A304118 -id:A304118 - cp's OEIS frontend

A322118 Number of non-isomorphic connected multiset partitions of weight n with no singletons that cannot be capped by a tree.

1, 1, 2, 3, 7, 11, 29, 55, 155, 386, 1171

Offset: 0

Gus Wiseman, Nov 26 2018

The density of a multiset partition is defined to be the sum of numbers of distinct elements in each part, minus the number of parts, minus the total number of distinct elements in the whole partition. A multiset partition is a tree if it has more than one part, is connected, and has density -1. A cap is a certain kind of non-transitive coarsening of a multiset partition. For example, the four caps of {{1,1},{1,2},{2,2}} are {{1,1},{1,2},{2,2}}, {{1,1},{1,2,2}}, {{1,1,2},{2,2}}, {{1,1,2,2}}. - Gus Wiseman, Feb 05 2021

			The multiset partition C = {{1,1},{1,2,3},{2,3,3}} is not a tree but has the cap {{1,1},{1,2,3,3}} which is a tree, so C is not counted under a(8).
Non-isomorphic representatives of the a(2) = 2 through a(6) = 29 multiset partitions:
  {{1,1}}  {{1,1,1}}  {{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}}    {{1,1,1,2,2,2}}
           {{1,2,3}}  {{1,2,2,2}}    {{1,2,2,2,2}}    {{1,1,2,2,2,2}}
                      {{1,2,3,3}}    {{1,2,2,3,3}}    {{1,1,2,2,3,3}}
                      {{1,2,3,4}}    {{1,2,3,3,3}}    {{1,2,2,2,2,2}}
                      {{1,1},{1,1}}  {{1,2,3,4,4}}    {{1,2,2,3,3,3}}
                      {{1,2},{1,2}}  {{1,2,3,4,5}}    {{1,2,3,3,3,3}}
                                     {{1,1},{1,1,1}}  {{1,2,3,3,4,4}}
                                     {{1,2},{1,2,2}}  {{1,2,3,4,4,4}}
                                     {{2,2},{1,2,2}}  {{1,2,3,4,5,5}}
                                     {{2,3},{1,2,3}}  {{1,2,3,4,5,6}}
                                                      {{1,1},{1,1,1,1}}
                                                      {{1,1,1},{1,1,1}}
                                                      {{1,1,2},{1,2,2}}
                                                      {{1,2},{1,1,2,2}}
                                                      {{1,2},{1,2,2,2}}
                                                      {{1,2},{1,2,3,3}}
                                                      {{1,2,2},{1,2,2}}
                                                      {{1,2,3},{1,2,3}}
                                                      {{1,2,3},{2,3,3}}
                                                      {{1,3,4},{2,3,4}}
                                                      {{2,2},{1,1,2,2}}
                                                      {{2,2},{1,2,2,2}}
                                                      {{2,3},{1,2,3,3}}
                                                      {{3,3},{1,2,3,3}}
                                                      {{3,4},{1,2,3,4}}
                                                      {{1,1},{1,1},{1,1}}
                                                      {{1,2},{1,2},{1,2}}
                                                      {{1,2},{1,3},{2,3}}

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

Non-isomorphic tree multiset partitions are counted by A321229, or A321231 without singletons.
The version with singletons is A322110.
The weak-antichain case is counted by A322138, or A322117 with singletons.
Cf. A002218, A007718, A013922, A030019, A275307, A293994, A304118, A304382, A304887, A305079, A319719, A319721, A321194.

Definition corrected by Gus Wiseman, Feb 05 2021

A305028 Number of unlabeled blobs spanning n vertices without singleton edges.

Original entry on oeis.org

1, 0, 1, 2, 10, 128

Offset: 0

Gus Wiseman, May 24 2018

nonn

A blob is a connected antichain of finite sets that cannot be capped by a hypertree with more than one branch.

			Non-isomorphic representatives of the a(4) = 10 blobs:
  {{1,2,3,4}}
  {{1,3,4},{2,3,4}}
  {{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}}
  {{1,2},{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}}

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

Cf. A030019, A035053, A048143, A054921, A134957, A144959, A275307, A286520, A293607, A304118, A304717, A304867.

A305081 Heinz numbers of z-trees. Heinz numbers of connected integer partitions with pairwise indivisible parts and z-density -1.

