A317655 - OEIS

%I #8 Aug 03 2018 08:17:06

%S 0,1,1,2,3,8,10,15,50,35,37,96,144,160,299,184,589,840,2483,578,1729,

%T 750,10746,1627,2246,3578,9357,3367,47420,6397,212668,3155,9818,17280,

%U 15666,18250,966324,84232,54990,12471,4439540,45015

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

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

%C 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.

%e The a(6) = 8 free pure symmetric multifunctions:

%e   1[1[2]]

%e   1[2[1]]

%e   2[1[1]]

%e   1[1][2]

%e   1[2][1]

%e   2[1][1]

%e   1[1,2]

%e   2[1,1]

%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];

%t mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];

%t 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]}]]]];

%t got[y_]:=Join@@Table[Table[i,{y[[i]]}],{i,Range[Length[y]]}];

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

%t Table[Length[exprUsing[got[Reverse[primeMS[n]]]]],{n,40}]

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

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

%K nonn

%O 1,4

%A _Gus Wiseman_, Aug 03 2018