A319226 Irregular triangle where T(n,k) is the number of acyclic spanning subgraphs of a cycle graph, where the sizes of the connected components are given by the integer partition with Heinz number A215366(n,k).

1, 2, 1, 3, 3, 1, 4, 2, 4, 4, 1, 5, 5, 5, 5, 5, 5, 1, 6, 6, 6, 3, 2, 6, 12, 9, 6, 6, 1, 7, 7, 7, 7, 14, 7, 7, 7, 7, 7, 21, 14, 7, 7, 1, 8, 8, 8, 4, 8, 8, 8, 16, 16, 8, 2, 24, 8, 24, 12, 16, 8, 32, 20, 8, 8, 1, 9, 9, 9, 9, 9, 9, 18, 9, 9, 9, 18, 18, 3, 27, 27

Offset: 1

Author

Gus Wiseman, Sep 13 2018

Keywords

• nonn
• tabf

Comments

A refinement of A135278, up the sign these are the coefficients appearing in the expansion of power-sum symmetric functions in terms of elementary or homogeneous symmetric functions.

Examples

			Triangle begins:
  1
  2  1
  3  3  1
  4  2  4  4  1
  5  5  5  5  5  5  1
  6  6  6  3  2  6 12  9  6  6  1
The fourth row corresponds to the symmetric function identities:
  p(4) = -4 e(4) + 2 e(22) + 4 e(31) - 4 e(211) + e(1111)
  p(4) =  4 h(4) - 2 h(22) - 4 h(31) + 4 h(211) - h(1111).
		

Links

• Wikipedia, Expressing power sums in terms of elementary symmetric polynomials
• Gus Wiseman, Enumeration of paths and cycles and e-coefficients of incomparability graphs, arXiv:0709.0430 [math.CO], 2007.

Crossrefs

Signed versions with different row-orderings are A115131, A210258, A263916.

Cf. A005651, A008480, A048994, A056239, A124794, A124795, A135278, A215366, A318762, A319191, A319193, A319225.

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[Union[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[Partition[Range[n],2,1,1],{n-PrimeOmega[m]}],Sort[Length/@csm[Union[#,List/@Range[n]]]]==primeMS[m]&]],{n,6},{m,Sort[Times@@Prime/@#&/@IntegerPartitions[n]]}]