OFFSET
1,9
COMMENTS
A vertex v in a rooted tree is said to be a star-vertex if all the children of v are leaves.
The Matula-Goebel number of a rooted tree can be 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.
LINKS
Emeric 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.
FORMULA
Let G(n)=G(n;t,z) be the bivariate generating polynomial of the star-vertices of the rooted tree with Matula-Goebel number n with respect to number of children (marked by t) and level (marked by z). Then G(1)=0; G(2)=t; if n = m-th prime (m>=2), then G(n)=z*G(m); if n=rs (r,s>=2) and both r and s are powers of 2, then G(n)=G(r)*G(s) (=t^{log[2](n)}); if n=rs (r,s>=2) and r is a power of 2 while s is not, then G(n)=G(s); if n=rs (r,s>=2) and s is a power of 2 while r is not, then G(n)=G(r); if n=rs (r,s>=2) and r and s are not powers of 2, then G(n)=G(r)+G(s). a(n)=G(n;1,1). The Maple program is based on these recurrence relations. With the given Maple program, the command G(n) yields the bivariate generating polynomial.
EXAMPLE
a(9)=2 because the rooted tree with Matula-Goebel number 9 is the path A-B-R-C-D with root R; B and C are the star-vertices.
a(5)=1 because the rooted tree with Matula-Goebel number 5 is the path R-A-B-C with root R; B is the only star-vertex.
a(16)=1 because the rooted tree with Matula-Goebel number 16 is the star K_{1,4}; the root is the only star-vertex.
G(987654321) =2*t*z + 3*t^2*z^2 + t*z^2 + t*z^4 + t^4*z^2 (reader may verify this on Fig. 2 of the Deutsch paper).
MAPLE
with(numtheory: G := proc (n) local r, s: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: if n = 1 then 0 elif n = 2 then t elif bigomega(n) = 1 then z*G(pi(n)) elif factorset(n) = {2} then t^log[2](n) elif factorset(r(n)) = {2} and factorset(s(n)) <> {2} then G(s(n)) elif factorset(r(n)) <> {2} and factorset(s(n)) = {2} then G(r(n)) else expand(G(r(n))+G(s(n))) end if end proc: a := proc (n) options operator, arrow: subs({t = 1, z = 1}, G(n)) end proc: seq(a(n), n = 1 .. 200);
MATHEMATICA
r[n_] := FactorInteger[n][[1, 1]];
s[n_] := n/r[n];
G[n_] := Which[n == 1, 0, n == 2, t, PrimeOmega[n] == 1, z*G[PrimePi[n]], FactorInteger[n][[All, 1]] == {2}, t^Log[2, n], FactorInteger[r[n]][[All, 1]] == {2} && FactorInteger[s[n]][[All, 1]] != {2}, G[s[n]], FactorInteger[r[n]][[All, 1]] != {2} && FactorInteger[s[n]][[All, 1]] == {2}, G[r[n]], True, Expand[G[r[n]] + G[s[n]]]];
a[n_] := G[n] /. {t -> 1, z -> 1};
Table[a[n], {n, 1, 85}] (* Jean-François Alcover, Jun 18 2024, after Maple code *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Jul 23 2012
STATUS
approved