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.

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

0,7

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.

Links

Table of n, a(n) for n=0..84.

Wikipedia, k-edge-connected graph

Example

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)

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}]

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.

Sequence in context: A114374 A111505 A109264 * A294265 A130507 A281449

Adjacent sequences: A322390 A322391 A322392 * A322394 A322395 A322396

Keyword

nonn,tabl

Author

Gus Wiseman, Dec 06 2018

Status

approved