OFFSET
1,2
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.
The entries in row n are the coefficients of the domination polynomial of the rooted tree with Matula-Goebel number n (see the Alikhani and Peng reference).
Sum of entries in row n = A212631(n) (number of dominating subsets).
The order of the first nonzero entry in row n = A212632(n) (the domination number).
LINKS
S. Alikhani and Y. H. Peng, Introduction to domination polynomial of a graph, arXiv:0905.2251 [math.CO], 2009.
É. Czabarka, L. Székely, and S. Wagner, The inverse problem for certain tree parameters, Discrete Appl. Math., 157, 2009, 3314-3319.
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 A(n)=A(n,x), B(n)=B(n,x), and C(n)=C(n,x) be the generating polynomial with respect to size of the dominating subsets which contain the root, of the dominating subsets which do not contain the root, and of the subsets which dominate all vertices except the root, respectively, of the rooted tree with Matula-Goebel number n. We have A(1)=x, B(1)=0, C(1)=1, A(t-th prime) = x [A(t)+B(t)+C(t)], B(t-th prime) = A(t), C(t-th prime) = B(t); A(rs) = A(r)A(s)/x, B(rs) = B(r)B(s) + B(r)C(s) + B(s)C(r) (r,s>=2). The generating polynomial of the dominating subsets with respect to size (i.e. the domination polynomial) is P(n)=P(n,x)=A(n)+B(n). The Maple program is based on these recurrence relations.
EXAMPLE
Row 3 is [1,3,1] because the rooted tree with Matula-Goebel number 3 is the path tree R - A - B; it has 1, 3, and 1 dominating subsets with 1, 2, and 3 vertices, respectively: [A], [RA, RB, AB], and [RAB].
Triangle begins:
1;
2,1;
1,3,1;
1,3,1;
0,4,4,1;
0,4,4,1;
1,3,4,1;
...
MAPLE
with(numtheory): P := proc (n) local r, s, A, B, C: r := n-> op(1, factorset(n)): s := n-> n/r(n): A := proc (n) if n = 1 then x elif bigomega(n) = 1 then x*(A(pi(n))+B(pi(n))+C(pi(n))) else A(r(n))*A(s(n))/x end if end proc: B := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then A(pi(n)) else sort(expand(B(r(n))*B(s(n))+B(r(n))*C(s(n))+B(s(n))*C(r(n)))) end if end proc: C := proc (n) if n = 1 then 1 elif bigomega(n) = 1 then B(pi(n)) else expand(C(r(n))*C(s(n))) end if end proc: sort(expand(A(n)+B(n))) end proc: for n to 20 do seq(coeff(P(n), x, j), j = 1 .. 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, PrimeOmega[n] == 1, x*(A[PrimePi[n]] + B[PrimePi[n]] + c[PrimePi[n]]), True, A[r[n]]*A[s[n]]/x];
B[n_] := Which[n == 1, 0, PrimeOmega[n] == 1, A[PrimePi[n]], True, Expand[B[r[n]]*B[s[n]] + B[r[n]]*c[s[n]] + B[s[n]]*c[r[n]]]];
c[n_] := Which[n == 1, 1, PrimeOmega[n] == 1, B[PrimePi[n]], True, Expand[c[r[n]]*c[s[n]]]];
T[n_] := Rest@CoefficientList[A[n] + B[n], x];
Table[T[n], {n, 1, 20}] // Flatten (* Jean-François Alcover, Jun 19 2024, after Maple code *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Jun 11 2012
STATUS
approved