A134954 -id:A134954 - 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.

A321228 Number of non-isomorphic hypertrees of weight n with singletons.

Original entry on oeis.org

1, 1, 1, 2, 4, 6, 13, 23, 49, 100, 220

Offset: 0

Gus Wiseman, Oct 31 2018

A hypertree with singletons is a connected set system (finite set of finite nonempty sets) with density -1, where the density of a set system is the sum of sizes of the parts (weight) minus the number of parts minus the number of vertices.

			Non-isomorphic representatives of the a(1) = 1 through a(7) = 23 hypertrees:
  {{1}}  {{1,2}}  {{1,2,3}}    {{1,2,3,4}}      {{1,2,3,4,5}}
                  {{2},{1,2}}  {{1,3},{2,3}}    {{1,4},{2,3,4}}
                               {{3},{1,2,3}}    {{4},{1,2,3,4}}
                               {{1},{2},{1,2}}  {{2},{1,3},{2,3}}
                                                {{2},{3},{1,2,3}}
                                                {{3},{1,3},{2,3}}
.
  {{1,2,3,4,5,6}}        {{1,2,3,4,5,6,7}}
  {{1,2,5},{3,4,5}}      {{1,2,6},{3,4,5,6}}
  {{1,5},{2,3,4,5}}      {{1,6},{2,3,4,5,6}}
  {{5},{1,2,3,4,5}}      {{6},{1,2,3,4,5,6}}
  {{1},{1,4},{2,3,4}}    {{1},{1,5},{2,3,4,5}}
  {{1,3},{2,4},{3,4}}    {{1,2},{2,5},{3,4,5}}
  {{1,4},{2,4},{3,4}}    {{1,4},{2,5},{3,4,5}}
  {{3},{1,4},{2,3,4}}    {{1,5},{2,5},{3,4,5}}
  {{3},{4},{1,2,3,4}}    {{4},{1,2,5},{3,4,5}}
  {{4},{1,4},{2,3,4}}    {{4},{1,5},{2,3,4,5}}
  {{1},{2},{1,3},{2,3}}  {{4},{5},{1,2,3,4,5}}
  {{1},{2},{3},{1,2,3}}  {{5},{1,2,5},{3,4,5}}
  {{2},{3},{1,3},{2,3}}  {{5},{1,5},{2,3,4,5}}
                         {{1},{3},{1,4},{2,3,4}}
                         {{1},{4},{1,4},{2,3,4}}
                         {{2},{1,3},{2,4},{3,4}}
                         {{2},{3},{1,4},{2,3,4}}
                         {{2},{3},{4},{1,2,3,4}}
                         {{3},{1,4},{2,4},{3,4}}
                         {{3},{4},{1,4},{2,3,4}}
                         {{4},{1,3},{2,4},{3,4}}
                         {{4},{1,4},{2,4},{3,4}}
                         {{1},{2},{3},{1,3},{2,3}}

Cf. A000272, A007716, A007718, A030019, A052888, A134954, A304867, A304887, A317672, A318697, A321155, A321227, A321229.

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 *)

A321253 Number of non-isomorphic strict connected weight-n multiset partitions with multiset density -1.

Original entry on oeis.org

0, 1, 2, 5, 12, 28, 78, 202, 578, 1650, 4904

Offset: 0

Gus Wiseman, Nov 01 2018

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.

			Non-isomorphic representatives of the a(1) = 1 through a(5) = 28 multiset partitions:
  {{1}}  {{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},{1,1}}  {{1,2,3,3}}      {{1,2,2,3,3}}
                  {{2},{1,2}}  {{1,2,3,4}}      {{1,2,3,3,3}}
                               {{1},{1,1,1}}    {{1,2,3,4,4}}
                               {{1},{1,2,2}}    {{1,2,3,4,5}}
                               {{1,2},{2,2}}    {{1},{1,1,1,1}}
                               {{1,3},{2,3}}    {{1,1},{1,1,1}}
                               {{2},{1,2,2}}    {{1,1},{1,2,2}}
                               {{3},{1,2,3}}    {{1},{1,2,2,2}}
                               {{1},{2},{1,2}}  {{1,2},{2,2,2}}
                                                {{1,2},{2,3,3}}
                                                {{1,3},{2,3,3}}
                                                {{1,4},{2,3,4}}
                                                {{2},{1,1,2,2}}
                                                {{2},{1,2,2,2}}
                                                {{2},{1,2,3,3}}
                                                {{2,2},{1,2,2}}
                                                {{3},{1,2,3,3}}
                                                {{3,3},{1,2,3}}
                                                {{4},{1,2,3,4}}
                                                {{1},{1,2},{2,2}}
                                                {{1},{2},{1,2,2}}
                                                {{2},{1,2},{2,2}}
                                                {{2},{1,3},{2,3}}
                                                {{2},{3},{1,2,3}}
                                                {{3},{1,3},{2,3}}

