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

1, 1, 2, 6, 22, 93, 421, 2010, 9926, 50357, 260728, 1372436, 7321982, 39504181, 215168221, 1181540841, 6534058589, 36357935615, 203414689462, 1143589234086, 6457159029573, 36602333187792, 208214459462774, 1188252476400972, 6801133579291811, 39032172166792887

Offset: 0

Author

Gus Wiseman, Aug 03 2018

Keywords

• nonn

Comments

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(4) = 22 free pure symmetric multifunctions:
  1[1[1[1]]]  1[1[2]]  1[3]  2[2]  4
  1[1[1][1]]  1[2[1]]  3[1]
  1[1][1[1]]  2[1[1]]
  1[1[1]][1]  1[1][2]
  1[1][1][1]  1[2][1]
  1[1[1,1]]   2[1][1]
  1[1,1[1]]   1[1,2]
  1[1][1,1]   2[1,1]
  1[1,1][1]
  1[1,1,1]
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..200

Crossrefs

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

Cf. A317653, A317654, A317655, 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]}]]]];
Table[Sum[Length[exprUsing[y]],{y,IntegerPartitions[n]}],{n,0,6}]

PARI

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
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

Extensions

Terms a(12) and beyond from Andrew Howroyd, Aug 28 2018