cp's OEIS Frontend

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

%I A317655 #8 Aug 03 2018 08:17:06
%S A317655 0,1,1,2,3,8,10,15,50,35,37,96,144,160,299,184,589,840,2483,578,1729,
%T A317655 750,10746,1627,2246,3578,9357,3367,47420,6397,212668,3155,9818,17280,
%U A317655 15666,18250,966324,84232,54990,12471,4439540,45015
%N A317655 Number of free pure symmetric multifunctions with leaves a multiset whose multiplicities are the integer partition with Heinz number n.
%C A317655 The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
%C A317655 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 A317655 The a(6) = 8 free pure symmetric multifunctions:
%e A317655   1[1[2]]
%e A317655   1[2[1]]
%e A317655   2[1[1]]
%e A317655   1[1][2]
%e A317655   1[2][1]
%e A317655   2[1][1]
%e A317655   1[1,2]
%e A317655   2[1,1]
%t A317655 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A317655 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A317655 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 A317655 got[y_]:=Join@@Table[Table[i,{y[[i]]}],{i,Range[Length[y]]}];
%t A317655 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A317655 Table[Length[exprUsing[got[Reverse[primeMS[n]]]]],{n,40}]
%Y A317655 Cf. A001003, A052893, A053492, A255906, A277996, A279944, A280000.
%Y A317655 Cf. A317652, A317653, A317654, A317656, A317658.
%K A317655 nonn
%O A317655 1,4
%A A317655 _Gus Wiseman_, Aug 03 2018