A322337 -id:A322337 - cp's OEIS frontend

A322338 Edge-connectivity of the integer partition with Heinz number n.

0, 1, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 3, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 1, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 3, 0, 2, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 4, 0, 1, 0, 0, 0, 2

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).

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.

			2093 is the Heinz number of (9,6,4), corresponding to the multiset partition {{1,1},{1,2},{2,2}}, which can be made disconnected by removing only the part {1,2}, so a(2093) = 1.

Wikipedia, k-edge-connected graph

Cf. A001222, A003963, A013922, A054921, A056239, A095983, A112798, A302242, A304714, A304716, A305078, A305079, A322335, A322336, A322337.

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]]]]]]]]];
Table[PrimeOmega[n]-Max@@PrimeOmega/@Select[Divisors[n],Length[csm[primeMS/@primeMS[#]]]!=1&],{n,100}]

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

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}]

A322393 Regular triangle read by rows where T(n,k) is the number of integer partitions of n with edge-connectivity k, for 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 1, 1, 0, 2, 1, 0, 0, 3, 1, 1, 0, 0, 6, 1, 0, 0, 0, 0, 7, 1, 2, 1, 0, 0, 0, 14, 1, 0, 0, 0, 0, 0, 0, 17, 1, 2, 1, 1, 0, 0, 0, 0, 27, 1, 1, 1, 0, 0, 0, 0, 0, 0, 34, 1, 3, 2, 1, 1, 0, 0, 0, 0, 0, 54, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 63, 1, 4, 4, 3, 1, 1

Offset: 0

Gus Wiseman, Dec 06 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.

			Triangle begins:
   1
   0  1
   1  1  0
   2  1  0  0
   3  1  1  0  0
   6  1  0  0  0  0
   7  1  2  1  0  0  0
  14  1  0  0  0  0  0  0
  17  1  2  1  1  0  0  0  0
  27  1  1  1  0  0  0  0  0  0
  34  1  3  2  1  1  0  0  0  0  0
  54  2  0  0  0  0  0  0  0  0  0  0
  63  1  4  4  3  1  1  0  0  0  0  0  0
Row 6 {7, 1, 2, 1} counts the following integer partitions:
  (51)      (6)  (33)  (222)
  (321)          (42)
  (411)
  (2211)
  (3111)
  (21111)
  (111111)

Wikipedia, k-edge-connected graph

Row sums are A000041. First column is A322367. Second column is A322391.

Cf. A013922, A054921, A095983, A304716, A305078, A305079, A322335, A322336, A322337, A322338, A322387.

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_]:=Length[y]-Max@@Length/@Select[Union[Subsets[y]],Length[csm[primeMS/@#]]!=1&]
Table[Length[Select[IntegerPartitions[n],edgeConn[#]==k&]],{n,10},{k,0,n}]

A322369 Number of strict disconnected or empty integer partitions of n.

Original entry on oeis.org

1, 0, 0, 1, 1, 2, 2, 4, 4, 6, 7, 10, 10, 16, 17, 22, 26, 33, 36, 48, 52, 64, 76, 90, 101, 125, 142, 166, 192, 225, 250, 302, 339, 393, 451, 515, 581, 675, 762, 866, 985, 1122, 1255, 1441, 1612, 1823, 2059, 2318, 2591, 2930, 3275, 3668, 4118, 4605, 5125, 5749

Offset: 0

Gus Wiseman, Dec 04 2018

nonn

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) = 1 through a(11) = 10 strict disconnected integer partitions:
  (2,1)  (3,1)  (3,2)  (5,1)    (4,3)    (5,3)    (5,4)    (7,3)      (6,5)
                (4,1)  (3,2,1)  (5,2)    (7,1)    (7,2)    (9,1)      (7,4)
                                (6,1)    (4,3,1)  (8,1)    (5,3,2)    (8,3)
                                (4,2,1)  (5,2,1)  (4,3,2)  (5,4,1)    (9,2)
                                                  (5,3,1)  (6,3,1)    (10,1)
                                                  (6,2,1)  (7,2,1)    (5,4,2)
                                                           (4,3,2,1)  (6,4,1)
                                                                      (7,3,1)
                                                                      (8,2,1)
                                                                      (5,3,2,1)

Cf. A054921, A218970, A286518, A304714, A304716, A305078, A305079, A322306, A322307, A322335, A322337, A322338, A322367, A322368.

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],And[UnsameQ@@#,Length[zsm[#]]!=1]&]],{n,30}]

A322401 Number of strict integer partitions of n with edge-connectivity 1.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 5, 1, 6, 2, 7, 2, 13, 3, 14, 6, 18, 8, 28, 11, 33, 19, 38, 22, 54, 28, 71, 44, 83, 53, 110, 68, 134, 98, 154, 120, 209, 145, 253, 191, 302, 244, 385, 299, 459, 390, 553, 483, 693, 578

Offset: 0

Gus Wiseman, Dec 06 2018

nonn

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(30) = 11 strict integer partitions with edge-connectivity 1:
  (30),
  (10,9,6,5), (12,10,5,3), (14,7,6,3), (15,6,5,4), (15,10,3,2),
  (9,8,6,4,3), (10,9,6,3,2), (12,9,4,3,2), (15,6,4,3,2),
  (10,6,5,4,3,2).

Wikipedia, k-edge-connected graph

Cf. A095983, A218970, A304714, A304716, A305078, A305079, A322335, A322337, A322387, A322390, A322391, A322393, A322394, A322400.

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],UnsameQ@@#&&edgeConn[#]==1&]],{n,30}]

cp's OEIS Frontend

A322338 Edge-connectivity of the integer partition with Heinz number n.

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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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A322367 Number of disconnected or empty integer partitions of n.

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A322393 Regular triangle read by rows where T(n,k) is the number of integer partitions of n with edge-connectivity k, for 0 <= k <= n.

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A322369 Number of strict disconnected or empty integer partitions of n.

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Mathematica

A322401 Number of strict integer partitions of n with edge-connectivity 1.

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