A321253 -id:A321253 - cp's OEIS frontend

A321254 Regular triangle where T(n,k) is the number of non-isomorphic connected multiset partitions of weight n with multiset density -1 <= k <= n-2.

1, 3, 0, 6, 0, 0, 16, 1, 0, 0, 37, 3, 0, 0, 0, 105, 18, 2, 0, 0, 0, 279, 68, 7, 0, 0, 0, 0, 817, 293, 46, 3, 0, 0, 0, 0, 2387, 1141, 228, 17, 1, 0, 0, 0, 0, 7269, 4511, 1189, 135, 9, 0, 0, 0, 0, 0

Offset: 1

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.

			Triangle begins:
     1
     3    0
     6    0    0
    16    1    0    0
    37    3    0    0    0
   105   18    2    0    0    0
   279   68    7    0    0    0    0
   817  293   46    3    0    0    0    0
  2387 1141  228   17    1    0    0    0    0
  7269 4511 1189  135    9    0    0    0    0    0

First column is A321229. Row sums are A007718.

Cf. A007716, A317672, A321155, A321194, A321253, A321255, A321256.

A321255 Number of connected multiset partitions with multiset density -1, of strongly normal multisets of size n, with no singletons.

Original entry on oeis.org

0, 0, 2, 3, 8, 19, 60, 183, 643, 2355, 9393

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.

A multiset is normal if it spans an initial interval of positive integers, and strongly normal if in addition its multiplicities are weakly decreasing.

			The a(2) = 2 through a(5) = 19 multiset partitions:
  {{1,1}}  {{1,1,1}}  {{1,1,1,1}}    {{1,1,1,1,1}}
  {{1,2}}  {{1,1,2}}  {{1,1,1,2}}    {{1,1,1,1,2}}
           {{1,2,3}}  {{1,1,2,2}}    {{1,1,1,2,2}}
                      {{1,1,2,3}}    {{1,1,1,2,3}}
                      {{1,2,3,4}}    {{1,1,2,2,3}}
                      {{1,1},{1,1}}  {{1,1,2,3,4}}
                      {{1,1},{1,2}}  {{1,2,3,4,5}}
                      {{1,2},{1,3}}  {{1,1},{1,1,1}}
                                     {{1,1},{1,1,2}}
                                     {{1,1},{1,2,2}}
                                     {{1,1},{1,2,3}}
                                     {{1,2},{1,1,1}}
                                     {{1,2},{1,1,3}}
                                     {{1,2},{1,3,4}}
                                     {{1,3},{1,1,2}}
                                     {{1,3},{1,2,2}}
                                     {{1,3},{1,2,4}}
                                     {{1,4},{1,2,3}}
                                     {{2,3},{1,1,2}}

Cf. A000272, A030019, A052888, A134954, A304867, A317672, A321228, A321229, A321253.

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.

Original entry on oeis.org

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.

A321271 Number of connected factorizations of n into positive integers > 1 with z-density -1.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 1, 2, 1, 2, 1, 1, 1, 4, 2, 1, 3, 2, 1, 1, 1, 7, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 2, 2, 1, 1, 7, 2, 2, 1, 2, 1, 4, 1, 4, 1, 1, 1, 3, 1, 1, 2, 11, 1, 1, 1, 2, 1, 1, 1, 8, 1, 1, 2, 2, 1, 1, 1, 7, 5, 1, 1, 3, 1, 1, 1

Offset: 1

Gus Wiseman, Nov 01 2018

nonn

These are z-trees (A303837, A305081, A305253, A321279) where we relax the requirement of pairwise indivisibility.
Given a finite multiset S of positive integers greater than 1, let G(S) be the simple labeled graph with vertices the distinct elements of S and with edges between any two vertices that have a common divisor greater than 1. For example, G({6,14,15,35}) is a 4-cycle. Then S is said to be connected if G(S) is a connected graph.
The z-density of a factorization S is defined to be Sum_{s in S} (omega(s) - 1) - omega(n), where omega = A001221 and n is the product of S.

			The a(72) = 8 factorizations are (2*2*3*6), (2*2*18), (2*3*12), (2*36), (3*4*6), (3*24), (4*18), (72). Missing from this list but still connected are (2*6*6),(6*12).

Cf. A001055, A001221, A030019, A286518, A303837, A304118, A304382, A305052, A305081, A305193, A305253, A319786, A321229, A321253.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
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[Times@@s];
Table[Length[Select[facs[n],And[zensity[#]==-1,Length[zsm[#]]==1]&]],{n,100}]

A321272 Number of connected multiset partitions with multiset density -1, of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 5, 1, 4, 4, 7, 3, 11, 7, 8, 1, 15, 8, 22, 7, 14, 12, 30, 5, 16, 19, 20, 14, 42, 18, 56, 1, 24, 30, 28, 18, 77, 45, 38, 14

Offset: 1

Gus Wiseman, Nov 01 2018

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.

			Non-isomorphic representatives of the a(2) = 1 through a(15) = 8 multiset partitions:
  {{1}}  {{11}}    {{12}}  {{111}}      {{112}}    {{1111}}
         {{1}{1}}          {{1}{11}}    {{1}{12}}  {{1}{111}}
                           {{1}{1}{1}}             {{11}{11}}
                                                   {{1}{1}{11}}
                                                   {{1}{1}{1}{1}}
.
  {{123}}  {{1122}}      {{1112}}      {{11111}}
           {{1}{122}}    {{1}{112}}    {{1}{1111}}
           {{2}{112}}    {{11}{12}}    {{11}{111}}
           {{1}{2}{12}}  {{1}{1}{12}}  {{1}{1}{111}}
                                       {{1}{11}{11}}
                                       {{1}{1}{1}{11}}
                                       {{1}{1}{1}{1}{1}}
.
  {{1123}}    {{111111}}            {{11112}}        {{11122}}
  {{1}{123}}  {{1}{11111}}          {{1}{1112}}      {{1}{1122}}
  {{12}{13}}  {{11}{1111}}          {{11}{112}}      {{11}{122}}
              {{111}{111}}          {{12}{111}}      {{2}{1112}}
              {{1}{1}{1111}}        {{1}{1}{112}}    {{1}{1}{122}}
              {{1}{11}{111}}        {{1}{11}{12}}    {{1}{2}{112}}
              {{11}{11}{11}}        {{1}{1}{1}{12}}  {{2}{11}{12}}
              {{1}{1}{1}{111}}                       {{1}{1}{2}{12}}
              {{1}{1}{11}{11}}
              {{1}{1}{1}{1}{11}}
              {{1}{1}{1}{1}{1}{1}}

Cf. A007718, A181821, A303837, A304382, A305081, A305936, A318284, A321155, A321229, A321253, A321270, A321271.

a(prime(n)) = A000041(n).

cp's OEIS Frontend

A321254 Regular triangle where T(n,k) is the number of non-isomorphic connected multiset partitions of weight n with multiset density -1 <= k <= n-2.

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A321255 Number of connected multiset partitions with multiset density -1, of strongly normal multisets of size n, with no singletons.

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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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A321271 Number of connected factorizations of n into positive integers > 1 with z-density -1.

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A321272 Number of connected multiset partitions with multiset density -1, of a multiset whose multiplicities are the prime indices of n.

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