Cf. A000272, A007716, A007718, A030019, A052888, A134954, A304867, A304887, A318697, A321155, A321194, A321228, A321229, A321254.

A304968 Number of labeled hypertrees spanning some subset of {1,...,n}, with singleton edges allowed.

Original entry on oeis.org

1, 2, 7, 48, 621, 12638, 351987, 12426060, 531225945, 26674100154, 1538781595999, 100292956964456, 7288903575373509, 584454485844541718, 51256293341752583499, 4880654469385955209092, 501471626403154217825457, 55300894427785157597436786

Offset: 0

Gus Wiseman, May 22 2018

nonn

			The a(2) = 7 hypertrees are the following:
{}
{{1}}
{{2}}
{{1,2}}
{{1},{1,2}}
{{2},{1,2}}
{{1},{2},{1,2}}

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A030019, A035053, A134954, A134955, A134956, A134957, A134958, A134959, A144959, A304386, A304867, A304911, A304912, A304918, A304968, A304970.

\\ here b(n) is A134958 with b(1)=1.
b(n)=if(n<2, n>=0, 2^n*sum(i=0, n, stirling(n-1, i, 2)*n^(i-1)));
a(n)=sum(k=0, n, binomial(n, k)*b(k)); \\ Andrew Howroyd, Aug 27 2018

Binomial transform of b(1) = 1, b(n) = A134958(n) otherwise.

A304970 Number of unlabeled hypertrees with up to n vertices and without singleton edges.

Original entry on oeis.org

1, 1, 2, 4, 8, 17, 39, 98, 263, 759, 2299, 7259, 23649, 79057, 269629, 935328, 3290260, 11714285, 42139053, 152963037, 559697097, 2062574000, 7649550572, 28534096988, 106994891146, 403119433266, 1525466082179, 5795853930652, 22102635416716, 84579153865570

Offset: 0

Gus Wiseman, May 22 2018

nonn

			Non-isomorphic representatives of the a(4) = 8 hypertrees are the following:
{}
{{1,2}}
{{1,2,3}}
{{1,2,3,4}}
{{1,3},{2,3}}
{{1,4},{2,3,4}}
{{1,3},{2,4},{3,4}}
{{1,4},{2,4},{3,4}}

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A030019, A035053, A134954, A134955, A134956, A134957, A134958, A134959, A144959, A304386, A304867, A304911, A304912, A304918, A304968, A304970.

\\ here b(n) is A007563 as vector
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
b(n)={my(v=[1]); for(i=2, n, v=concat([1], EulerT(EulerT(v)))); v}
seq(n)={my(u=b(n)); Vec(1 + (x*Ser(EulerT(u))*(1-x*Ser(u)))/(1-x))} \\ Andrew Howroyd, Aug 27 2018

Partial sums of A035053 if we assume A035053(1) = 0.

a(n) = A304937(n) + 1 for n > 0.

A322111 Number of non-isomorphic self-dual connected multiset partitions of weight n with multiset density -1.

