login
Number of dominating subsets of the rooted tree with Matula-Goebel number n.
13

%I #26 Jun 19 2024 09:37:18

%S 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,

%T 27,53,25,29,51,31,33,57,29,53,45,27,27,51,53,27,45,25,53,97,53,51,51,

%U 49,97,53,45,33,81,105,45,53,51,29,87,45,57,89,65,93,93,27

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

%C 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.

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

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

%D 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.

%H S. Alikhani and Y. H. Peng, <a href="http://arxiv.org/abs/0905.2251">Introduction to domination polynomial of a graph</a>, arXiv:0905.2251 [math.CO], 2009.

%H É. Czabarka, L. Székely, and S. Wagner, <a href="http://dx.doi.org/10.1016/j.dam.2009.07.004">The inverse problem for certain tree parameters</a>, Discrete Appl. Math., 157, 2009, 3314-3319.

%H Emeric Deutsch, <a href="http://arxiv.org/abs/1111.4288">Rooted tree statistics from Matula numbers</a>, arXiv:1111.4288 [math.CO], 2011.

%H F. Goebel, <a href="http://dx.doi.org/10.1016/0095-8956(80)90049-0">On a 1-1-correspondence between rooted trees and natural numbers</a>, J. Combin. Theory, B 29 (1980), 141-143.

%H I. Gutman and A. Ivic, <a href="http://dx.doi.org/10.1016/0012-365X(95)00182-V">On Matula numbers</a>, Discrete Math., 150, 1996, 131-142.

%H I. Gutman and Yeong-Nan Yeh, <a href="http://www.emis.de/journals/PIMB/067/3.html">Deducing properties of trees from their Matula numbers</a>, Publ. Inst. Math., 53 (67), 1993, 17-22.

%H D. W. Matula, <a href="http://www.jstor.org/stable/2027327">A natural rooted tree enumeration by prime factorization</a>, SIAM Rev. 10 (1968) 273.

%H <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>

%F 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).

%e 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}.

%p 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);

%t r[n_] := FactorInteger[n][[1, 1]];

%t s[n_] := n/r[n];

%t 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];

%t 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]]]];

%t c[n_] := Which[n == 1, 1, PrimeOmega[n] == 1, B[PrimePi[n]], True, Expand[c[r[n]]*c[s[n]]]];

%t a[n_] := A[n] + B[n] /. x -> 1;

%t Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, Jun 19 2024, after Maple code *)

%Y Cf. A212618 - A212632.

%K nonn

%O 1,2

%A _Emeric Deutsch_, Jun 11 2012