A304717 -id:A304717 - cp's OEIS frontend

A304918 Number of labeled antichain hyperforests spanning a subset of {1,...,n}.

1, 2, 5, 18, 104, 943, 12133, 203038, 4177755, 101922814, 2874725600, 92009680557, 3294276613933, 130446181101044, 5660055256165565, 267044522107706072, 13611243187516647324, 745329728016955480687, 43636132793651444511809, 2719977663069107176768790

Offset: 0

Gus Wiseman, May 21 2018

nonn

			The a(3) = 18 hyperforests are the following:
{{1,2,3}}      {{2,3}}    {{1,3}}    {{1,2}}    {{3}}   {{2}}   {{1}}   {}
{{1,3},{2,3}}  {{2},{3}}  {{1},{3}}  {{1},{2}}
{{1,2},{2,3}}
{{1,2},{1,3}}
{{3},{1,2}}
{{2},{1,3}}
{{1},{2,3}}
{{1},{2},{3}}

Cf. A030019, A035053, A048143, A134954, A134955, A134957, A134956, A134957, A134959, A144959, A303838, A304716, A304717, A304867, A304911, A304912.

Binomial transform of A134954.

A305028 Number of unlabeled blobs spanning n vertices without singleton edges.

Original entry on oeis.org

1, 0, 1, 2, 10, 128

Offset: 0

Gus Wiseman, May 24 2018

nonn

A blob is a connected antichain of finite sets that cannot be capped by a hypertree with more than one branch.

			Non-isomorphic representatives of the a(4) = 10 blobs:
  {{1,2,3,4}}
  {{1,3,4},{2,3,4}}
  {{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}}
  {{1,2},{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}}

Gus Wiseman, Every Clutter Is a Tree of Blobs, The Mathematica Journal, Vol. 19, 2017.

Cf. A030019, A035053, A048143, A054921, A134957, A144959, A275307, A286520, A293607, A304118, A304717, A304867.

A329632 Number of connected integer partitions of n whose distinct parts are pairwise indivisible.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 1, 3, 2, 4, 1, 5, 1, 6, 4, 6, 1, 9, 2, 10, 6, 13, 3, 15, 6, 18, 8, 22, 9, 29, 10, 30, 20, 40, 22, 48, 24, 57, 36, 68

Offset: 0

Gus Wiseman, Nov 18 2019

Given an integer partition y of length k, let G(y) be the simple labeled graph with vertices {1..k} and edges between any two vertices i, j such that GCD(y_i, y_j) > 1. For example, G(6,14,15,35) is a 4-cycle. A partition y is said to be connected if G(y) is a connected graph.

			The a(n) partitions for n = 1, 4, 6, 10, 12, 14:
  (1)  (4)    (6)      (10)         (12)           (14)
       (2,2)  (3,3)    (5,5)        (6,6)          (7,7)
              (2,2,2)  (6,4)        (4,4,4)        (8,6)
                       (2,2,2,2,2)  (3,3,3,3)      (10,4)
                                    (2,2,2,2,2,2)  (6,4,4)
                                                   (2,2,2,2,2,2,2)

The Heinz numbers of these partitions are given by A329559.
The strict version is A304717.
Connected partitions are A218970.
Pairwise indivisible partitions are A305148.
Cf. A048143, A286518, A286520, A304714, A304716, A305078.

stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
zsm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}],GCD@@s[[#]]>1&]},If[c=={},s,zsm[Sort[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
Table[Length[Select[IntegerPartitions[n],stableQ[#,Divisible]&&Length[zsm[#]]<=1&]],{n,0,30}]

A304919 Number of labeled hyperforests spanning {1,...,n} and allowing singleton edges.

Original entry on oeis.org

1, 1, 5, 45, 665, 14153, 399421, 14137301, 603647601, 30231588689, 1738713049013, 112976375651901, 8186616300733321, 654642360222892057, 57267075701210437229, 5440407421313402397541, 557802495215406348358113, 61393838258161429159571873, 7220049654850517272144419941, 903546142463635579042416518989

Offset: 0

Gus Wiseman, May 21 2018

nonn

			The a(2) = 5 hyperforests are the following:
{{1,2}}
{{1},{2}}
{{1},{1,2}}
{{2},{1,2}}
{{1},{2},{1,2}}

Cf. A030019, A035053, A048143, A134954, A134955, A134956, A134957, A134959, A144959, A303838, A304717, A304867, A304911, A304912, A304918.

Inverse binomial transform of A134956.

cp's OEIS Frontend

A304918 Number of labeled antichain hyperforests spanning a subset of {1,...,n}.

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A305028 Number of unlabeled blobs spanning n vertices without singleton edges.

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A329632 Number of connected integer partitions of n whose distinct parts are pairwise indivisible.

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Mathematica

A304919 Number of labeled hyperforests spanning {1,...,n} and allowing singleton edges.

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