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

1,2

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.

Links

Table of n, a(n) for n=1..50.

Example

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)))

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}]]

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.

Sequence in context: A261476 A119840 A298033 * A358526 A069171 A071611

Adjacent sequences: A358504 A358505 A358506 * A358508 A358509 A358510

Keyword

nonn

Author

Gus Wiseman, Nov 20 2022

Status

approved