A218970 -id:A218970 - cp's OEIS frontend

A304716 Number of integer partitions of n whose distinct parts are connected.

1, 2, 2, 3, 2, 5, 2, 6, 4, 9, 3, 15, 4, 18, 12, 25, 11, 41, 17, 54, 36, 72, 44, 113, 69, 145, 113, 204, 153, 302, 220, 394, 343, 541, 475, 771, 662, 1023, 968, 1398, 1314, 1929, 1822, 2566, 2565, 3440, 3446, 4677, 4688, 6187, 6407, 8216, 8544, 10975, 11436

Offset: 1

Gus Wiseman, May 17 2018

nonn

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. 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 a(12) = 15 connected integer partitions and their corresponding connected multiset multisystems (see A112798, A302242) are the following.
                     (12): {{1,1,2}}
                    (6 6): {{1,2},{1,2}}
                    (8 4): {{1,1,1},{1,1}}
                    (9 3): {{2,2},{2}}
                   (10 2): {{1,3},{1}}
                  (4 4 4): {{1,1},{1,1},{1,1}}
                  (6 3 3): {{1,2},{2},{2}}
                  (6 4 2): {{1,2},{1,1},{1}}
                  (8 2 2): {{1,1,1},{1},{1}}
                (3 3 3 3): {{2},{2},{2},{2}}
                (4 4 2 2): {{1,1},{1,1},{1},{1}}
                (6 2 2 2): {{1,2},{1},{1},{1}}
              (4 2 2 2 2): {{1,1},{1},{1},{1},{1}}
            (2 2 2 2 2 2): {{1},{1},{1},{1},{1},{1}}
(1 1 1 1 1 1 1 1 1 1 1 1): {{},{},{},{},{},{},{},{},{},{},{},{}}

Cf. A000009, A003963, A048143, A054921, A218970, A285572, A286518, A302242, A304714, A305078, A305079, A322306, A322307.

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[Select[IntegerPartitions[n],Length[zsm[Union[#]]]===1&]],{n,30}]

For n > 1, a(n) = A218970(n) + 1. - Gus Wiseman, Dec 04 2018

Name changed to distinguish from A218970 by Gus Wiseman, Dec 04 2018

A322335 Number of 2-edge-connected integer partitions of n.

Original entry on oeis.org

0, 0, 0, 1, 0, 3, 0, 4, 2, 7, 0, 13, 0, 15, 8, 21, 1, 37, 2, 45, 18, 58, 8, 95, 19, 109, 45, 150, 38, 232, 59, 268, 129, 357, 155, 523, 203, 633, 359, 852, 431, 1185, 609, 1464, 969

Offset: 1

Gus Wiseman, Dec 04 2018

First differs from A108572 at a(17) = 1, A108572(17) = 0.

An integer partition is 2-edge-connected if the hypergraph of prime factorizations of its parts is connected and cannot be disconnected by removing any single part. For example (6,6,3,2) is 2-edge-connected but (6,3,2) is not.

			The a(14) = 15 2-edge-connected integer partitions of 14:
  (7,7)   (6,4,4)   (4,4,4,2)  (4,4,2,2,2)  (4,2,2,2,2,2)  (2,2,2,2,2,2,2)
  (8,6)   (6,6,2)   (6,4,2,2)  (6,2,2,2,2)
  (10,4)  (8,4,2)   (8,2,2,2)
  (12,2)  (10,2,2)

Wikipedia, k-edge-connected graph

Cf. A007718, A013922, A054921, A095983, A218970, A275307, A286518, A304714, A304716, A322336, A322337, A322338.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
twoedQ[sys_]:=And[Length[csm[sys]]==1,And@@Table[Length[csm[Delete[sys,i]]]==1,{i,Length[sys]}]];
Table[Length[Select[IntegerPartitions[n],twoedQ[primeMS/@#]&]],{n,30}]

a(42)-a(45) from Jinyuan Wang, Jun 20 2020

A322336 Heinz numbers of 2-edge-connected integer partitions.

