A319762 -id:A319762 - cp's OEIS frontend

A319759 Number of non-isomorphic intersecting multiset partitions of weight n with empty intersection.

1, 0, 0, 0, 0, 0, 1, 2, 13, 49, 199

Offset: 0

Gus Wiseman, Sep 27 2018

A multiset partition is intersecting if no two parts are disjoint. 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(6) = 1 through a(8) = 13 multiset partitions:
6: {{1,2},{1,3},{2,3}}
7: {{1,2},{1,3},{2,3,3}}
   {{1,3},{1,4},{2,3,4}}
8: {{1,2},{1,3},{2,2,3,3}}
   {{1,2},{1,3},{2,3,3,3}}
   {{1,2},{1,3},{2,3,4,4}}
   {{1,2},{1,3,3},{2,3,3}}
   {{1,2},{1,3,4},{2,3,4}}
   {{1,3},{1,4},{2,3,4,4}}
   {{1,3},{1,1,2},{2,3,3}}
   {{1,3},{1,2,2},{2,3,3}}
   {{1,4},{1,5},{2,3,4,5}}
   {{2,3},{1,2,4},{3,4,4}}
   {{2,4},{1,2,3},{3,4,4}}
   {{2,4},{1,2,5},{3,4,5}}
   {{1,2},{1,3},{2,3},{2,3}}

Cf. A007716, A283877, A305854, A306006, A317757, A318715, A318717.

Cf. A319752, A319762, A319763, A319764, A319765, A319779, A319781.

A326912 BII-numbers of pairwise intersecting set-systems with empty intersection.

Original entry on oeis.org

0, 52, 116, 772, 832, 836, 1072, 1076, 1136, 1140, 1796, 1856, 1860, 2320, 2368, 2384, 2592, 2624, 2656, 2880, 3088, 3104, 3120, 3136, 3152, 3168, 3184, 3344, 3392, 3408, 3616, 3648, 3680, 3904, 4132, 4148, 4196, 4212, 4612, 4640, 4644, 4672, 4676, 4704, 4708

Offset: 1

Gus Wiseman, Aug 04 2019

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every finite set of finite nonempty sets has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges.

			The sequence of all pairwise intersecting set-systems with empty intersection, together with their BII-numbers, begins:
     0: {}
    52: {{1,2},{1,3},{2,3}}
   116: {{1,2},{1,3},{2,3},{1,2,3}}
   772: {{1,2},{1,4},{2,4}}
   832: {{1,2,3},{1,4},{2,4}}
   836: {{1,2},{1,2,3},{1,4},{2,4}}
  1072: {{1,3},{2,3},{1,2,4}}
  1076: {{1,2},{1,3},{2,3},{1,2,4}}
  1136: {{1,3},{2,3},{1,2,3},{1,2,4}}
  1140: {{1,2},{1,3},{2,3},{1,2,3},{1,2,4}}
  1796: {{1,2},{1,4},{2,4},{1,2,4}}
  1856: {{1,2,3},{1,4},{2,4},{1,2,4}}
  1860: {{1,2},{1,2,3},{1,4},{2,4},{1,2,4}}
  2320: {{1,3},{1,4},{3,4}}
  2368: {{1,2,3},{1,4},{3,4}}
  2384: {{1,3},{1,2,3},{1,4},{3,4}}
  2592: {{2,3},{2,4},{3,4}}
  2624: {{1,2,3},{2,4},{3,4}}
  2656: {{2,3},{1,2,3},{2,4},{3,4}}
  2880: {{1,2,3},{1,4},{2,4},{3,4}}

Cf. A048793, A051185, A305843, A317752, A317755, A317757, A319077, A326031, A326910, A326911.

Cf. A318715, A319759, A319762, A319763, A319764.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[0,1000],(#==0||Intersection@@bpe/@bpe[#]=={})&&stableQ[bpe/@bpe[#],Intersection[#1,#2]=={}&]&]

A319763 Number of non-isomorphic strict intersecting multiset partitions (sets of multisets) of weight n with empty intersection.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 2, 12, 46, 181

Offset: 0

Gus Wiseman, Sep 27 2018

A multiset partition is intersecting if no two parts are disjoint. 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(6) = 1 through a(8) = 12 multiset partitions:
6: {{1,2},{1,3},{2,3}}
7: {{1,2},{1,3},{2,3,3}}
   {{1,3},{1,4},{2,3,4}}
8: {{1,2},{1,3},{2,2,3,3}}
   {{1,2},{1,3},{2,3,3,3}}
   {{1,2},{1,3},{2,3,4,4}}
   {{1,2},{1,3,3},{2,3,3}}
   {{1,2},{1,3,4},{2,3,4}}
   {{1,3},{1,4},{2,3,4,4}}
   {{1,3},{1,1,2},{2,3,3}}
   {{1,3},{1,2,2},{2,3,3}}
   {{1,4},{1,5},{2,3,4,5}}
   {{2,3},{1,2,4},{3,4,4}}
   {{2,4},{1,2,3},{3,4,4}}
   {{2,4},{1,2,5},{3,4,5}}

Cf. A007716, A281116, A283877, A305854, A306006, A316980, A317757, A318715, A318717.

Cf. A319752, A319759, A319762, A319764, A319765, A319779, A319781.

A319764 Number of non-isomorphic intersecting set systems of weight n with empty intersection.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 1, 3, 8, 18

Offset: 0

Gus Wiseman, Sep 27 2018

A set system is a finite set of finite nonempty sets. It is intersecting if no two parts are disjoint. The weight of a set system is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

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

Cf. A007716, A281116, A283877, A305854, A306006, A316980, A317752, A317755, A317757, A318715, A318717.

Cf. A319752, A319759, A319762, A319763, A319765, A319779, A319781.

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A319759 Number of non-isomorphic intersecting multiset partitions of weight n with empty intersection.

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A326912 BII-numbers of pairwise intersecting set-systems with empty intersection.

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A319763 Number of non-isomorphic strict intersecting multiset partitions (sets of multisets) of weight n with empty intersection.

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A319764 Number of non-isomorphic intersecting set systems of weight n with empty intersection.

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