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Irregular triangle read by rows: T(n,k) is the number of independent vertex subsets with k vertices of the rooted tree with Matula-Goebel number n (n>=1, k>=0).
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%I #26 Jun 20 2024 07:57:22

%S 1,1,1,2,1,3,1,1,3,1,1,4,3,1,4,3,1,4,3,1,1,4,3,1,1,5,6,1,1,5,6,1,1,5,

%T 6,1,1,5,6,2,1,5,6,2,1,5,6,2,1,6,10,4,1,5,6,4,1,1,5,6,2,1,6,10,5,1,5,

%U 6,4,1,1,6,10,5,1,1,6,10,5,1,1,6,10,4,1,6,10,5,1,6,10,7,2,1,7,15

%N Irregular triangle read by rows: T(n,k) is the number of independent vertex subsets with k vertices of the rooted tree with Matula-Goebel number n (n>=1, k>=0).

%C 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.

%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 Sum of entries in row n = A184165(n) = number of independent vertex subset (the Merrifield-Simmons index).

%C Sum(k*T(n,k), k>=0) = A212624(n) = number of vertices in all independent vertex subsets.

%C Number of entries in row n = 1 + number of vertices in the largest independent vertex subset = 1 + A212625(n).

%C Last entry in row n = A212626(n) = number of largest independent vertex subsets.

%C With the given Maple program, the command P(n) yields the generating polynomial of row n.

%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 H. Prodinger and R. F. Tichy, <a href="http://www.fq.math.ca/Scanned/20-1/prodinger.pdf">Fibonacci numbers of graphs</a>, Fibonacci Quarterly, 20, 1982, 16-21.

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

%F 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).

%e 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}.

%e Triangle starts:

%e 1, 1;

%e 1, 2;

%e 1, 3, 1;

%e 1, 3, 1;

%e 1, 4, 3;

%e ...

%p 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

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

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

%t 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]]}];

%t P[n_] := A[n] // Total;

%t T[n_] := CoefficientList[P[n], x];

%t Table[T[n], {n, 1, 35}] // Flatten (* _Jean-François Alcover_, Jun 20 2024, after Maple code *)

%Y Cf. A212618, A212619, A212620, A212621, A212622, A212624, A212625, A212626, A212627, A212628, A212629, A212630, A212631, A212632.

%K nonn,tabf

%O 1,4

%A _Emeric Deutsch_, Jun 01 2012