Original entry on oeis.org

9, 21, 25, 27, 39, 49, 57, 63, 65, 81, 87, 91, 111, 115, 117, 121, 125, 129, 133, 147, 159, 169, 171, 183, 185, 189, 203, 213, 235, 237, 243, 247, 259, 261, 267, 273, 289, 299, 301, 303, 305, 319, 321, 325, 333, 339, 343, 351, 361, 365, 371, 377, 387, 393, 399

Offset: 1

Gus Wiseman, Dec 04 2018

nonn

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

An integer partition is 2-edge-connected if the hypergraph of prime factorizations of its parts is connected and cannot be disconnected by removing any single part. For example (6,6,3,2) is 2-edge-connected but (6,3,2) is not.

			The sequence of all 2-edge-connected integer partitions begins: (2,2), (4,2), (3,3), (2,2,2), (6,2), (4,4), (8,2), (4,2,2), (6,3), (2,2,2,2), (10,2), (6,4), (12,2), (9,3), (6,2,2), (5,5), (3,3,3), (14,2), (8,4), (4,4,2).

Wikipedia, k-edge-connected graph

Cf. A001222, A003963, A013922, A054921, A056239, A095983, A112798, A218970, A275307, A304714, A304716, A305078, A305079, A322335, A322337, A322338.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
twoedQ[sys_]:=And[Length[csm[sys]]==1,And@@Table[Length[csm[Delete[sys,i]]]==1,{i,Length[sys]}]];
Select[Range[100],twoedQ[primeMS/@primeMS[#]]&]

A322387 Number of 2-vertex-connected integer partitions of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 1, 3, 1, 6, 2, 10, 8, 13, 9, 26, 14, 35, 28, 50, 37, 77, 54, 101, 84, 138, 110, 205, 149, 252, 222, 335, 287, 455, 375, 577, 522, 740, 657, 985

Offset: 1

Gus Wiseman, Dec 05 2018

An integer partition is 2-vertex-connected if the prime factorizations of the parts form a connected hypergraph that is still connected if any single prime number is divided out of all the parts (and any parts then equal to 1 are removed).

			The a(14) = 10 2-vertex-connected integer partitions:
  (14)  (8,6)   (6,4,4)   (6,3,3,2)  (6,2,2,2,2)
        (10,4)  (6,6,2)   (6,4,2,2)
        (12,2)  (10,2,2)

Wikipedia, k-vertex-connected graph

Cf. A013922, A095983, A218970, A275307, A304714, A304716, A305078, A305079, A322335, A322336, A322337, A322338, A322388, A322389, A322390.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
vertConn[y_]:=If[Length[csm[primeMS/@y]]!=1,0,Min@@Length/@Select[Subsets[Union@@primeMS/@y],Function[del,Length[csm[DeleteCases[DeleteCases[primeMS/@y,Alternatives@@del,{2}],{}]]]!=1]]];
Table[Length[Select[IntegerPartitions[n],vertConn[#]>1&]],{n,30}]

a(41)-a(42) from Jinyuan Wang, Jun 20 2020

A322337 Number of strict 2-edge-connected integer partitions of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 0, 4, 0, 4, 3, 5, 0, 9, 0, 10, 5, 11, 1, 18, 3, 17, 8, 22, 3, 35, 5, 32, 17, 39, 16, 59, 14, 58, 33, 75, 28, 103, 35, 106, 71, 125, 63, 174, 81, 192, 127, 220, 130, 294, 170, 325, 237, 378, 257, 504

Offset: 1

Gus Wiseman, Dec 04 2018

nonn

An integer partition is 2-edge-connected if the hypergraph of prime factorizations of its parts is connected and cannot be disconnected by removing any single part.