Original entry on oeis.org

2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 25, 27, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 81, 83, 89, 91, 97, 101, 103, 107, 109, 113, 121, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 203, 211, 223, 227, 229

Offset: 1

Gus Wiseman, May 25 2018

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
Given a finite set S of positive integers greater than one, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices with a common divisor greater than 1. For example, G({6,14,15,35}) is a 4-cycle. A multiset S is said to be connected if G(S) is a connected graph.
The clutter density of a multiset S of positive integers is Sum_{s in S} (omega(s) - 1) - omega(lcm(S)) where omega = A001221.

			4331 is the Heinz number of {18,20}, which is a z-tree corresponding to the multiset multisystem {{1,2,2},{1,1,3}}.
17927 is the Heinz number of {4,6,45}, which is a z-tree corresponding to the multiset multisystem {{1,1},{1,2},{2,2,3}}.
27391 is the Heinz number of {4,4,6,14}, which is a z-tree corresponding to the multiset multisystem {{1,1},{1,1},{1,2},{1,4}}.

Cf. A001221, A030019, A056239, A112798, A286520, A302242, A303362, A303837, A304118, A304714, A304716, A305052, A305078, A305079.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{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]]]]]]]]];
zens[n_]:=If[n==1,0,Total@Cases[FactorInteger[n],{p_,k_}:>k*(PrimeNu[PrimePi[p]]-1)]-PrimeNu[LCM@@Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]]]];
Select[Range[300],And[zens[#]==-1,Length[zsm[primeMS[#]]]==1,Select[Tuples[primeMS[#],2],UnsameQ@@#&&Divisible@@#&]=={}]&]

A322139 Number of labeled 2-connected simple graphs with n edges (the vertices are {1,2,...,k} for some k).

Original entry on oeis.org

1, 1, 0, 1, 3, 18, 131, 1180, 12570, 154535, 2151439, 33431046, 573197723, 10743619285, 218447494812, 4787255999220, 112454930390211, 2818138438707516, 75031660452368001, 2114705500316025737, 62890323682634277951, 1967901134191778583146, 64623905086814216468839

Offset: 0

Gus Wiseman, Nov 27 2018

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..100
Gus Wiseman, The a(6) = 131 labeled 2-connected graphs with 6 edges.

Cf. A002218, A013922, A123534, A275307, A291841, A304118, A304887, A322110, A322117, A322118, A322137, A322138.

seq(n)={Vec(1 + vecsum(Vec(serlaplace(log(x/serreverse(x*deriv(log(sum(k=0, n, (1 + y + O(y*y^n))^binomial(k, 2) * x^k / k!) + O(x*x^n)))))))))} \\ Andrew Howroyd, Nov 29 2018

a(n) = Sum_{i=3..n} A123534(i, n). - Andrew Howroyd, Nov 30 2018

Terms a(7) and beyond from Andrew Howroyd, Nov 29 2018

A305501 Number of connected components of the integer partition y + 1 where y is the integer partition with Heinz number n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 1, 2, 2, 2, 2, 3, 1, 1, 1, 1, 1, 3, 1, 2, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 2, 1, 1, 1, 2, 1, 2, 3

Offset: 1

Gus Wiseman, Jun 03 2018

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
Given a finite set S of positive integers greater than one, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices with a common divisor greater than 1. For example, G({6,14,15,35}) is a 4-cycle. A partition y is said to be connected if G(U(y + 1)) is a connected graph, where U(y + 1) is the set of distinct successors of the parts of y.
This is intended to be a cleaner form of A305079, where the treatment of empty multisets is arbitrary.

			The "prime index plus 1" multiset of 7410 is {2,3,4,7,9}, with connected components {{2,4},{3,9},{7}}, so a(7410) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to Heinz numbers

Cf. A001221, A048143, A056239, A112798, A275024, A286518, A302242, A303837, A304118, A304714, A304716, A305078, A305079, A305504.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{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]]]]]]]]];
Table[Length[zsm[primeMS[n]+1]],{n,100}]

zero_first_elem_and_connected_elems(ys) = { my(cs = List([ys[1]]), i=1); ys[1] = 0; while(i<=#cs, for(j=2,#ys,if(ys[j]&&(1!=gcd(cs[i],ys[j])), listput(cs,ys[j]); ys[j] = 0)); i++); (ys); };
A305501(n) = { my(cs = apply(p -> 1+primepi(p),factor(n)[,1]~), s=0); while(#cs, cs = select(c -> c, zero_first_elem_and_connected_elems(cs)); s++); (s); }; \\ Antti Karttunen, Nov 09 2018

More terms from Antti Karttunen, Nov 09 2018

A322138 Number of non-isomorphic weight-n blobs (2-connected weak antichains) of multisets with no singletons.

