cp's OEIS Frontend

A212630 Irregular triangle read by rows: T(n,k) is the number of dominating subsets with k vertices of the rooted tree with Matula-Goebel number n (n>=1, k>=1).

%I A212630 #26 Jun 19 2024 09:32:34
%S A212630 1,2,1,1,3,1,1,3,1,0,4,4,1,0,4,4,1,1,3,4,1,1,3,4,1,0,3,8,5,1,0,3,8,5,
%T A212630 1,0,3,8,5,1,0,2,7,5,1,0,2,7,5,1,0,2,7,5,1,0,1,10,13,6,1,1,4,6,5,1,0,
%U A212630 2,7,5,1,0,0,8,12,6,1,1,4,6,5,1,0,2,8,12,6
%N A212630 Irregular triangle read by rows: T(n,k) is the number of dominating subsets with k vertices of the rooted tree with Matula-Goebel number n (n>=1, k>=1).
%C A212630 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 A212630 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).
%C A212630 Sum of entries in row n = A212631(n) (number of dominating subsets).
%C A212630 The order of the first nonzero entry in row n = A212632(n) (the domination number).
%H A212630 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 A212630 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 A212630 Emeric Deutsch, <a href="http://arxiv.org/abs/1111.4288">Rooted tree statistics from Matula numbers</a>, arXiv:1111.4288 [math.CO], 2011.
%H A212630 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 A212630 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 A212630 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 A212630 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 A212630 <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F A212630 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.
%e A212630 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].
%e A212630 Triangle begins:
%e A212630   1;
%e A212630   2,1;
%e A212630   1,3,1;
%e A212630   1,3,1;
%e A212630   0,4,4,1;
%e A212630   0,4,4,1;
%e A212630   1,3,4,1;
%e A212630   ...
%p A212630 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
%t A212630 r[n_] := FactorInteger[n][[1, 1]];
%t A212630 s[n_] := n/r[n];
%t A212630 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 A212630 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 A212630 c[n_] := Which[n == 1, 1, PrimeOmega[n] == 1, B[PrimePi[n]], True,  Expand[c[r[n]]*c[s[n]]]];
%t A212630 T[n_] := Rest@CoefficientList[A[n] + B[n], x];
%t A212630 Table[T[n], {n, 1, 20}] // Flatten (* _Jean-François Alcover_, Jun 19 2024, after Maple code *)
%Y A212630 Cf. A212618 - A212632.
%K A212630 nonn,tabf
%O A212630 1,2
%A A212630 _Emeric Deutsch_, Jun 11 2012