			The a(24) = 18 strict 2-edge-connected integer partitions of 24:
  (15,9)   (10,8,6)   (10,8,4,2)
  (16,8)   (12,8,4)   (12,6,4,2)
  (18,6)   (12,9,3)
  (20,4)   (14,6,4)
  (21,3)   (14,8,2)
  (22,2)   (15,6,3)
  (14,10)  (16,6,2)
           (18,4,2)
           (12,10,2)

Wikipedia, k-edge-connected graph

Cf. A007718, A013922, A054921, A095983, A218970, A275307, A286518, A304714, A304716, A305078, A305079, A322335, A322336.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
twoedQ[sys_]:=And[Length[csm[sys]]==1,And@@Table[Length[csm[Delete[sys,i]]]==1,{i,Length[sys]}]];
Table[Length[Select[IntegerPartitions[n],And[UnsameQ@@#,twoedQ[primeMS/@#]]&]],{n,30}]

A322307 Number of multisets in the swell of the n-th multiset multisystem.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Dec 03 2018

nonn

First differs from A001221 at a(91) = 3, A001221(91) = 2.
The swell of a multiset partition is the set of possible joins of its connected submultisets, where the multiplicity of a vertex in the join of a set of multisets is the maximum multiplicity of the same vertex among the parts. For example the swell of {{1,1},{1,2},{2,2}} is:
  {1,1}
  {1,2}
  {2,2}
  {1,1,2}
  {1,2,2}
  {1,1,2,2}

Cf. A003963, A054921, A056239, A112798, A218970, A286518, A290103, A301957, A302242, A304716, A305078, A305079, A316556, A322306.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
zwell[y_]:=Union[y,Join@@Cases[Subsets[Union[y],{2}],{x_,z_}?(GCD@@#>1&):>zwell[Sort[Append[Fold[DeleteCases[#1,#2,{1},1]&,y,{x,z}],LCM[x,z]]]]]];
Table[Length[zwell[primeMS[n]]],{n,100}]

A322388 Heinz numbers of 2-vertex-connected integer partitions.

Original entry on oeis.org

13, 29, 37, 39, 43, 47, 61, 65, 71, 73, 79, 87, 89, 91, 101, 107, 111, 113, 117, 129, 137, 139, 149, 151, 163, 167, 169, 173, 181, 183, 185, 193, 195, 197, 199, 203, 213, 223, 229, 233, 235, 237, 239, 247, 251, 257, 259, 261, 263, 267, 269, 271, 273, 281

Offset: 1

Gus Wiseman, Dec 05 2018

nonn

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

An integer partition is 2-vertex-connected if the prime factorizations of the parts form a connected hypergraph that is still connected if any single prime number is divided out of all the parts (and any parts then equal to 1 are removed).

			The sequence of all 2-vertex-connected integer partitions begins: (1), (6), (10), (12), (6,2), (14), (15), (18), (6,3), (20), (21), (22), (10,2), (24), (6,4), (26), (28), (12,2), (30), (6,2,2), (14,2), (33), (34), (35), (36), (38), (39), (6,6), (40), (42), (18,2), (12,3), (44), (6,3,2), (45), (46).

Wikipedia, k-vertex-connected graph

Cf. A003963, A013922, A056239, A095983, A112798, A218970, A275307, A304716, A305078, A305079, A322336, A322338, A322387, A322389, A322390.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
vertConn[y_]:=If[Length[csm[primeMS/@y]]!=1,0,Min@@Length/@Select[Subsets[Union@@primeMS/@y],Function[del,Length[csm[DeleteCases[DeleteCases[primeMS/@y,Alternatives@@del,{2}],{}]]]!=1]]]
Select[Range[100],vertConn[primeMS[#]]>1&]

A322391 Number of integer partitions of n with edge-connectivity 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 2, 3, 3, 9, 3, 14, 8, 17, 13, 35, 17, 49, 35, 67, 53, 114, 69

Offset: 1

Gus Wiseman, Dec 05 2018

The edge-connectivity of an integer partition is the minimum number of parts that must be removed so that the prime factorizations of the remaining parts form a disconnected (or empty) hypergraph.

			The a(20) = 8 integer partitions:
  (20),
  (12,3,3,2), (9,6,3,2), (8,6,3,3),
  (6,4,4,3,3),
  (6,4,3,3,2,2), (6,3,3,3,3,2),
  (6,3,3,2,2,2,2).

