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A317880 Number of series-reduced free pure symmetric identity multifunctions (with empty expressions allowed) with one atom and n positions. 8
1, 1, 1, 1, 2, 4, 8, 16, 33, 70, 152, 333, 735, 1635, 3668, 8285, 18823, 42970, 98535, 226870, 524290, 1215641, 2827203, 6593432, 15416197, 36129894, 84860282, 199719932, 470930802, 1112388190, 2631903295, 6236669381, 14800078408, 35169529363, 83680908692 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

A series-reduced free pure symmetric identity multifunction (with empty expressions allowed) (SROI) is either (case 1) the leaf symbol "o", or (case 2) a possibly empty expression of the form h[g_1, ..., g_k] where h is an SROI, k is an integer greater than or equal to 0 but not equal to 1, each of the g_i for i = 1, ..., k is an SROI, and for i < j we have g_i < g_j under a canonical total ordering such as the Mathematica ordering of expressions. The number of positions in an SROI is the number of brackets [...] plus the number of o's.

Also the number of series-reduced orderless identity Mathematica expressions with one atom and n positions.

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..200

EXAMPLE

The a(7) = 8 SROIs:

  o[o,o[][][]]

  o[o[],o[][]]

  o[][o,o[][]]

  o[][][o,o[]]

  o[o,o[][]][]

  o[][o,o[]][]

  o[o,o[]][][]

  o[][][][][][]

MATHEMATICA

allIdExprSR[n_]:=If[n==1, {"o"}, Join@@Cases[Table[PR[k, n-k-1], {k, n-1}], PR[h_, g_]:>Join@@Table[Apply@@@Tuples[{allIdExprSR[h], Select[Union[Sort/@Tuples[allIdExprSR/@p]], UnsameQ@@#&]}], {p, If[g==0, {{}}, Rest[IntegerPartitions[g]]]}]]];

Table[Length[allIdExprSR[n]], {n, 12}]

PROG

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

seq(n)={my(v=[1]); for(n=2, n, my(t=WeighT(v)-v); v=concat(v, v[n-1] + sum(k=1, n-2, v[k]*t[n-k-1]))); v} \\ Andrew Howroyd, Aug 19 2018

CROSSREFS

Cf. A001003, A001678, A052893, A053492, A067824, A167865, A277996, A280000, A317875.

Cf. A317876, A317877, A317878, A317879, A317880, A317881.

Sequence in context: A177794 A004149 A129986 * A299271 A110334 A084636

Adjacent sequences:  A317877 A317878 A317879 * A317881 A317882 A317883

KEYWORD

nonn

AUTHOR

Gus Wiseman, Aug 09 2018

EXTENSIONS

Terms a(13) and beyond from Andrew Howroyd, Aug 19 2018

STATUS

approved

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Last modified June 21 04:10 EDT 2021. Contains 345354 sequences. (Running on oeis4.)