A317880 Number of series-reduced free pure symmetric identity multifunctions (with empty expressions allowed) with one atom and n positions.

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

Offset: 1

Author

Gus Wiseman, Aug 09 2018

Keywords

• nonn

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.

Examples

			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[][][][][][]
		

Links

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

Crossrefs

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

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

Programs

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}]

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

Extensions

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