OFFSET
1,7
COMMENTS
The escape distance of a vertex v in a rooted tree T is the distance from v to the nearest leaf of T that is a descendant of v. For the rooted tree ARBCDEF, rooted at R, the escape distance of B is 4 (the leaf A is not a descendant of B).
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.
Each row is nonincreasing (each vertex with escape distance k (k>=1) is the parent of some vertex with escape distance k-1).
REFERENCES
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 Review, 10, 1968, 273.
LINKS
Emeric Deutsch, Tree statistics from Matula numbers, arXiv preprint arXiv:1111.4288 [math.CO], 2011.
FORMULA
We give the recursive construction of the row generating polynomials P(n)=P(n,x): if n = prime(t), then P(n)=P(t)+x^{1+LLL(t)}; if n=r*s (r,s>=2), then P(n)=P(r)+P(s)-x^{max(LLL(r),LLL(s))}; LLL denotes the level of the lowest leaf (computed recursively and programmed in A184166) (2nd Maple program yields P(n)).
EXAMPLE
Row n=7 is [2,1,1] because the rooted tree with Matula-Goebel number 7 is the rooted tree Y, having 2 leaves and 1 (1) vertex at distance 1 (2) from either of the leaves.
Triangle starts:
1;
1, 1;
1, 1, 1;
2, 1;
1, 1, 1, 1;
2, 2;
2, 1, 1;
...
MAPLE
with(numtheory): P := proc (n) local r, s, LLL: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: LLL := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then 1+LLL(pi(n)) else min(LLL(r(n)), LLL(s(n))) end if end proc: if n = 1 then 1 elif bigomega(n) = 1 then P(pi(n))+x^(1+LLL(pi(n))) else P(r(n))+P(s(n))-x^max(LLL(r(n)), LLL(s(n))) end if end proc: for n to 30 do seq(coeff(P(n), x, k), k = 0 .. degree(P(n))) end do; # yields sequence in triangular form
with(numtheory): P := proc (n) local r, s, LLL: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: LLL := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then 1+LLL(pi(n)) else min(LLL(r(n)), LLL(s(n))) end if end proc: if n = 1 then 1 elif bigomega(n) = 1 then P(pi(n))+x^(1+LLL(pi(n))) else P(r(n))+P(s(n))-x^max(LLL(r(n)), LLL(s(n))) end if end proc: P(998877665544);
MATHEMATICA
r[n_] := FactorInteger[n][[1, 1]];
s[n_] := n/r[n];
LLL[n_] := Which[n == 1, 0, PrimeOmega[n] == 1, 1 + LLL[PrimePi[n]], True, Min[LLL[r[n]], LLL[s[n]]]];
P[n_] := Which[n == 1, 1, PrimeOmega[n] == 1, P[PrimePi[n]] + x^(1 + LLL[PrimePi[n]]), True, P[r[n]] + P[s[n]] - x^Max[LLL[r[n]], LLL[s[n]]]];
T[n_] := CoefficientList[P[n], x];
Table[T[n], {n, 1, 40}] // Flatten (* Jean-François Alcover, Jun 24 2024, after Maple code *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Oct 23 2011
STATUS
approved