cp's OEIS Frontend

A317654 Number of free pure symmetric multifunctions whose leaves are a strongly normal multiset of size n.

%I A317654 #9 Jan 01 2021 18:12:50
%S A317654 1,3,26,375,6696,159837,4389226,144915350,5377002075,227624621051,
%T A317654 10632808475596,550932945236121,31062550998284221,1907051034025848314,
%U A317654 126052420069459211076,8956882232940915920404,679298518935625486287703,54868537321267493152151502,4696952405203792017289469056
%N A317654 Number of free pure symmetric multifunctions whose leaves are a strongly normal multiset of size n.
%C A317654 A multiset is strongly normal if it spans an initial interval of positive integers with weakly decreasing multiplicities. 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.
%e A317654 The a(3) = 26 free pure symmetric multifunctions:
%e A317654 1[1[1]], 1[1,1], 1[1][1],
%e A317654 1[1[2]], 1[2[1]], 1[1,2], 2[1[1]], 2[1,1], 1[1][2], 1[2][1], 2[1][1],
%e A317654 1[2[3]], 1[3[2]], 1[2,3], 2[1[3]], 2[3[1]], 2[1,3], 3[1[2]], 3[2[1]], 3[1,2], 1[2][3], 2[1][3], 1[3][2], 3[1][2], 2[3][1], 3[2][1].
%t A317654 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A317654 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A317654 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 A317654 got[y_]:=Join@@Table[Table[i,{y[[i]]}],{i,Range[Length[y]]}];
%t A317654 Table[Sum[Length[exprUsing[got[y]]],{y,IntegerPartitions[n]}],{n,6}]
%o A317654 (PARI) \\ See links in A339645 for combinatorial species functions.
%o A317654 cycleIndexSeries(n)={my(p=O(x)); for(n=1, n, p = x*sv(1) + p*(sExp(p)-1)); p}
%o A317654 StronglyNormalLabelingsSeq(cycleIndexSeries(15)) \\ _Andrew Howroyd_, Jan 01 2021
%Y A317654 Cf. A001003, A052893, A053492, A255906, A277996, A279944, A280000.
%Y A317654 Cf. A317652, A317653, A317655, A317656, A317658.
%K A317654 nonn
%O A317654 1,2
%A A317654 _Gus Wiseman_, Aug 03 2018
%E A317654 Terms a(8) and beyond from _Andrew Howroyd_, Jan 01 2021