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A317652 Number of free pure symmetric multifunctions whose leaves are an integer partition of n. 9
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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.

LINKS

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

EXAMPLE

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]

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}]

PROG

(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

CROSSREFS

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

Cf. A317653, A317654, A317655, A317656, A317658.

Sequence in context: A229741 A261518 A185349 * A150274 A109317 A109153

Adjacent sequences:  A317649 A317650 A317651 * A317653 A317654 A317655

KEYWORD

nonn

AUTHOR

Gus Wiseman, Aug 03 2018

EXTENSIONS

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

STATUS

approved

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Last modified May 12 13:46 EDT 2021. Contains 343823 sequences. (Running on oeis4.)