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%I #5 Jan 07 2013 13:05:56
%S 0,1,1,1,2,1,1,1,0,2,1,3,2,2,0,1,1,1,1,0,0,0,0,4,2,1,1,0,3,1,0,0,0,2,
%T 2,1,0,2,2,2,0,3,3,0,0,0,1,1,1,1,1,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U 4,2,0,0,0,4,1,0,0,0,0,0,2,1,1,1,0,0,0,0,6,0,0,3,1,1
%N Irregular triangle read by rows: if the rooted tree with Matula-Goebel number n is a generalized Bethe tree, then row n is the sequence of the associated partition numbers; otherwise, row n consists of a single 0.
%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 generalized Bethe tree is a rooted tree in which vertices at the same level have the same degree; they are called uniform trees in the Goldberg & Livshits reference.
%C There is a simple bijection between generalized Bethe trees with n edges (n>=1) and partitions of n in which each part is divisible by the next (the parts are given by the number of edges at the successive levels). We have the correspondences: number of edges --- sum of parts; root degree --- last part; number of leaves --- first part; height --- number of parts.
%D F. Goebel, On a 1-1 correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
%D I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
%D I. Gutman and Y-N. Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
%D D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.
%D E. Deutsch, Rooted tree statistics from Matula numbers, Discrete Applied Math., 160, 2012, 2314-2322.
%D O. Rojo, Spectra of weighted generalized Bethe trees joined at the root, Linear Algebra and its Appl., 428, 2008, 2961-2979.
%D M. K. Goldberg and E. M. Livshits, On minimal universal trees, Mathematical Notes of the Acad. of Sciences of the USSR, 4, 1968, 713-717 (translation from the Russian Mat. Zametki 4 1968 371-379).
%H E. Deutsch, <a href="http://arxiv.org/abs/1111.4288"> Rooted tree statistics from Matula numbers</a>, arXiv:1111.4288.
%F If n>=1 and the rooted tree with Matula-Goebel number n is not a generalized Bethe tree then we define Q(n)=0; otherwise let Q(n) be the polynomial (in x) whose coefficients are the parts of the partition associated to the generalized Bethe tree. We have Q(1)=0; Q(2)=1; if n = p(t) (=the t-th prime) and Q(t)=0, then Q(n)=0; if n=p(t) and Q(t) =/ 0, then Q(n)=1+xQ(t); if n=rs , r, s >=2, Q(r)=/0, Q(s)=/0, and Q(r)/Q(s) = const, then Q(n) = Q(r)+Q(s); otherwise, Q(n) = 0.
%F With the given Maple program we obtain, for example, Q(529) = 4x^2 + 4x + 2, showing that the corresponding rooted tree is a generalized Bethe tree; also Q(987654321)=0, showing that the corresponding rooted tree is not a generalized Bethe tree.
%e Row 7 is 2,1 because the rooted tree with Matula-Goebel number 7 is Y; it is a generalized Bethe tree with corresponding partition 2,1 (number of edges at the various levels).
%e Triangle starts:
%e 0;
%e 1;
%e 1,1;
%e 2;
%e 1,1,1;
%e 0;
%e 2,1;
%p with(numtheory): Q := 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 0 elif n = 2 then 1 elif bigomega(n) = 1 and Q(pi(n)) = 0 then 0 elif bigomega(n) = 1 then sort(expand(1+x*Q(pi(n)))) elif Q(r(n)) <> 0 and Q(s(n)) <> 0 and type(simplify(Q(r(n))/Q(s(n))), constant) = true then sort(Q(r(n))+Q(s(n))) else 0 end if end proc: for n to 20 do if Q(n) = 0 then print(0) else print(seq(coeff(Q(n), x, degree(Q(n))-j), j = 0 .. degree(Q(n)))) end if end do; # yields sequence in triangular form
%K nonn,tabf
%O 1,5
%A _Emeric Deutsch_, Aug 18 2012
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