A322393 - OEIS

%I #8 Dec 06 2018 16:35:55

%S 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,

%T 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,

%U 54,2,0,0,0,0,0,0,0,0,0,0,63,1,4,4,3,1,1

%N 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.

%C 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.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/K-edge-connected_graph">k-edge-connected graph</a>

%e Triangle begins:

%e    1

%e    0  1

%e    1  1  0

%e    2  1  0  0

%e    3  1  1  0  0

%e    6  1  0  0  0  0

%e    7  1  2  1  0  0  0

%e   14  1  0  0  0  0  0  0

%e   17  1  2  1  1  0  0  0  0

%e   27  1  1  1  0  0  0  0  0  0

%e   34  1  3  2  1  1  0  0  0  0  0

%e   54  2  0  0  0  0  0  0  0  0  0  0

%e   63  1  4  4  3  1  1  0  0  0  0  0  0

%e Row 6 {7, 1, 2, 1} counts the following integer partitions:

%e   (51)      (6)  (33)  (222)

%e   (321)          (42)

%e   (411)

%e   (2211)

%e   (3111)

%e   (21111)

%e   (111111)

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t 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]]]]]]]]];

%t edgeConn[y_]:=Length[y]-Max@@Length/@Select[Union[Subsets[y]],Length[csm[primeMS/@#]]!=1&]

%t Table[Length[Select[IntegerPartitions[n],edgeConn[#]==k&]],{n,10},{k,0,n}]

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

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

%K nonn,tabl

%O 0,7

%A _Gus Wiseman_, Dec 06 2018