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%I #4 Mar 26 2014 15:43:55
%S 1,2,2,2,1,2,1,3,3,2,2,2,2,2,2,1,2,1,1,2,1,1,2,1,2,3,4,1,2,1,2,1,2,4,
%T 1,1,2,1,1,1,2,1,2,3,2,1,2
%N Row n contains the numbers obtained from the rooted tree with Matula number n in the following manner: partition the edges of the tree into subsets so that in each subset each edge has the same pair of endpoint degrees; row n contains the sizes of these subsets, listed in an arbitrary order.
%C Very often the terms of the sequence are denoted by m_{i,j}; see, for example, the Vukicevic reference.
%C The Maple program yields a more precise information, For example, for n=5 it gives the polynomial P(5) = 2xy^2 + x^2 y^2. A term bx^i y^j in such a polynomial means that the tree has b edges with endpoint degrees i,j.
%C The Matula 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.
%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 D. Vukicevic, On the edge degrees of trees, Glasnik Matem., 44(64), 2009, 259-266.
%H E. Deutsch, <a href="http://arxiv.org/abs/1111.4288"> Rooted tree statistics from Matula numbers</a>, arXiv1111.4288.
%e Row 5 is 1,2. Indeed, the rooted tree with Matula number 5 is the path ABCD. The edge BC has endpoint degrees 2,2; both edges AB and CD have endpoint degrees 1,2.
%e Row 8 is 3. Indeed the rooted tree with Matula number 8 is the star with 4 vertices. Each of the 3 edges has endpoint degrees 1,3.
%e Row 987654321 is 4,7,3,1,4,5,2,2. Indeed, the Maple program yields P(987654321) = 4xy^2 + 7xy^3 + 3x^2y^3 + x^2y^2 + 4xy^5 + 5x^2y^5 + 2x^3y^5 + 2x^5y^5. The tree is shown in Fig. 2 of the Deutsch reference.
%p with(numtheory): P := proc (n) local DL, r, s: DL := proc (n) if n = 2 then [1] elif bigomega(n) = 1 then [1+bigomega(pi(n))] else [op(DL(op(1, factorset(n)))), op(DL(n/op(1, factorset(n))))] end if end proc: 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 = 2 then x*y elif bigomega(n) = 1 then P(pi(n))+x*y^(1+bigomega(pi(n)))-(sum(x^min(bigomega(pi(n)), DL(pi(n))[j])*y^max(bigomega(pi(n)), DL(pi(n))[j]), j = 1 .. bigomega(pi(n))))+sum(x^min(1+bigomega(pi(n)), DL(pi(n))[j])*y^max(1+bigomega(pi(n)), DL(pi(n))[j]), j = 1 .. bigomega(pi(n))) else P(r(n))+P(s(n))-(sum(x^min(bigomega(r(n)), DL(r(n))[j])*y^max(bigomega(r(n)), DL(r(n))[j]), j = 1 .. bigomega(r(n))))-(sum(x^min(bigomega(s(n)), DL(s(n))[j])*y^max(bigomega(s(n)), DL(s(n))[j]), j = 1 .. bigomega(s(n))))+sum(x^min(bigomega(n), DL(r(n))[j])*y^max(bigomega(n), DL(r(n))[j]), j = 1 .. bigomega(r(n)))+sum(x^min(bigomega(n), DL(s(n))[j])*y^max(bigomega(n), DL(s(n))[j]), j = 1 .. bigomega(s(n))) end if end proc:
%K nonn,tabf
%O 2,2
%A _Emeric Deutsch_, Mar 26 2014
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