A317671 -id:A317671 - 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}]

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}]

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 *)

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

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A317677 Fixed point of a shifted hypertree transform.

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