cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-15 of 15 results.

A321231 Number of non-isomorphic connected weight-n multiset partitions with no singletons and multiset density -1.

Original entry on oeis.org

1, 0, 2, 3, 8, 15, 42, 94, 256, 656, 1807
Offset: 0

Views

Author

Gus Wiseman, Oct 31 2018

Keywords

Comments

The multiset density of a multiset partition is the sum of the numbers of distinct vertices in each part minus the number of parts minus the number of vertices.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

Examples

			Non-isomorphic representatives of the a(2) = 2 through a(5) = 15 multiset partitions:
  {{1,1}}  {{1,1,1}}  {{1,1,1,1}}    {{1,1,1,1,1}}
  {{1,2}}  {{1,2,2}}  {{1,1,2,2}}    {{1,1,2,2,2}}
           {{1,2,3}}  {{1,2,2,2}}    {{1,2,2,2,2}}
                      {{1,2,3,3}}    {{1,2,2,3,3}}
                      {{1,2,3,4}}    {{1,2,3,3,3}}
                      {{1,1},{1,1}}  {{1,2,3,4,4}}
                      {{1,2},{2,2}}  {{1,2,3,4,5}}
                      {{1,3},{2,3}}  {{1,1},{1,1,1}}
                                     {{1,1},{1,2,2}}
                                     {{1,2},{2,2,2}}
                                     {{1,2},{2,3,3}}
                                     {{1,3},{2,3,3}}
                                     {{1,4},{2,3,4}}
                                     {{2,2},{1,2,2}}
                                     {{3,3},{1,2,3}}

Crossrefs

Cf. A000272, A007716, A007718, A030019, A052888, A134954, A304867, A304887, A317672, A318697, A321155, A321228, A321229.

A321255 Number of connected multiset partitions with multiset density -1, of strongly normal multisets of size n, with no singletons.

Original entry on oeis.org

0, 0, 2, 3, 8, 19, 60, 183, 643, 2355, 9393
Offset: 0

Views

Author

Gus Wiseman, Nov 01 2018

Keywords

Comments

The multiset density of a multiset partition is the sum of the numbers of distinct vertices in each part minus the number of parts minus the number of vertices.
A multiset is normal if it spans an initial interval of positive integers, and strongly normal if in addition its multiplicities are weakly decreasing.

Examples

			The a(2) = 2 through a(5) = 19 multiset partitions:
  {{1,1}}  {{1,1,1}}  {{1,1,1,1}}    {{1,1,1,1,1}}
  {{1,2}}  {{1,1,2}}  {{1,1,1,2}}    {{1,1,1,1,2}}
           {{1,2,3}}  {{1,1,2,2}}    {{1,1,1,2,2}}
                      {{1,1,2,3}}    {{1,1,1,2,3}}
                      {{1,2,3,4}}    {{1,1,2,2,3}}
                      {{1,1},{1,1}}  {{1,1,2,3,4}}
                      {{1,1},{1,2}}  {{1,2,3,4,5}}
                      {{1,2},{1,3}}  {{1,1},{1,1,1}}
                                     {{1,1},{1,1,2}}
                                     {{1,1},{1,2,2}}
                                     {{1,1},{1,2,3}}
                                     {{1,2},{1,1,1}}
                                     {{1,2},{1,1,3}}
                                     {{1,2},{1,3,4}}
                                     {{1,3},{1,1,2}}
                                     {{1,3},{1,2,2}}
                                     {{1,3},{1,2,4}}
                                     {{1,4},{1,2,3}}
                                     {{2,3},{1,1,2}}

Crossrefs

Cf. A000272, A030019, A052888, A134954, A304867, A317672, A321228, A321229, A321253.

A322397 Number of 2-edge-connected clutters spanning n vertices.

Original entry on oeis.org

0, 0, 4, 71, 5927
Offset: 1

Views

Author

Gus Wiseman, Dec 06 2018

Keywords

Comments

A clutter is a connected antichain of sets. It is 2-edge-connected if it cannot be disconnected by removing any single edge. Compare to blobs or 2-vertex-connected clutters (A275307).

Examples

			The a(3) = 4 clutters:
  {{1,3},{2,3}}
  {{1,2},{2,3}}
  {{1,2},{1,3}}
  {{1,2},{1,3},{2,3}}

Crossrefs

Cf. A001187, A013922, A030019, A048143, A095983, A275307, A317634, A317635, A317672, A322335, A322393, A322395, A322399.

A322399 Number of non-isomorphic 2-edge-connected clutters spanning n vertices.

Original entry on oeis.org

0, 0, 2, 12, 149
Offset: 1

Views

Author

Gus Wiseman, Dec 06 2018

Keywords

Comments

A clutter is a connected antichain of sets. It is 2-edge-connected if it cannot be disconnected by removing any single edge. Compare to blobs or 2-vertex-connected clutters (A304887).

Examples

			Non-isomorphic representatives of the a(4) = 12 clutters:
  {{1,4},{2,3,4}}
  {{1,3,4},{2,3,4}}
  {{1,4},{2,4},{3,4}}
  {{1,3},{1,4},{2,3,4}}
  {{1,2},{1,3,4},{2,3,4}}
  {{1,2,4},{1,3,4},{2,3,4}}
  {{1,2},{1,3},{2,4},{3,4}}
  {{1,4},{2,3},{2,4},{3,4}}
  {{1,2},{1,3},{1,4},{2,3,4}}
  {{1,3},{1,4},{2,3},{2,4},{3,4}}
  {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}
  {{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}}

Crossrefs

Cf. A002218, A007718, A013922, A030019, A048143, A095983, A304887, A317634, A317635, A317672, A322335, A322394, A322396, A322397.

A322134 Regular tetrangle where T(n,k,i) is the number of unlabeled connected multiset partitions of weight n with k vertices and i edges.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 2, 1, 1, 2, 4, 2, 1, 2, 1, 0, 0, 0, 0, 0, 0, 1, 2, 2, 1, 1, 2, 7, 6, 2, 2, 6, 4, 1, 2, 1, 0, 0, 0, 0, 0, 0, 0, 1, 3, 3, 2, 1, 1, 3, 14, 17, 9, 3, 3, 17, 18, 7, 2, 9, 7, 1, 3, 1, 0, 0
Offset: 0

Views

Author

Gus Wiseman, Nov 27 2018

Keywords

Examples

			Tetrangle begins:
  1
.
  0 0
  1
.
  0 0 0
  1 1
  1
.
  0 0 0 0
  1 1 1
  1 1
  1
.
  0 0 0 0 0
  1 2 1 1
  2 4 2
  1 2
  1
.
  0 0 0 0 0 0
  1 2 2 1 1
  2 7 6 2
  2 6 4
  1 2
  1
.
  0  0  0  0  0  0  0
  1  3  3  2  1  1
  3 14 17  9  3
  3 17 18  7
  2  9  7
  1  3
  1
.
  0  0  0  0  0  0  0  0
  1  3  4  3  2  1  1
  3 20 33 24 11  3
  4 33 59 35 10
  3 24 35 14
  2 11 10
  1  3
  1

Crossrefs

Triangle sums are A007718.
Cf. A007716, A054923, A191646, A191970, A275421, A286520, A317533, A317672, A321155, A321254, A322114.
Previous Showing 11-15 of 15 results.

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