A293607 -id:A293607 - cp's OEIS frontend

A321256 Regular triangle where T(n,k) is the number of non-isomorphic connected set systems of weight n with density -1 <= k <= n-2.

1, 1, 0, 2, 0, 0, 4, 0, 0, 0, 6, 1, 0, 0, 0, 13, 5, 0, 0, 0, 0, 23, 12, 2, 0, 0, 0, 0, 49, 36, 11, 0, 0, 0, 0, 0, 100, 95, 39, 5, 0, 0, 0, 0, 0, 220, 262, 143, 32, 1, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Nov 01 2018

A set system is a finite set of finite nonempty sets. The density of a set system is the sum of sizes of each part (weight) minus the number of parts minus the number of vertices.

			Triangle begins:
    1
    1   0
    2   0   0
    4   0   0   0
    6   1   0   0   0
   13   5   0   0   0   0
   23  12   2   0   0   0   0
   49  36  11   0   0   0   0   0
  100  95  39   5   0   0   0   0   0
  220 262 143  32   1   0   0   0   0   0

First column is A321228. Row sums are A007718.

Cf. A007716, A030019, A056156, A283877, A293607, A300913, A319557, A321155, A321253, A321254.

A321279 Number of z-trees with product A181821(n). Number of connected antichains of multisets with multiset density -1, of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

0, 1, 2, 1, 2, 1, 3, 1, 1, 2, 2, 2, 4, 2, 2, 1, 2, 3, 4, 4, 2, 4, 3, 4, 4, 3, 4, 6, 4, 6, 2, 1, 4, 6, 4, 9, 6, 5, 3, 9, 2, 8, 4, 9, 8, 7, 4, 8, 4, 12, 6, 12, 5, 16, 8, 17, 5, 7, 2, 19, 6, 10, 10, 1, 6, 13, 2, 16, 7, 16, 6, 27, 4, 7, 16, 20, 8, 15, 4, 22

Offset: 1

Gus Wiseman, Nov 01 2018

nonn

This multiset (row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

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 sequence of antichains begins:
   2: {{1}}
   3: {{1,1}}
   3: {{1},{1}}
   4: {{1,2}}
   5: {{1,1,1}}
   5: {{1},{1},{1}}
   6: {{1,1,2}}
   7: {{1,1,1,1}}
   7: {{1,1},{1,1}}
   7: {{1},{1},{1},{1}}
   8: {{1,2,3}}
   9: {{1,1,2,2}}
  10: {{1,1,1,2}}
  10: {{1,1},{1,2}}
  11: {{1,1,1,1,1}}
  11: {{1},{1},{1},{1},{1}}
  12: {{1,1,2,3}}
  12: {{1,2},{1,3}}
  13: {{1,1,1,1,1,1}}
  13: {{1,1,1},{1,1,1}}
  13: {{1,1},{1,1},{1,1}}
  13: {{1},{1},{1},{1},{1},{1}}
  14: {{1,1,1,1,2}}
  14: {{1,2},{1,1,1}}
  15: {{1,1,1,2,2}}
  15: {{1,1},{1,2,2}}
  16: {{1,2,3,4}}

Cf. A001055, A007718, A030019, A181821, A293607, A303837, A304382, A305081, A305936, A318284, A321229, A321270, A321271, A321272.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c=={},s,zsm[Union[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
zensity[s_]:=Total[(PrimeNu[#]-1&)/@s]-PrimeNu[LCM@@s];
Table[Length[Select[facs[Times@@Prime/@nrmptn[n]],And[zensity[#]==-1,Length[zsm[#]]==1,Select[Tuples[#,2],UnsameQ@@#&&Divisible@@#&]=={}]&]],{n,50}]

cp's OEIS Frontend

A321256 Regular triangle where T(n,k) is the number of non-isomorphic connected set systems of weight n with density -1 <= k <= n-2.

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