A293606 -id:A293606 - cp's OEIS frontend

A327808 Number of unlabeled, disconnected, nonempty antichains of subsets of {1..n}.

0, 0, 1, 3, 9, 32, 233, 16578

Offset: 0

Gus Wiseman, Sep 26 2019

An antichain is a set of nonempty sets, none of which is a subset of any other. A singleton is considered to be connected.

			Non-isomorphic representatives of the a(2) = 1 through a(4) = 9 antichains:
   {{1},{2}}  {{1},{2}}      {{1},{2}}
              {{1},{2,3}}    {{1},{2,3}}
              {{1},{2},{3}}  {{1},{2},{3}}
                             {{1},{2,3,4}}
                             {{1,2},{3,4}}
                             {{1},{2},{3,4}}
                             {{1},{2},{3},{4}}
                             {{2},{1,3},{1,4}}
                             {{4},{1,2},{1,3},{2,3}}

The labeled version is A327354 - 1.
The covering case is A327426.
Unlabeled antichains that are either not connected or not covering are A327437.
The version with empty antichains allowed is A327424.
Cf. A000372, A014466, A120338, A293606, A326704, A327062, A327355.

a(n) = A327424(n) - 1.

A319638 Number of non-isomorphic weight-n antichains of distinct sets whose dual is also an antichain of distinct sets.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 3, 4, 7

Offset: 0

Gus Wiseman, Sep 25 2018

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},{2,2}} is {{1},{1,2,2}}.

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(1) = 1 through a(10) = 7 antichains:
1:  {{1}}
2:  {{1},{2}}
3:  {{1},{2},{3}}
4:  {{1},{2},{3},{4}}
5:  {{1},{2},{3},{4},{5}}
6:  {{1,2},{1,3},{2,3}}
    {{1},{2},{3},{4},{5},{6}}
7:  {{1},{2,3},{2,4},{3,4}}
    {{1},{2},{3},{4},{5},{6},{7}}
8:  {{1,2},{1,3},{2,4},{3,4}}
    {{1},{2},{3,4},{3,5},{4,5}}
    {{1},{2},{3},{4},{5},{6},{7},{8}}
9:  {{1,2},{1,3},{1,4},{2,3,4}}
    {{1},{2,3},{2,4},{3,5},{4,5}}
    {{1},{2},{3},{4,5},{4,6},{5,6}}
    {{1},{2},{3},{4},{5},{6},{7},{8},{9}}
10: {{1,3},{2,4},{1,2,5},{3,4,5}}
    {{1},{2,3},{2,4},{2,5},{3,4,5}}
    {{1,2},{1,3},{2,4},{3,5},{4,5}}
    {{1,3},{1,4},{2,3},{2,4},{3,4}}
    {{1},{2},{3,4},{3,5},{4,6},{5,6}}
    {{1},{2},{3},{4},{5,6},{5,7},{6,7}}
    {{1},{2},{3},{4},{5},{6},{7},{8},{9},{10}}

Cf. A006126, A007716, A049311, A059201, A283877, A293606, A316980, A316983, A318099, A319558, A319616-A319646, A300913.

Euler transform of A319625.

A319641 Number of non-isomorphic weight-n antichains of distinct multisets whose dual is also an antichain of (not necessarily distinct) multisets.

Original entry on oeis.org

1, 1, 3, 5, 11, 18, 41, 70, 159, 323, 778

Offset: 0

Gus Wiseman, Sep 25 2018

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},{2,2}} is {{1},{1,2,2}}.

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

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

Cf. A006126, A007716, A049311, A059201, A283877, A293606, A316980, A316983, A318099, A319558, A319616-A319646.

Euler transform of A319628.

A319643 Number of non-isomorphic weight-n strict multiset partitions whose dual is an antichain of (not necessarily distinct) multisets.

Original entry on oeis.org

1, 1, 3, 6, 15, 29, 82, 179, 504, 1302, 3822

Offset: 0

Gus Wiseman, Sep 25 2018

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},{2,2}} is {{1},{1,2,2}}.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.
From Gus Wiseman, Aug 15 2019: (Start)
Also the number of non-isomorphic T_0 weak antichains of weight n. The T_0 condition means that the dual is strict (no repeated edges). A weak antichain is a multiset of multisets, none of which is a proper submultiset of any other. For example, non-isomorphic representatives of the a(0) = 1 through a(4) = 15 T_0 weak antichains are:
  {}  {{1}}  {{1,1}}    {{1,1,1}}      {{1,1,1,1}}
             {{1},{1}}  {{1,2,2}}      {{1,2,2,2}}
             {{1},{2}}  {{1},{2,2}}    {{1,1},{1,1}}
                        {{1},{1},{1}}  {{1,1},{2,2}}
                        {{1},{2},{2}}  {{1},{2,2,2}}
                        {{1},{2},{3}}  {{1,2},{2,2}}
                                       {{1},{2,3,3}}
                                       {{1,3},{2,3}}
                                       {{1},{1},{2,2}}
                                       {{1},{2},{3,3}}
                                       {{1},{1},{1},{1}}
                                       {{1},{1},{2},{2}}
                                       {{1},{2},{2},{2}}
                                       {{1},{2},{3},{3}}
                                       {{1},{2},{3},{4}}
(End)