Wikipedia, k-edge-connected graph

Cf. A007718, A013922, A054921, A095983, A218970, A275307, A304716, A305078, A305079, A322335, A322336, A322387, A322390.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
edgeConn[y_]:=If[Length[csm[primeMS/@y]]!=1,0,Length[y]-Max@@Length/@Select[Union[Subsets[y]],Length[csm[primeMS/@#]]!=1&]];
Table[Length[Select[IntegerPartitions[n],edgeConn[#]==1&]],{n,20}]

A322394 Heinz numbers of integer partitions with edge-connectivity 1.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 195, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269

Offset: 1

Gus Wiseman, Dec 06 2018

nonn

The first nonprime term is 195, which is the Heinz number of (6,3,2).
The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
An integer partition has edge-connectivity 1 if the prime factorizations of the parts form a connected hypergraph that can be disconnected (or made empty) by removing a single part.

Wikipedia, k-edge-connected graph

Cf. A007718, A013922, A054921, A056239, A095983, A112798, A218970, A304716, A305078, A305079, A322336, A322391, A322393.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
edgeConn[y_]:=If[Length[csm[primeMS/@y]]!=1,0,Length[y]-Max@@Length/@Select[Union[Subsets[y]],Length[csm[primeMS/@#]]!=1&]];
Select[Range[100],edgeConn[primeMS[#]]==1&]

A322367 Number of disconnected or empty integer partitions of n.

Original entry on oeis.org

1, 0, 1, 2, 3, 6, 7, 14, 17, 27, 34, 54, 63, 98, 118, 165, 207, 287, 345, 474, 574, 757, 931, 1212, 1463, 1890, 2292, 2898, 3515, 4413, 5303

Offset: 0

Gus Wiseman, Dec 04 2018

An integer partition is connected if the prime factorizations of its parts form a connected hypergraph. It is disconnected if it can be separated into two or more integer partitions with relatively prime products. For example, the integer partition (654321) has three connected components: (6432)(5)(1).

			The a(3) = 2 through a(9) = 27 disconnected integer partitions:
  (21)   (31)    (32)     (51)      (43)       (53)        (54)
  (111)  (211)   (41)     (321)     (52)       (71)        (72)
         (1111)  (221)    (411)     (61)       (332)       (81)
                 (311)    (2211)    (322)      (431)       (432)
                 (2111)   (3111)    (331)      (521)       (441)
                 (11111)  (21111)   (421)      (611)       (522)
                          (111111)  (511)      (3221)      (531)
                                    (2221)     (3311)      (621)
                                    (3211)     (4211)      (711)
                                    (4111)     (5111)      (3222)
                                    (22111)    (22211)     (3321)
                                    (31111)    (32111)     (4221)
                                    (211111)   (41111)     (4311)
                                    (1111111)  (221111)    (5211)
                                               (311111)    (6111)
                                               (2111111)   (22221)
                                               (11111111)  (32211)
                                                           (33111)
                                                           (42111)
                                                           (51111)
                                                           (222111)
                                                           (321111)
                                                           (411111)
                                                           (2211111)
                                                           (3111111)
                                                           (21111111)
                                                           (111111111)

Cf. A054921, A218970, A286518, A322335, A304714, A304716, A305078, A305079, A322306, A322307, A322337, A322338, A322368, A322369.

zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c=={},s,zsm[Sort[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
Table[Length[Select[IntegerPartitions[n],Length[zsm[#]]!=1&]],{n,20}]

cp's OEIS Frontend

A304716 Number of integer partitions of n whose distinct parts are connected.

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A322335 Number of 2-edge-connected integer partitions of n.

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A322336 Heinz numbers of 2-edge-connected integer partitions.

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A322387 Number of 2-vertex-connected integer partitions of n.

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A322337 Number of strict 2-edge-connected integer partitions of n.

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A322307 Number of multisets in the swell of the n-th multiset multisystem.

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A322388 Heinz numbers of 2-vertex-connected integer partitions.

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A322391 Number of integer partitions of n with edge-connectivity 1.

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