proposed

approved

editing

proposed

Regular triangle where T(n,k) is the number of distinct orderless Mathematica free pure symmetric multifunctions (with empty expressions allowed) with one atom, n positions, and k leaves.

approved

editing

proposed

approved

editing

proposed

Cf. A317652, A317653, A317654, A317655, A317656, A317676.

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

allocated for Gus WisemanRegular triangle where T(n,k) is the number of distinct orderless Mathematica expressions with one atom, n positions, and k leaves.

1, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 5, 1, 0, 1, 10, 17, 7, 1, 0, 1, 15, 43, 33, 9, 1, 0, 1, 21, 92, 118, 55, 11, 1, 0, 1, 28, 174, 341, 252, 82, 13, 1, 0, 1, 36, 302, 845, 935, 463, 115, 15, 1, 0, 1, 45, 490, 1864, 2921, 2103, 769, 153, 17, 1, 0, 1, 55, 755

1,8

The T(5,3) = 5 expressions are o[o[o]], o[o,o[]], o[][o,o], o[o][o], o[o,o][].

Triangle begins:

1

1 0

1 1 0

1 3 1 0

1 6 5 1 0

1 10 17 7 1 0

1 15 43 33 9 1 0

1 21 92 118 55 11 1 0

1 28 174 341 252 82 13 1 0

1 36 302 845 935 463 115 15 1 0

1 45 490 1864 2921 2103 769 153 17 1 0

1 55 755 3755 7981 8012 4145 1187 197 19 1 0

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]}]]];

Table[Length[Select[maxUsing[n], Length[Position[#, "o"]]==k&]], {n, 12}, {k, n}]

Cf. A001003, A052893, A053492, A255906, A277996, A279944, A280000.

Cf. A317652, A317653, A317654, A317655, A317656.

allocated

nonn,tabl

Gus Wiseman, Aug 03 2018

approved

editing

allocated for Gus Wiseman

allocated

approved