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 15 multiset partitions:
1: {{1}}
2: {{1,1}}
   {{1,2}}
   {{1},{2}}
3: {{1,1,1}}
   {{1,2,3}}
   {{1},{1,1}}
   {{1},{2,2}}
   {{1},{2,3}}
   {{1},{2},{3}}
4: {{1,1,1,1}}
   {{1,1,2,2}}
   {{1,2,3,4}}
   {{1},{1,1,1}}
   {{1},{1,2,2}}
   {{1},{2,2,2}}
   {{1},{2,3,4}}
   {{1,1},{2,2}}
   {{1,2},{3,3}}
   {{1,2},{3,4}}
   {{1},{2},{1,2}}
   {{1},{2},{2,2}}
   {{1},{2},{3,3}}
   {{1},{2},{3,4}}
   {{1},{2},{3},{4}}

Cf. A006126, A007716, A059201, A283877, A293606, A316980, A316983, A318099, A319558, A319616-A319646.

Cf. A245567, A293993, A319721, A326704, A326950, A326973, A326978.

A319644 Number of non-isomorphic weight-n antichains of distinct multisets whose dual is also an antichain of distinct multisets.

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 18, 31, 73, 162, 413

Offset: 0

Gus Wiseman, Sep 25 2018

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},{2,2}} is {{1},{1,2,2}}.

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(1) = 1 through a(5) = 8 antichains:
1: {{1}}
2: {{1,1}}
   {{1},{2}}
3: {{1,1,1}}
   {{1},{2,2}}
   {{1},{2},{3}}
4: {{1,1,1,1}}
   {{1},{2,2,2}}
   {{1,1},{2,2}}
   {{1},{2},{3,3}}
   {{1},{2},{3},{4}}
5: {{1,1,1,1,1}}
   {{1},{2,2,2,2}}
   {{1,1},{1,2,2}}
   {{1,1},{2,2,2}}
   {{1},{2},{3,3,3}}
   {{1},{2,2},{3,3}}
   {{1},{2},{3},{4,4}}
   {{1},{2},{3},{4},{5}}

Cf. A006126, A007716, A049311, A059201, A283877, A293606, A316980, A316983, A318099, A319558, A319616-A319646.

Euler transform of A319629.

A320291 Number of singleton-free multiset partitions of integer partitions of n with no 1's.

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 3, 3, 7, 8, 15, 19, 36, 46, 79, 110, 181, 254, 407, 580, 907, 1309, 2004, 2909, 4410, 6407, 9599, 13984, 20782, 30252, 44677, 64967, 95414, 138563, 202527, 293583, 427442, 618337, 897023, 1295020, 1872696, 2697777, 3889964, 5591917, 8041593, 11535890

Offset: 0

Gus Wiseman, Oct 09 2018

nonn

			The a(4) = 1 through a(10) = 15 multiset partitions:
  ((22))  ((23))  ((24))   ((25))   ((26))      ((27))      ((28))
                  ((33))   ((34))   ((35))      ((36))      ((37))
                  ((222))  ((223))  ((44))      ((45))      ((46))
                                    ((224))     ((225))     ((55))
                                    ((233))     ((234))     ((226))
                                    ((2222))    ((333))     ((235))
                                    ((22)(22))  ((2223))    ((244))
                                                ((22)(23))  ((334))
                                                            ((2224))
                                                            ((2233))
                                                            ((22222))
                                                            ((22)(24))
                                                            ((22)(33))
                                                            ((23)(23))
                                                            ((22)(222))

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Cf. A002865, A007716, A049311, A083751, A283877, A293606, A302545, A304966, A304967, A320294, A320295, A320296.

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]]]];
Table[Length[Select[Join@@mps/@Select[IntegerPartitions[n],FreeQ[#,1]&],FreeQ[Length/@#,1]&]],{n,20}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(v=vector(n,i,i>1)); concat([1], EulerT(EulerT(v)-v))} \\ Andrew Howroyd, Oct 25 2018

Euler transform of A083751. - Andrew Howroyd, Oct 25 2018

Terms a(21) and beyond from Andrew Howroyd, Oct 25 2018

A327807 Triangle read by rows where T(n,k) is the number of unlabeled antichains of sets with n vertices and vertex-connectivity >= k.

