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%I #7 Nov 21 2022 09:47:56
%S 1,6,12,24,30,48,60,72,104,120,144,148,156,180,192,222,288,312,360,
%T 390,432,444,480,576,712,720,780,832,864,900,1080,1110,1248,1260,1296,
%U 1440,1560,1680,2136,2160,2262,2304,2340,2496,2520,2592,2738,2880,2886,3072
%N Sorted list of positions of first appearances in the sequence counting permutations of Matula-Goebel trees (A206487).
%C To get a permutation of a tree, we choose a permutation of the multiset of branches of each node.
%C 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.
%e The terms together with their corresponding trees begin:
%e 1: o
%e 6: (o(o))
%e 12: (oo(o))
%e 24: (ooo(o))
%e 30: (o(o)((o)))
%e 48: (oooo(o))
%e 60: (oo(o)((o)))
%e 72: (ooo(o)(o))
%e 104: (ooo(o(o)))
%e 120: (ooo(o)((o)))
%e 144: (oooo(o)(o))
%e 148: (oo(oo(o)))
%e 156: (oo(o)(o(o)))
%e 180: (oo(o)(o)((o)))
%e 192: (oooooo(o))
%e 222: (o(o)(oo(o)))
%e 288: (ooooo(o)(o))
%e 312: (ooo(o)(o(o)))
%t primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]
%t MGTree[n_Integer]:=If[n===1,{},MGTree/@primeMS[n]]
%t treeperms[t_]:=Times@@Cases[t,b:{__}:>Length[Permutations[b]],{0,Infinity}];
%t fir[q_]:=Select[Range[Length[q]],!MemberQ[Take[q,#-1],q[[#]]]&];
%t fir[Table[treeperms[MGTree[n]],{n,100}]]
%Y Positions of first appearances in A206487.
%Y The unsorted version is A358508.
%Y A000081 counts rooted trees, ordered A000108.
%Y A214577 and A358377 rank trees with no permutations.
%Y Cf. A001263, A032027, A061775, A127301, A196050, A358378, A358506, A358521, A358522.
%K nonn
%O 1,2
%A _Gus Wiseman_, Nov 20 2022
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