A317635 -id:A317635 - cp's OEIS frontend

A317672 Regular triangle where T(n,k) is the number of clutters (connected antichains) on n + 1 vertices with k maximal blobs (2-connected components).

1, 2, 3, 44, 24, 16, 4983, 940, 300, 125, 7565342, 154770, 18000, 4320, 1296, 2414249587694, 318926314, 3927105, 363580, 72030, 16807, 56130437054842366160898, 135200580256336, 10244647168, 99187200, 8028160, 1376256, 262144

Offset: 1

Gus Wiseman, Aug 03 2018

			Triangle begins:
        1
        2       3
       44      24      16
     4983     940     300     125
  7565342  154770   18000    4320    1296

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

Row sums are A048143. First column is A275307. Last column is A030019.

Cf. A000272, A001187, A002218, A013922, A134954, A293510, A317631, A317632, A317634, A317635, A317671.

blg={0,1,2,44,4983,7565342,2414249587694,56130437054842366160898} (* A275307 *);
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Sum[n^(k-1)*Product[blg[[Length[s]+1]],{s,spn}],{spn,Select[sps[Range[n-1]],Length[#]==k&]}],{n,Length[blg]},{k,n-1}]

A317631 Number of connected set partitions of the vertices of labeled graphs with n vertices.

Original entry on oeis.org

1, 1, 1, 8, 200, 15901

Offset: 0

Gus Wiseman, Aug 02 2018

Gus Wiseman, All 200 connected set partitions of labeled graphs with 4 vertices.

Cf. A000110, A001187, A006125, A048143, A293510, A304717, A317632, A317634, A317635.

A317634 Number of caps (also clutter partitions) of clutters (connected antichains) spanning n vertices.

Original entry on oeis.org

1, 0, 1, 9, 315, 64880

Offset: 0

Gus Wiseman, Aug 02 2018

A kernel of a clutter is the restriction of the edge set to all edges contained within some connected vertex set. A clutter partition is a set partition of the edge set using kernels.

			The a(3) = 9 clutter partitions:
  {{{1,2,3}}}
  {{{1,3},{2,3}}}
  {{{1,2},{2,3}}}
  {{{1,2},{1,3}}}
  {{{1,3}},{{2,3}}}
  {{{1,2}},{{2,3}}}
  {{{1,2}},{{1,3}}}
  {{{1,2},{1,3},{2,3}}}
  {{{1,2}},{{1,3}},{{2,3}}}

Gus Wiseman, Every Clutter Is a Tree of Blobs, The Mathematica Journal, Vol. 19, 2017.
Gus Wiseman, All clutter partitions of non-isomorphic clutters on 4 vertices.

Cf. A001187, A030019, A048143, A275307, A286520,, A293510, A304717, A317631, A317632, A317635.

A317632 Number of connected induced nonempty non-singleton subgraphs of labeled connected graphs with n vertices.

Original entry on oeis.org

0, 0, 1, 13, 294, 12198, 946712, 140168924, 40223263760, 22598607583376, 24999757695984960, 54630901092648916704, 236304498092496715916416, 2026201628540583716863002880, 34482826679730591694177065948928, 1166004710785628820717860509317415168

Offset: 0

Gus Wiseman, Aug 02 2018

nonn

The edges of an induced subgraph G|S are those edges of G with both ends contained in S, where S is a subset of the vertices.

Andrew Howroyd, Table of n, a(n) for n = 0..50
Gus Wiseman, All 294 connected induced subgraphs of labeled connected graphs with 4 vertices.

Cf. A001187, A006125, A048143, A293510, A304717, A317631, A317634, A317635.

seq(n)={
  my(p=sum(k=0, n, 2^binomial(k, 2)*x^k/k!, O(x*x^n)));
  my(g=Vec(serlaplace(log(p))));
  my(q=sum(k=0, n, sum(j=2, k, binomial(k,j)*g[j]*2^(binomial(k-j, 2) + j*(k-j)))*x^k/k!, O(x*x^n)));
  Vec(serlaplace(q/p), -n-1)
} \\ Andrew Howroyd, Dec 10 2018

a(6) from Gus Wiseman, Dec 10 2018

Terms a(7) and beyond from Andrew Howroyd, Dec 10 2018

A317674 Regular triangle where T(n,k) is the number of antichains covering n vertices with k connected components.

Original entry on oeis.org

1, 1, 1, 5, 3, 1, 84, 23, 6, 1, 6348, 470, 65, 10, 1, 7743728, 39598, 1575, 145, 15, 1, 2414572893530, 54354104, 144403, 4095, 280, 21, 1, 56130437190053299918162, 19316801997024, 218033088, 402073, 9100, 490, 28, 1

Offset: 1

Gus Wiseman, Aug 03 2018

			Triangle begins:
        1
        1       1
        5       3       1
       84      23       6       1
     6348     470      65      10       1
  7743728   39598    1575     145      15       1

Gus Wiseman, Sequences enumerating clutters, antichains, hypertrees, and hyperforests, organized by labeling, spanning, and allowance of singletons.

First column is A048143. Row sums are A006126.
Cf. A000272, A001187, A013922, A030019, A134954, A143543, A275307, A293510.
Cf. A317634, A317635, A317671, A317672, A317677.

blg={1,1,5,84,6348,7743728,2414572893530,56130437190053299918162} (*A048143*);
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Sum[Product[blg[[Length[s]]],{s,spn}],{spn,Select[sps[Range[n]],Length[#]==k&]}],{n,Length[blg]},{k,n}]

A318697 Number of ways to partition a hypertree spanning n vertices into hypertrees.

