A212631 Number of dominating subsets of the rooted tree with Matula-Goebel number n.

1, 3, 5, 5, 9, 9, 9, 9, 17, 17, 17, 15, 15, 15, 31, 17, 15, 27, 17, 29, 29, 31, 27, 27, 57, 27, 53, 25, 29, 51, 31, 33, 57, 29, 53, 45, 27, 27, 51, 53, 27, 45, 25, 53, 97, 53, 51, 51, 49, 97, 53, 45, 33, 81, 105, 45, 53, 51, 29, 87, 45, 57, 89, 65, 93, 93, 27

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.

a(n) = Sum(A212630(n,k), k>=1).

a(n) is odd (see the Brouwer-Csorba-Schrijver reference).

References

A. E. Brouwer, P. Csorba, and A. Schrijver, The number of dominating sets of a finite graph is odd. Preprint available on A. E. Brouwer's homepage.

Links

Table of n, a(n) for n=1..67.

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.

Index entries for sequences related to Matula-Goebel numbers

Formula

In A212630 one gives the domination polynomial P(n)=P(n,x) of the rooted tree with Matula-Goebel number n. We have a(n) = P(n,1).

Example

a(3)=5 because the rooted tree with Matula-Goebel number 3 is the path tree R - A - B; its dominating subsets are {A}, {R,A}, {R,B}, {A,B}, and {R,A,B}.

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: seq(subs(x = 1, P(n)), n = 1 .. 100);

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]]]];
a[n_] := A[n] + B[n] /. x -> 1;
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jun 19 2024, after Maple code *)

Crossrefs

Cf. A212618 - A212632.

Sequence in context: A348374 A029639 A087349 * A090792 A076877 A332775

Adjacent sequences: A212628 A212629 A212630 * A212632 A212633 A212634

Keyword

nonn

Author

Emeric Deutsch, Jun 11 2012

Status

approved