OFFSET
1,4
COMMENTS
A vertex subset in a tree is said to be independent if no pair of vertices is connected by an edge. The empty set is considered to be independent.
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.
Sum of entries in row n = A184165(n) = number of independent vertex subset (the Merrifield-Simmons index).
Sum(k*T(n,k), k>=0) = A212624(n) = number of vertices in all independent vertex subsets.
Number of entries in row n = 1 + number of vertices in the largest independent vertex subset = 1 + A212625(n).
Last entry in row n = A212626(n) = number of largest independent vertex subsets.
With the given Maple program, the command P(n) yields the generating polynomial of row n.
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.
H. Prodinger and R. F. Tichy, Fibonacci numbers of graphs, Fibonacci Quarterly, 20, 1982, 16-21.
FORMULA
Define R(n) =R(n,x) (S(n)=S(n,x)) the generating polynomial of the independent vertex subsets that contain (do not contain) the root of the rooted tree with Matula-Goebel number n. Then R(1)=x, S(1)=1, R(the t-th prime) = x*S(t), S(the t-th prime) = R(t) + S(t); R(rs) = R(r)R(s)/x, S(rs) = S(r)S(s), (r,s>=2).
EXAMPLE
Row 5 is [1,4,3] because the rooted tree with Matula-Goebel number 5 is the path tree R - A - B - C with independent vertex subsets: {}, {R}, {A}, {B}, {C}, {R,B}, {R,C}, {A,C}.
Triangle starts:
1, 1;
1, 2;
1, 3, 1;
1, 3, 1;
1, 4, 3;
...
MAPLE
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 [x, 1] elif bigomega(n) = 1 then [expand(x*A(pi(n))[2]), expand(A(pi(n))[1])+A(pi(n))[2]] else [sort(expand(A(r(n))[1]*A(s(n))[1]/x)), sort(expand(A(r(n))[2]*A(s(n))[2]))] end if end proc: P := proc (n) options operator, arrow: sort(A(n)[1]+A(n)[2]) end proc: for n to 35 do seq(coeff(P(n), x, k), k = 0 .. degree(P(n))) end do; % yields sequence in triangular form
MATHEMATICA
r[n_] := FactorInteger[n][[1, 1]];
s[n_] := n/r[n];
A [n_] := Which[n == 1, {x, 1}, PrimeOmega[n] == 1, {x*A[PrimePi[n]][[2]], A[PrimePi[n]][[1]] + A[PrimePi[n]][[2]]}, True, {A[r[n]][[1]]*A[s[n]][[1]]/x, A[r[n]][[2]]*A[s[n]][[2]]}];
P[n_] := A[n] // Total;
T[n_] := CoefficientList[P[n], x];
Table[T[n], {n, 1, 35}] // Flatten (* Jean-François Alcover, Jun 20 2024, after Maple code *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Jun 01 2012
STATUS
approved