%I #8 Sep 11 2018 21:14:29

%S 1,2,3,4,5,6,8,9,16,7,10,12,13,21,25,27,32,36,64,81,128,256,11,14,17,

%T 18,28,33,35,41,45,49,75,93,100,125,144,145,169,216,243,279,441,512,

%U 625,729,1024,1296,2048,2187,4096,6561,8192,16384,65536,524288,8388608,9007199254740992

%N Triangle whose n-th row lists in order all e-numbers of free pure symmetric multifunctions (with empty expressions allowed) with one atom and n positions.

%C Given a positive integer n we construct a unique free pure symmetric multifunction e(n) by expressing n as a power of a number that is not a perfect power to a product of prime numbers: n = rad(x)^(prime(y_1) * ... * prime(y_k)) where rad = A007916. Then e(n) = e(x)[e(y_1), ..., e(y_k)].

%C Every free pure symmetric multifunction (with empty expressions allowed) f with one atom and n positions has a unique e-number n such that e(n) = f, and vice versa, so this sequence is a permutation of the positive integers.

%H Mathematica Reference, <a href="http://reference.wolfram.com/mathematica/ref/Orderless.html">Orderless</a>

%e Triangle begins:

%e 1

%e 2

%e 3 4

%e 5 6 8 9 16

%e 7 10 12 13 21 25 27 32 36 64 81 128 256

%e Corresponding triangle of free pure symmetric multifunctions (with empty expressions allowed) begins:

%e o,

%e o[],

%e o[][], o[o],

%e o[][][], o[o][], o[o[]], o[][o], o[o,o].

%t maxUsing[n_]:=If[n==1,{"o"},Join@@Cases[Table[PR[k,n-k-1],{k,n-1}],PR[h_,g_]:>Join@@Table[Apply@@@Tuples[{maxUsing[h],Union[Sort/@Tuples[maxUsing/@p]]}],{p,IntegerPartitions[g]}]]];

%t radQ[n_]:=And[n>1,GCD@@FactorInteger[n][[All,2]]==1];

%t Clear[rad];rad[n_]:=rad[n]=If[n==0,1,NestWhile[#+1&,rad[n-1]+1,Not[radQ[#]]&]];

%t ungo[x_?AtomQ]:=1;ungo[h_[g___]]:=rad[ungo[h]]^(Times@@Prime/@ungo/@{g});

%t Table[Sort[ungo/@maxUsing[n]],{n,5}]

%Y Row lengths are A277996.

%Y Cf. A007916, A052409, A052410, A052893, A053492, A215366, A279944, A280000, A299759, A317658, A317659, A317677.

%K nonn,tabf

%O 1,2

%A _Gus Wiseman_, Aug 03 2018