cp's OEIS Frontend

A317652 Number of free pure symmetric multifunctions whose leaves are an integer partition of n.

%I A317652 #9 Aug 28 2018 20:20:43
%S A317652 1,1,2,6,22,93,421,2010,9926,50357,260728,1372436,7321982,39504181,
%T A317652 215168221,1181540841,6534058589,36357935615,203414689462,
%U A317652 1143589234086,6457159029573,36602333187792,208214459462774,1188252476400972,6801133579291811,39032172166792887
%N A317652 Number of free pure symmetric multifunctions whose leaves are an integer partition of n.
%C A317652 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.
%H A317652 Andrew Howroyd, <a href="/A317652/b317652.txt">Table of n, a(n) for n = 0..200</a>
%e A317652 The a(4) = 22 free pure symmetric multifunctions:
%e A317652   1[1[1[1]]]  1[1[2]]  1[3]  2[2]  4
%e A317652   1[1[1][1]]  1[2[1]]  3[1]
%e A317652   1[1][1[1]]  2[1[1]]
%e A317652   1[1[1]][1]  1[1][2]
%e A317652   1[1][1][1]  1[2][1]
%e A317652   1[1[1,1]]   2[1][1]
%e A317652   1[1,1[1]]   1[1,2]
%e A317652   1[1][1,1]   2[1,1]
%e A317652   1[1,1][1]
%e A317652   1[1,1,1]
%t A317652 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A317652 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A317652 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 A317652 Table[Sum[Length[exprUsing[y]],{y,IntegerPartitions[n]}],{n,0,6}]
%o A317652 (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
%o A317652 seq(n)={my(v=[]); for(n=1, n, my(t=EulerT(v)); v=concat(v, 1 + sum(k=1, n-1, v[k]*t[n-k]))); concat([1],v)} \\ _Andrew Howroyd_, Aug 28 2018
%Y A317652 Cf. A001003, A052893, A053492, A277996, A279944, A280000.
%Y A317652 Cf. A317653, A317654, A317655, A317656, A317658.
%K A317652 nonn
%O A317652 0,3
%A A317652 _Gus Wiseman_, Aug 03 2018
%E A317652 Terms a(12) and beyond from _Andrew Howroyd_, Aug 28 2018