Original entry on oeis.org

1, 0, 2, 3, 7, 7, 20, 26, 78, 184, 553

Offset: 0

Gus Wiseman, Nov 27 2018

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(7) = 26 blobs:
  {{11}}  {{111}}  {{1111}}    {{11111}}  {{111111}}      {{1111111}}
  {{12}}  {{122}}  {{1122}}    {{11222}}  {{111222}}      {{1112222}}
          {{123}}  {{1222}}    {{12222}}  {{112222}}      {{1122222}}
                   {{1233}}    {{12233}}  {{112233}}      {{1122333}}
                   {{1234}}    {{12333}}  {{122222}}      {{1222222}}
                   {{11}{11}}  {{12344}}  {{122333}}      {{1222333}}
                   {{12}{12}}  {{12345}}  {{123333}}      {{1223333}}
                                          {{123344}}      {{1223344}}
                                          {{123444}}      {{1233333}}
                                          {{123455}}      {{1233444}}
                                          {{123456}}      {{1234444}}
                                          {{111}{111}}    {{1234455}}
                                          {{112}{122}}    {{1234555}}
                                          {{122}{122}}    {{1234566}}
                                          {{123}{123}}    {{1234567}}
                                          {{123}{233}}    {{112}{1222}}
                                          {{134}{234}}    {{122}{1233}}
                                          {{11}{11}{11}}  {{123}{2233}}
                                          {{12}{12}{12}}  {{123}{2333}}
                                          {{12}{13}{23}}  {{123}{2344}}
                                                          {{134}{2344}}
                                                          {{145}{2345}}
                                                          {{223}{1233}}
                                                          {{344}{1234}}
                                                          {{12}{13}{233}}
                                                          {{13}{14}{234}}

Cf. A002218, A007718, A013922, A275307, A286520, A293994, A304118, A304887, A319719, A322110, A322117, A322118.

A305055 Numbers n such that the z-density of the integer partition with Heinz number n is 0.

Original entry on oeis.org

1, 169, 481, 507, 793, 841, 845, 1157, 1183, 1369, 1443, 1469, 1521, 1849, 1963, 2059, 2209, 2257, 2353, 2379, 2405, 2523, 2535, 2899, 3211, 3263, 3277, 3293, 3367, 3471, 3549, 3653, 3721, 3887, 3965, 4107, 4121, 4181, 4225, 4329, 4394, 4407, 4563, 4601, 4667

Offset: 1

Gus Wiseman, May 24 2018

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

The z-density of a multiset S of positive integers is Sum_{s in S} (omega(s) - 1) - omega(lcm(S)) where omega = A001221 is number of distinct prime factors.

Cf. A001221, A030019, A048143, A056239, A112798, A290103, A302242, A303837, A304118, A304714, A304716, A305001, A305052, A305053, A305054.

zens[n_]:=If[n==1,0,Total@Cases[FactorInteger[n],{p_,k_}:>k*(PrimeNu[PrimePi[p]]-1)]-PrimeNu[LCM@@Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]]]];
Select[Range[1000],zens[#]==0&]

A305194 Number of z-forests summing to n. Number of strict integer partitions of n with pairwise indivisible parts and all connected components having clutter density -1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 2, 3, 4, 5, 4, 6, 7, 7, 9, 11, 12, 13, 15, 17, 20, 23, 25, 27, 32, 35, 40, 45, 49, 54, 58, 67, 78, 82, 95, 99, 111, 123, 135, 150, 164, 177, 194, 214, 236, 260, 282, 309, 330

Offset: 1

Gus Wiseman, May 27 2018

Given a finite set S of positive integers greater than 1, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices that have a common divisor greater than 1. For example, G({6,14,15,35}) is a 4-cycle. A set S is said to be connected if G(S) is a connected graph. The clutter density of S is defined to be Sum_{s in S} (omega(s) - 1) - omega(lcm(S)), where omega = A001221 and lcm is least common multiple. Then a z-forest is a strict integer partition with pairwise indivisible parts greater than 1 such that all connected components have clutter density -1.

