A358507 Sorted list of positions of first appearances in the sequence counting permutations of Matula-Goebel trees (A206487).

1, 6, 12, 24, 30, 48, 60, 72, 104, 120, 144, 148, 156, 180, 192, 222, 288, 312, 360, 390, 432, 444, 480, 576, 712, 720, 780, 832, 864, 900, 1080, 1110, 1248, 1260, 1296, 1440, 1560, 1680, 2136, 2160, 2262, 2304, 2340, 2496, 2520, 2592, 2738, 2880, 2886, 3072

Offset: 1

Author

Gus Wiseman, Nov 20 2022

Keywords

• nonn

Comments

To get a permutation of a tree, we choose a permutation of the multiset of branches of each node.

The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of the branches of its root, which gives a bijective correspondence between positive integers and unlabeled rooted trees.

Examples

			The terms together with their corresponding trees begin:
    1: o
    6: (o(o))
   12: (oo(o))
   24: (ooo(o))
   30: (o(o)((o)))
   48: (oooo(o))
   60: (oo(o)((o)))
   72: (ooo(o)(o))
  104: (ooo(o(o)))
  120: (ooo(o)((o)))
  144: (oooo(o)(o))
  148: (oo(oo(o)))
  156: (oo(o)(o(o)))
  180: (oo(o)(o)((o)))
  192: (oooooo(o))
  222: (o(o)(oo(o)))
  288: (ooooo(o)(o))
  312: (ooo(o)(o(o)))
		

Crossrefs

Positions of first appearances in A206487.

The unsorted version is A358508.

A000081 counts rooted trees, ordered A000108.

A214577 and A358377 rank trees with no permutations.

Cf. A001263, A032027, A061775, A127301, A196050, A358378, A358506, A358521, A358522.

Programs

Mathematica

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]
MGTree[n_Integer]:=If[n===1,{},MGTree/@primeMS[n]]
treeperms[t_]:=Times@@Cases[t,b:{}:>Length[Permutations[b]],{0,Infinity}];
fir[q_]:=Select[Range[Length[q]],!MemberQ[Take[q,#-1],q[[#]]]&];
fir[Table[treeperms[MGTree[n]],{n,100}]]