Original entry on oeis.org

1, 2, 0, 4, 1, 0, 9, 3, 2, 0, 29, 14, 10, 6, 0, 209, 157, 128, 91, 54, 0

Offset: 0

Gus Wiseman, Sep 26 2019

An antichain is a set of sets, none of which is a subset of any other.

The vertex-connectivity of a set-system is the minimum number of vertices that must be removed (along with any resulting empty edges) to obtain a non-connected set-system or singleton. Note that this means a single node has vertex-connectivity 0.

			Triangle begins:
    1
    2   0
    4   1   0
    9   3   2   0
   29  14  10   6   0
  209 157 128  91  54   0

Column k = 0 is A306505.
Column k = 1 is A261006 (clutters), if we assume A261006(0) = A261006(1) = 0.
Column k = 2 is A305028 (blobs), if we assume A305028(0) = A305028(2) = 0.
Except for the first column, same as A327358 (the covering case).
The labeled version is A327806.
Cf. A014466, A293606, A326704, A326950, A327062, A327334, A327350, A327351, A327805, A327808.

A319631 Number of non-isomorphic weight-n antichains of multisets whose dual is a chain of distinct multisets.

Original entry on oeis.org

1, 1, 2, 3, 5, 5, 13, 11, 25, 31, 54

Offset: 0

Gus Wiseman, Sep 25 2018

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},{2,2}} is {{1},{1,2,2}}.

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(1) = 1 through a(5) = 5 antichains:
1: {{1}}
2: {{1,1}}
   {{1},{1}}
3: {{1,1,1}}
   {{1,2,2}}
   {{1},{1},{1}}
4: {{1,1,1,1}}
   {{1,2,2,2}}
   {{1,1},{1,1}}
   {{1,2},{2,2}}
   {{1},{1},{1},{1}}
5: {{1,1,1,1,1}}
   {{1,1,2,2,2}}
   {{1,2,2,2,2}}
   {{1,2},{2,2,2}}
   {{1},{1},{1},{1},{1}}

Cf. A000219, A006126, A007716, A059201, A293606, A316980, A316983, A318099, A319558, A319616-A319646, A300913.

A319632 Number of non-isomorphic weight-n antichains of (not necessarily distinct) sets whose dual is also an antichain of (not necessarily distinct) sets.

Original entry on oeis.org

1, 1, 3, 5, 11, 17, 35, 53, 100, 154, 275

Offset: 0

Gus Wiseman, Sep 25 2018

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},{2,2}} is {{1},{1,2,2}}.

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

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

Cf. A006126, A007716, A049311, A283877, A293606, A316980, A316983, A318099, A319558, A319616-A319646, A300913.

A319633 Number of antichain covers of n vertices by distinct sets whose dual is also an antichain of (not necessarily distinct) sets.

Original entry on oeis.org

1, 1, 2, 6, 40, 2309

Offset: 0

Gus Wiseman, Sep 25 2018

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},{2,2}} is {{1},{1,2,2}}.

			The a(3) = 6 antichain covers:
   {{1,2,3}}
   {{3},{1,2}}
   {{2},{1,3}}
   {{1},{2,3}}
   {{1},{2},{3}}
   {{1,2},{1,3},{2,3}}

Cf. A006126, A007716, A049311, A059201, A283877, A293606, A316980, A316983, A318099, A319558, A319616-A319646, A300913.

cp's OEIS Frontend

A327808 Number of unlabeled, disconnected, nonempty antichains of subsets of {1..n}.

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A319638 Number of non-isomorphic weight-n antichains of distinct sets whose dual is also an antichain of distinct sets.

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A319641 Number of non-isomorphic weight-n antichains of distinct multisets whose dual is also an antichain of (not necessarily distinct) multisets.

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A319643 Number of non-isomorphic weight-n strict multiset partitions whose dual is an antichain of (not necessarily distinct) multisets.

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A319644 Number of non-isomorphic weight-n antichains of distinct multisets whose dual is also an antichain of distinct multisets.

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A320291 Number of singleton-free multiset partitions of integer partitions of n with no 1's.

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A327807 Triangle read by rows where T(n,k) is the number of unlabeled antichains of sets with n vertices and vertex-connectivity >= k.

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A319631 Number of non-isomorphic weight-n antichains of multisets whose dual is a chain of distinct multisets.

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A319632 Number of non-isomorphic weight-n antichains of (not necessarily distinct) sets whose dual is also an antichain of (not necessarily distinct) sets.

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A319633 Number of antichain covers of n vertices by distinct sets whose dual is also an antichain of (not necessarily distinct) sets.

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