A206487 Number of ordered trees isomorphic (as rooted trees) to the rooted tree having Matula number n.

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 2, 2, 2, 1, 1, 3, 1, 3, 2, 2, 1, 4, 1, 4, 1, 3, 2, 6, 1, 1, 2, 2, 2, 6, 3, 2, 4, 4, 2, 6, 2, 3, 3, 2, 2, 5, 1, 3, 2, 6, 1, 4, 2, 4, 2, 4, 1, 12, 3, 2, 3, 1, 4, 6, 1, 3, 2, 6, 3, 10, 2, 6, 3, 3, 2, 12, 2, 5, 1, 4, 1, 12, 2, 4, 4, 4, 4, 12, 4, 3, 2, 4, 2, 6, 1, 3, 3, 6, 4, 6, 1, 8, 6, 2, 3, 10, 2, 6, 6, 5, 6, 6, 2, 6, 6, 2, 2, 20, 1, 6, 4, 3, 1, 12, 1, 1, 4, 12, 1, 12, 2, 2, 4, 4, 2, 6, 2, 12, 4, 6, 4, 15, 4, 4, 3, 9, 2, 12, 6, 4, 3, 6, 2, 24, 3, 4, 2, 6

Offset: 1

Author

Emeric Deutsch, Apr 14 2012

Keywords

• nonn

Comments

The Matula-Goebel number of a rooted tree is defined in the following recursive manner: to the one-vertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the t-th prime number, where t is the Matula-Goebel number of the tree obtained from T by deleting the edge emanating from the root; to a tree T with root degree m>=2 there corresponds the product of the Matula-Goebel numbers of the m branches of T.

a(n) = the number of times n occurs in A127301. - Antti Karttunen, Jan 03 2013

Examples

			a(4)=1 because the rooted tree with Matula number 4 is V and there is no other ordered tree isomorphic to V. a(6)=2 because the rooted tree corresponding to n = 6 is obtained by joining the trees A - B and C - D - E at their roots A and C. Interchanging their order, we obtain another ordered tree, isomorphic (as rooted tree) to the first one.
		

Links

• E. Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288 [math.CO], 2011.
• F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
• I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
• I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
• D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
• P. Schultz, Enumeration of rooted trees with an application to group presentations, Discrete Math., 41, 1982, 199-214.
• Index entries for sequences related to Matula-Goebel numbers

Crossrefs

Cf. A127301.

Positions of 1's are 1 and A214577.

Positions of first appearances are A358507, unsorted A358508.

A000108 counts ordered rooted trees, unordered A000081.

A061775 and A196050 count nodes and edges in Matula-Goebel trees.

Cf. A001263, A003238, A014486, A358506.

Programs

Maple

with(numtheory): F := proc (n) options operator, arrow: factorset(n) end proc: PD := proc (n) local k, m, j: for k to nops(F(n)) do m[k] := 0: for j while is(n/F(n)[k]^j, integer) = true do m[k] := m[k]+1 end do end do: [seq([F(n)[q], m[q]], q = 1 .. nops(F(n)))] end proc: a := proc (n) if n = 1 then 1 elif bigomega(n) = 1 then a(pi(n)) else mul(a(PD(n)[j][1])^PD(n)[j][2], j = 1 .. nops(F(n)))*factorial(add(PD(n)[k][2], k = 1 .. nops(F(n))))/mul(factorial(PD(n)[k][2]), k = 1 .. nops(F(n))) end if end proc: seq(a(n), n = 1 .. 160);

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}];
Table[treeperms[MGTree[n]],{n,100}] (* Gus Wiseman, Nov 21 2022 *)

Formula

a(1)=1; denoting by p(t) the t-th prime, if n = p(n_1)^{k_1}...p(n_r)^{k_r}, then a(n) = a(n_1)^{k_1}...a(n_r)^{k_r}*(k_1 + ... + k_r)!/[(k_1)!...(k_r)!] (see Theorem 1 in the Schultz reference, where the exponents k_j of N(n_j) have been inadvertently omitted).