			The a(17) = 11 z-forests together with the corresponding multiset systems:
       (17): {{7}}
     (15,2): {{2,3},{1}}
     (14,3): {{1,4},{2}}
     (13,4): {{6},{1,1}}
     (12,5): {{1,1,2},{3}}
     (11,6): {{5},{1,2}}
     (10,7): {{1,3},{4}}
      (9,8): {{2,2},{1,1,1}}
   (10,4,3): {{1,3},{1,1},{2}}
    (7,6,4): {{4},{1,2},{1,1}}
  (7,5,3,2): {{4},{3},{2},{1}}

Roland Bacher, On the enumeration of labelled hypertrees and of labelled bipartite trees, arXiv:1102.2708 [math.CO], 2011.

Cf. A030019, A048143, A134954, A275307, A285572, A293510, A293993, A293994, A303362, A303837, A303838, A304118, A304382, A305078, A305195.

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];
zreeQ[s_]:=And[Length[s]>=2,zensity[s]==-1];
Table[Length[Select[IntegerPartitions[n],Function[s,UnsameQ@@s&&And@@(Length[#]==1||zreeQ[#]&)/@Table[Select[s,Divisible[m,#]&],{m,zsm[s]}]&&Select[Tuples[s,2],UnsameQ@@#&&Divisible@@#&]=={}]]],{n,50}]

A305195 Number of z-blobs summing to n. Number of connected strict integer partitions of n, with pairwise indivisible parts, that cannot be capped by a z-tree.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 2, 1, 1, 1, 3, 2, 2, 2, 1, 1, 3, 3, 3, 1, 1, 1, 4, 5, 6, 2, 1, 1, 4, 6, 7, 2, 2, 6

Offset: 1

Gus Wiseman, May 27 2018

nonn

Caps of a clutter are defined in the link, and the generalization to "multiclutters," where edges can be multisets, is straightforward.

			The a(30) = 2 z-blobs together with the corresponding multiset systems:
     (30): {{1,2,3}}
  (18,12): {{1,2,2},{1,1,2}}
The a(47) = 3 z-blobs together with the corresponding multiset systems:
        (47): {{15}}
  (21,14,12): {{2,4},{1,4},{1,1,2}}
  (20,15,12): {{1,1,3},{2,3},{1,1,2}}
The a(60) = 5 z-blobs together with the corresponding multiset systems:
           (60): {{1,1,2,3}}
        (42,18): {{1,2,4},{1,2,2}}
        (36,24): {{1,1,2,2},{1,1,1,2}}
     (30,18,12): {{1,2,3},{1,2,2},{1,1,2}}
  (21,15,14,10): {{2,4},{2,3},{1,4},{1,3}}
The a(67) = 7 z-blobs together with the corresponding multiset systems:
           (67): {{19}}
     (45,12,10): {{2,2,3},{1,1,2},{1,3}}
     (42,15,10): {{1,2,4},{2,3},{1,3}}
     (40,15,12): {{1,1,1,3},{2,3},{1,1,2}}
     (33,22,12): {{2,5},{1,5},{1,1,2}}
     (28,21,18): {{1,1,4},{2,4},{1,2,2}}
  (24,18,15,10): {{1,1,1,2},{1,2,2},{2,3},{1,3}}

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

Cf. A030019, A048143, A275307, A285572, A293510, A303362, A303837, A303838, A304118, A304382, A304887, A305028, A305078, A305194.

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}]

cp's OEIS Frontend

A322118 Number of non-isomorphic connected multiset partitions of weight n with no singletons that cannot be capped by a tree.

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A305028 Number of unlabeled blobs spanning n vertices without singleton edges.

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A305081 Heinz numbers of z-trees. Heinz numbers of connected integer partitions with pairwise indivisible parts and z-density -1.

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A322139 Number of labeled 2-connected simple graphs with n edges (the vertices are {1,2,...,k} for some k).

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A305501 Number of connected components of the integer partition y + 1 where y is the integer partition with Heinz number n.

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A322138 Number of non-isomorphic weight-n blobs (2-connected weak antichains) of multisets with no singletons.

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A305055 Numbers n such that the z-density of the integer partition with Heinz number n is 0.

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A305194 Number of z-forests summing to n. Number of strict integer partitions of n with pairwise indivisible parts and all connected components having clutter density -1.

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A305195 Number of z-blobs summing to n. Number of connected strict integer partitions of n, with pairwise indivisible parts, that cannot be capped by a z-tree.

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