A317655 Number of free pure symmetric multifunctions with leaves a multiset whose multiplicities are the integer partition with Heinz number n.

0, 1, 1, 2, 3, 8, 10, 15, 50, 35, 37, 96, 144, 160, 299, 184, 589, 840, 2483, 578, 1729, 750, 10746, 1627, 2246, 3578, 9357, 3367, 47420, 6397, 212668, 3155, 9818, 17280, 15666, 18250, 966324, 84232, 54990, 12471, 4439540, 45015

Offset: 1

Author

Gus Wiseman, Aug 03 2018

Keywords

• nonn

Comments

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

A free pure symmetric multifunction f in EPSM is either (case 1) a positive integer, or (case 2) an expression of the form h[g_1, ..., g_k] where k > 0, h is in EPSM, each of the g_i for i = 1, ..., k is in EPSM, and for i < j we have g_i <= g_j under a canonical total ordering of EPSM, such as the Mathematica ordering of expressions.

Examples

			The a(6) = 8 free pure symmetric multifunctions:
  1[1[2]]
  1[2[1]]
  2[1[1]]
  1[1][2]
  1[2][1]
  2[1][1]
  1[1,2]
  2[1,1]
		

Crossrefs

Cf. A001003, A052893, A053492, A255906, A277996, A279944, A280000.

Cf. A317652, A317653, A317654, A317656, A317658.

Programs

Mathematica

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
exprUsing[m_]:=exprUsing[m]=If[Length[m]==0,{},If[Length[m]==1,{First[m]},Join@@Cases[Union[Table[PR[m[[s]],m[[Complement[Range[Length[m]],s]]]],{s,Take[Subsets[Range[Length[m]]],{2,-2}]}]],PR[h_,g_]:>Join@@Table[Apply@@@Tuples[{exprUsing[h],Union[Sort/@Tuples[exprUsing/@p]]}],{p,mps[g]}]]]];
got[y_]:=Join@@Table[Table[i,{y[[i]]}],{i,Range[Length[y]]}];
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[exprUsing[got[Reverse[primeMS[n]]]]],{n,40}]