A214578 - OEIS

%I #13 Aug 03 2024 19:08:25

%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 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 Emeric Deutsch, <a href="http://dx.doi.org/10.1016/j.dam.2012.05.012">Rooted tree statistics from Matula numbers</a>, Discrete Appl. Math., 160, 2012, 2314-2322; <a href="http://arxiv.org/abs/1111.4288">arXiv preprint</a>.

%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.

%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;

%e   ...

%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

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

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

%t Q[n_Integer] := Cancel@ Together@ Simplify@ Which[n == 1, 0, n == 2, 1, PrimeOmega[n] == 1 && Q[PrimePi[n]] === 0, 0, PrimeOmega[n] == 1, 1 + x * Q[PrimePi[n]], Q[r[n]] =!= 0 && Q[s[n]] =!= 0 && FreeQ[Q[r[n]]/Q[s[n]], x], Q[r[n]] + Q[s[n]], True, 0];

%t Table[If[Q[n] =!= 0, Reverse@ CoefficientList[Q[n], x], {0}], {n, 1, 100}] // Flatten (* _Jean-François Alcover_, Aug 03 2024, after _Emeric Deutsch_ *)

%K nonn,tabf

%O 1,5

%A _Emeric Deutsch_, Aug 18 2012