Original entry on oeis.org

1, 1, 1, 2, 2, 5, 5, 13, 13, 37, 37

Offset: 0

Gus Wiseman, Nov 26 2018

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 dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}. A multiset partition is self-dual if it is isomorphic to its dual. For example, {{1,1},{1,2,2},{2,3,3}} is self-dual, as it is isomorphic to its dual {{1,1,2},{2,2,3},{3,3}}.
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(8) = 13 multiset partitions:
  {{1}}                    {{1,1}}
.
  {{1,1,1}}                {{1,1,1,1}}
  {{2},{1,2}}              {{2},{1,2,2}}
.
  {{1,1,1,1,1}}            {{1,1,1,1,1,1}}
  {{1,1},{1,2,2}}          {{2},{1,2,2,2,2}}
  {{2},{1,2,2,2}}          {{2,2},{1,1,2,2}}
  {{2},{1,3},{2,3}}        {{2},{1,3},{2,3,3}}
  {{3},{3},{1,2,3}}        {{3},{3},{1,2,3,3}}
.
  {{1,1,1,1,1,1,1}}        {{1,1,1,1,1,1,1,1}}
  {{1,1,1},{1,2,2,2}}      {{1,1,1},{1,1,2,2,2}}
  {{2},{1,2,2,2,2,2}}      {{2},{1,2,2,2,2,2,2}}
  {{2,2},{1,1,2,2,2}}      {{2,2},{1,1,2,2,2,2}}
  {{1,1},{1,2},{2,3,3}}    {{1,1},{1,2,2},{2,3,3}}
  {{2},{1,3},{2,3,3,3}}    {{2},{1,3},{2,3,3,3,3}}
  {{2},{2,2},{1,2,3,3}}    {{2},{1,3,3},{2,2,3,3}}
  {{3},{1,2,2},{2,3,3}}    {{3},{3},{1,2,3,3,3,3}}
  {{3},{3},{1,2,3,3,3}}    {{3},{3,3},{1,2,2,3,3}}
  {{1},{1},{1,4},{2,3,4}}  {{2},{1,3},{2,4},{3,4,4}}
  {{2},{1,3},{2,4},{3,4}}  {{3},{3},{1,2,4},{3,4,4}}
  {{3},{4},{1,4},{2,3,4}}  {{3},{4},{1,4},{2,3,4,4}}
  {{4},{4},{4},{1,2,3,4}}  {{4},{4},{4},{1,2,3,4,4}}

Cf. A000272, A007716, A007718, A030019, A052888, A134954, A304867, A304887, A316983, A318697, A319616, A321155, A321255.

A304937 Number of unlabeled nonempty hypertrees with up to n vertices and no singleton edges.

Original entry on oeis.org

1, 0, 1, 3, 7, 16, 38, 97, 262, 758, 2298, 7258, 23648, 79056, 269628, 935327, 3290259, 11714284, 42139052, 152963036, 559697096, 2062573999, 7649550571, 28534096987, 106994891145, 403119433265, 1525466082178, 5795853930651, 22102635416715, 84579153865569

Offset: 0

Gus Wiseman, May 21 2018

nonn

			Non-isomorphic representatives of the a(5) = 16 hypertrees are the following:
{{1,2}}
{{1,2,3}}
{{1,2,3,4}}
{{1,2,3,4,5}}
{{1,3},{2,3}}
{{1,4},{2,3,4}}
{{1,5},{2,3,4,5}}
{{1,2,5},{3,4,5}}
{{1,2},{2,5},{3,4,5}}
{{1,3},{2,4},{3,4}}
{{1,4},{2,4},{3,4}}
{{1,4},{2,5},{3,4,5}}
{{1,5},{2,5},{3,4,5}}
{{1,3},{2,4},{3,5},{4,5}}
{{1,4},{2,5},{3,5},{4,5}}
{{1,5},{2,5},{3,5},{4,5}}

Andrew Howroyd, Table of n, a(n) for n = 0..200

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

\\ here b(n) is A007563 as vector
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
b(n)={my(v=[1]); for(i=2, n, v=concat([1], EulerT(EulerT(v)))); v}
seq(n)={my(u=b(n)); Vec(1 + (x*Ser(EulerT(u))*(1-x*Ser(u)) - x)/(1-x))} \\ Andrew Howroyd, Aug 27 2018

a(n) = a(n-1) + A035053(n) for n > 1, a(n) = 1 - n for n < 2.

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}]

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

Gus Wiseman, Oct 31 2018

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.

			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}}

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

cp's OEIS Frontend

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

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A321228 Number of non-isomorphic hypertrees of weight n with singletons.

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A317677 Fixed point of a shifted hypertree transform.

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Maple

Mathematica

A321253 Number of non-isomorphic strict connected weight-n multiset partitions with multiset density -1.

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A304968 Number of labeled hypertrees spanning some subset of {1,...,n}, with singleton edges allowed.

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PARI

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A304970 Number of unlabeled hypertrees with up to n vertices and without singleton edges.

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PARI

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A322111 Number of non-isomorphic self-dual connected multiset partitions of weight n with multiset density -1.

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A304937 Number of unlabeled nonempty hypertrees with up to n vertices and no singleton edges.

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PARI

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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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Mathematica

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

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