OFFSET
1,9
COMMENTS
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.
A. Ilic and M. Ilic, Generalizations of Wiener polarity index and terminal Wiener index, arXiv:11106.2986 [math.CO], 2011-2012.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
FORMULA
We give recurrence formulas for the more general case of vertices of degree k (k>=2). Let bigomega(n) denote the number of prime divisors of n, counted with multiplicities. Let g(n)=g(n,k,x) be the generating polynomial of the vertices of degree k of the rooted tree with Matula-Goebel number n with respect to level. We have a(1) = 0; if n = prime(t) and bigomega(t)=k-1 then a(n) = a(t)+[dg(t)/dx]_{x=1}; if n = prime(t) and bigomega(t)=k, then a(n) = a(t)-[dg(t)/dx]_{x=1}; if n = prime(t) and bigomega(t) =/ k and =/ k-1, then a(n)= a(t); if n = r*s with r prime, s>=2, bigomega(s)=k-1, then a(n) = a(r) + a(s) + [d[g(r)g(s)]/dx]_{x=1} +[dg(r)/dx]_{x=1} +[dg(s)/dx]_{x=1}; if n = r*s with r prime, s>=2, bigomega(s) =k, then a(n) = a(r) + a(s) + [d[g(r)g(s)]/dx]_{x=1} - [dg(r)/dx]_{x=1} - [dg(s)/dx]_{x=1}; if n = r*s with r prime, s>=2, bigomega(s) != k-1 and != k, then a(n) = a(r) + a(s) + [d[g(r)g(s)]/dx]_{x=1}.
EXAMPLE
a(11)=4 because the rooted tree with Matula-Goebel number 11 is the path tree A - B - C - D - E; the vertices of degree 2 are B, C, and D; we have dist(B,C)+dist(B,D)+dist(C,D) = 1+2+1 = 4.
a(987654321) = 68, as given by the Maple program; the reader can verify this on the rooted tree of Fig. 2 of the Deutsch reference.
MAPLE
k := 2: 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 bigomega(n) = 1 and bigomega(pi(n)) = k-1 then sort(expand(x+x*g(pi(n)))) elif bigomega(n) = 1 and bigomega(pi(n)) = k then sort(expand(-x+x*g(pi(n)))) elif bigomega(n) = 1 and bigomega(pi(n)) <> k-1 and bigomega(pi(n)) <> k then sort(expand(x*g(pi(n)))) elif bigomega(s(n)) = k-1 then sort(expand(1+g(r(n))+g(s(n)))) elif bigomega(s(n)) = k then sort(expand(-1+g(r(n))+g(s(n)))) else sort(g(r(n))+g(s(n))) end if end proc; with(numtheory): a := 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 bigomega(n) = 1 and bigomega(pi(n)) = k-1 then a(pi(n))+subs(x = 1, diff(g(pi(n)), x)) elif bigomega(n) = 1 and bigomega(pi(n)) = k then a(pi(n))-subs(x = 1, diff(g(pi(n)), x)) elif bigomega(n) = 1 and bigomega(pi(n)) <> k and bigomega(pi(n)) <> k-1 then a(pi(n)) elif bigomega(s(n)) = k-1 then a(r(n))+a(s(n))+subs(x = 1, diff(g(r(n))*g(s(n)), x))+subs(x = 1, diff(g(r(n)), x))+subs(x = 1, diff(g(s(n)), x)) elif bigomega(s(n)) = k then a(r(n))+a(s(n))+subs(x = 1, diff(g(r(n))*g(s(n)), x))-subs(x = 1, diff(g(r(n)), x))-subs(x = 1, diff(g(s(n)), x)) else a(r(n))+a(s(n))+subs(x = 1, diff(g(r(n))*g(s(n)), x)) end if end proc: seq(a(n), n = 1 .. 120);
MATHEMATICA
k = 2;
r[n_] := FactorInteger[n][[1, 1]];
s[n_] := n/r[n];
g[n_] := Which[n == 1, 0,
PrimeOmega[n] == 1 && PrimeOmega[PrimePi[n]] == k - 1,
Expand[x + x*g[PrimePi[n]]] ,
PrimeOmega[n] == 1 && PrimeOmega[PrimePi[n]] == k,
Expand[-x + x*g[PrimePi[n]]],
PrimeOmega[n] == 1 && PrimeOmega[PrimePi[n]] != k - 1 &&
PrimeOmega[PrimePi[n]] != k, Expand[x*g[PrimePi[n]]],
PrimeOmega[s[n]] == k - 1, Expand[1 + g[r[n]] + g[s[n]]],
PrimeOmega[s[n]] == k, Expand[-1 + g[r[n]] + g[s[n]]], True,
g[r[n]] + g[s[n]]];
a[n_] := Which[n == 1, 0,
PrimeOmega[n] == 1 && PrimeOmega[PrimePi[n]] == k - 1,
a[PrimePi[n]] + (D[g[PrimePi[n]], x] /. x -> 1),
PrimeOmega[n] == 1 && PrimeOmega[PrimePi[n]] == k,
a[PrimePi[n]] - (D[g[PrimePi[n]], x] /. x -> 1),
PrimeOmega[n] == 1 && PrimeOmega[PrimePi[n]] != k &&
PrimeOmega[PrimePi[n]] != k - 1, a[PrimePi[n]],
PrimeOmega[s[n]] == k - 1,
a[r[n]] + a[s[n]] + (D[g[r[n]]*g[s[n]], x] /. x -> 1) +
(D[g[r[n]], x] /. x -> 1) + (D[g[s[n]], x] /. x -> 1) ,
PrimeOmega[s[n]] == k,
a[r[n]] + a[s[n]] + (D[g[r[n]]*g[s[n]], x] /. x -> 1) -
(D[g[r[n]], x] /. x -> 1) - (D[g[s[n]], x] /. x -> 1), True,
a[r[n]] + a[s[n]] + (D[g[r[n]]*g[s[n]], x] /. x -> 1) ];
Table[a[n], {n, 1, 120}] (* Jean-François Alcover, Jun 19 2024, after Maple code *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, May 22 2012
STATUS
approved