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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.
3
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.
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.
Sequence in context: A114374 A111505 A109264 * A294265 A130507 A281449
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Dec 06 2018
STATUS
approved