cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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.

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

Views

Author

Gus Wiseman, Dec 06 2018

Keywords

Comments

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.

Examples

			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)

Links

Crossrefs

Row sums are A000041. First column is A322367. Second column is A322391.
Cf. A013922, A054921, A095983, A304716, A305078, A305079, A322335, A322336, A322337, A322338, A322387.

Programs

  • Mathematica
    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}]

A322400 Heinz numbers of integer partitions with vertex-connectivity 1.

Original entry on oeis.org

3, 5, 7, 9, 11, 17, 19, 21, 23, 25, 27, 31, 41, 49, 53, 57, 59, 63, 67, 81, 83, 97, 103, 109, 115, 121, 125, 127, 131, 133, 147, 157, 159, 171, 179, 189, 191, 211, 227, 241, 243, 277, 283, 289, 311, 331, 343, 353, 361, 367, 371, 377, 393, 399, 401, 419, 431
Offset: 1

Views

Author

Gus Wiseman, Dec 06 2018

Keywords

Comments

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
The vertex-connectivity of an integer partition is the minimum number of primes that must be divided out (and any parts then equal to 1 removed) so that the prime factorizations of the remaining parts form a disconnected (or empty) hypergraph.

Examples

			The sequence of all integer partitions with vertex-connectivity 1 begins: (2), (3), (4), (2,2), (5), (7), (8), (4,2), (9), (3,3), (2,2,2), (11), (13), (4,4), (16), (8,2), (17), (4,2,2), (19), (2,2,2,2), (23), (25), (27), (29), (9,3), (5,5), (3,3,3), (31), (32), (8,4), (4,4,2), (37), (16,2), (8,2,2), (41), (4,2,2,2), (43).

Links

Crossrefs

Cf. A003963, A013922, A056239, A112798, A302242, A304716, A305078, A305079, A322387, A322388, A322389, A322390, A322390, A322394.

Programs

  • Mathematica
    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&]

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

Views

Author

Gus Wiseman, Dec 06 2018

Keywords

Comments

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.

Examples

			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).

Links

Crossrefs

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

Programs

  • Mathematica
    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}]
Previous Showing 11-13 of 13 results.

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