Original entry on oeis.org

1, 1, 7, 93, 1856, 49753, 1679441, 68463769, 3273695758, 179710285011, 11141016392749, 769939840667473, 58695964339179805, 4893452980658819151, 442915168219228586581, 43255083632741702266097, 4533695508041747494704359, 507638249638364368312476913

Offset: 1

Gus Wiseman, Aug 31 2018

nonn

			The a(3) = 7 hypertree partitions:
  {{{1,2,3}}}
  {{{1,2},{1,3}}}
  {{{1,2},{2,3}}}
  {{{1,3},{2,3}}}
  {{{1,2}},{{1,3}}}
  {{{1,2}},{{2,3}}}
  {{{1,3}},{{2,3}}}

Cf. A000272, A030019, A048143, A134954, A275307, A317631, A317632, A317634, A317635, A317677.

trct[n_]:=Sum[StirlingS2[n-1,i]*n^(i-1),{i,0,n-1}];
numSetPtnsOfType[ptn_]:=Total[ptn]!/Times@@Factorial/@ptn/Times@@Factorial/@Length/@Split[ptn];
Table[Sum[n^(Length[ptn]-1)*Product[trct[s+1],{s,ptn}]*numSetPtnsOfType[ptn],{ptn,IntegerPartitions[n-1]}],{n,20}]

A317677 Fixed point of a shifted hypertree transform.

Original entry on oeis.org

1, 1, 4, 32, 402, 7038, 160114, 4522578, 153640590, 6132546770, 282517271694, 14812447505646, 873934551644074, 57486823088667270, 4183353479821220130, 334572221351085006242, 29242220614539638127294, 2779426070382982579163202, 286058737295150226682469518

Offset: 1

Gus Wiseman, Aug 04 2018

The hypertree transform H(a) of a sequence a is given by H(a)(n) = Sum_p n^(k-1) Prod_i a(|p_i|+1), where the sum is over all set partitions U(p_1, ..., p_k) = {1, ..., n-1}.

Alois P. Heinz, Table of n, a(n) for n = 1..305

Cf. A000272, A030019, A048143, A134954, A275307, A293510, A317631, A317632, A317634, A317635, A317671.

b:= proc(n, k) option remember; `if`(n=0, 1/k, add(
      a(j)*b(n-j, k)*binomial(n-1, j-1)*k, j=1..n))
    end:
a:= n-> b(n-1, n):
seq(a(n), n=1..20);  # Alois P. Heinz, Aug 21 2019

numSetPtnsOfType[ptn_]:=Total[ptn]!/Times@@Factorial/@ptn/Times@@Factorial/@Length/@Split[ptn];
a[n_]:=a[n]=Sum[n^(Length[ptn]-1)*numSetPtnsOfType[ptn]*Product[a[s],{s,ptn}],{ptn,IntegerPartitions[n-1]}];
Array[a,20]
(* Second program: *)
b[n_, k_] := b[n, k] = If[n == 0, 1/k, Sum[
     a[j]*b[n - j, k]*Binomial[n - 1, j - 1]*k, {j, 1, n}]];
a[n_] := b[n - 1, n];
Array[a, 20] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)

A317671 Regular triangle where T(n,k) is the number of labeled connected graphs on n + 1 vertices with k maximal blobs (2-connected components).

Original entry on oeis.org

1, 1, 3, 10, 12, 16, 238, 215, 150, 125, 11368, 7740, 4140, 2160, 1296, 1014888, 509446, 205065, 84035, 36015, 16807, 166537616, 59409952, 17393152, 5393920, 1863680, 688128, 262144, 50680432112, 12321597708, 2516756508, 563570217, 148803480, 45467730

Offset: 1

Gus Wiseman, Aug 03 2018

			Triangle begins:
        1
        1       3
       10      12      16
      238     215     150     125
    11368    7740    4140    2160    1296
  1014888  509446  205065   84035   36015   16807

Row sums are A001187. First column is A013922. Last column is A000272.

Cf. A002218, A030019, A048143, A134954, A275307, A293510, A317631, A317632, A317634, A317635, A317672, A317677.

blg={0,1,1,10,238,11368,1014888,166537616,50680432112,29107809374336} (*A013922*);
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Sum[n^(k-1)*Product[blg[[Length[s]+1]],{s,spn}],{spn,Select[sps[Range[n-1]],Length[#]==k&]}],{n,Length[blg]},{k,n-1}]

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

Original entry on oeis.org

0, 0, 4, 71, 5927

Offset: 1

Gus Wiseman, Dec 06 2018

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).

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

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

Gus Wiseman, Dec 06 2018

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).

			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}}

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

cp's OEIS Frontend

A317672 Regular triangle where T(n,k) is the number of clutters (connected antichains) on n + 1 vertices with k maximal blobs (2-connected components).

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Mathematica

A317631 Number of connected set partitions of the vertices of labeled graphs with n vertices.

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A317634 Number of caps (also clutter partitions) of clutters (connected antichains) spanning n vertices.

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A317632 Number of connected induced nonempty non-singleton subgraphs of labeled connected graphs with n vertices.

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PARI

Extensions

A317674 Regular triangle where T(n,k) is the number of antichains covering n vertices with k connected components.

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Mathematica

A318697 Number of ways to partition a hypertree spanning n vertices into hypertrees.

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Mathematica

A317677 Fixed point of a shifted hypertree transform.

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Maple

Mathematica

A317671 Regular triangle where T(n,k) is the number of labeled connected graphs on n + 1 vertices with k maximal blobs (2-connected components).

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Programs

Mathematica

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

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

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