-12-23     1-2-3-23-123
       12-23   2-3-13     1-2-13-23     1-2-12-13-123
       13-23   1-12-13    1-2-3-123     1-2-12-23-123
       2-123   1-12-23    1-3-13-23     1-2-13-23-123
       3-123   1-13-23    2-3-13-23     1-3-13-23-123
       12-123  1-2-123    1-12-13-23    2-3-13-23-123
       13-123  1-3-123    1-2-12-123    1-12-13-23-123
       23-123  2-12-13    1-2-13-123    2-12-13-23-123
               2-12-23    1-2-23-123
               2-13-23    1-3-13-123
               2-3-123    1-3-23-123
               3-13-23    2-12-13-23
               1-12-123   2-3-13-123
               1-13-123   2-3-23-123
               12-13-23   1-12-13-123
               1-23-123   1-12-23-123
               2-12-123   1-13-23-123
               2-13-123   2-12-13-123
               2-23-123   2-12-23-123
               3-13-123   2-13-23-123
               3-23-123   3-13-23-123
               12-13-123  12-13-23-123
               12-23-123
               13-23-123

Andrew Howroyd, Table of n, a(n) for n = 0..20
Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

The antichain case is A326572.
The graphical case is A327904.
Cf. A003465, A006126, A035470, A275780, A321469, A326030, A326519, A326535, A326565, A326571, A326573.

stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
qes[n_]:=Select[stableSets[Subsets[Range[n],{1,n}],Total[#1]==Total[#2]&],Union@@#==Range[n]&];
Table[Length[qes[n]],{n,0,4}]

\\ by inclusion/exclusion on covered vertices.
C(v)={my(u=Vecrev(-1 + prod(k=1, #v, 1 + x^v[k]))); prod(i=1, #u, 1 + u[i])}
a(n)={my(s=0); forsubset(n, v, s += (-1)^(n-#v)*C(v)); s} \\ Andrew Howroyd, Oct 02 2019

Terms a(4) and beyond from Andrew Howroyd, Oct 02 2019

A327904 Number of labeled simple graphs with vertices {1..n} such that every edge has a different sum.

Original entry on oeis.org

1, 1, 2, 8, 48, 432, 5184, 82944, 1658880, 41472000, 1244160000, 44789760000, 1881169920000, 92177326080000, 5161930260480000, 330363536670720000, 23786174640291840000, 1926680145863639040000, 173401213127727513600000, 17340121312772751360000000

Offset: 0

Gus Wiseman, Sep 30 2019

nonn

			The graph with edge-set {{1,2},{1,3},{1,4},{2,3}}, which looks like a triangle with a tail, has edges {1,4} and {2,3} with equal sum, so is not counted under a(4).

Andrew Howroyd, Table of n, a(n) for n = 0..100
Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.
Gus Wiseman, The a(4) = 48 graphs where every edge has a different sum.

The generalization to antichains is A326030.

Cf. A006125, A006129, A275780, A321469, A326519, A326535, A327903.

a:= proc(n) option remember; `if`(n=0, 1,
      a(n-1)*ceil(n/2)*ceil(n/2+1/4))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Oct 03 2019

stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
qes[n_]:=stableSets[Subsets[Range[n],{2}],Total[#1]==Total[#2]&];
Table[Length[qes[n]],{n,0,5}]

a(n) = {prod(k=1, 2*n+1, ceil(k/4))} \\ Andrew Howroyd, Oct 02 2019

a(n) = Product_{k=1..2*n+1} ceiling(k/4). - Andrew Howroyd, Oct 02 2019

Terms a(8) and beyond from Andrew Howroyd, Oct 02 2019

cp's OEIS Frontend

A326950 Number of T_0 antichains of nonempty subsets of {1..n}.

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A326574 Number of antichains of subsets of {1..n} with equal edge-sums.

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A327903 Number of set-systems covering n vertices where every edge has a different sum.

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A327904 Number of labeled simple graphs with vertices {1..n} such that every edge has a different sum.

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