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%I #39 Jul 21 2019 08:17:17
%S 1,1,1,2,1,3,2,6,6,4,1,12,3,8,10,24,2,30,6,20,20,5,6,60,20,15,90,40,4,
%T 60,1,120,15,10,40,180,12,30,45,120,3,120,8,30,210,36,10,360,80,140,
%U 30,90,24,630,35,240,90,24,2,420,30,6,420,720,105,105,6,60,126,280,20,1260
%N Number of ways to take apart the rooted tree corresponding to the Matula-Goebel number n by sequentially removing terminal edges.
%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 Number of ways to label the vertices of the rooted tree corresponding to the Matula-Goebel number n with the elements of {1,2,...,n} so that the label of each vertex is less than that of its descendants. Example: a(8)=6 because the rooted tree with Matula-Goebel number 8 is the star \|/; the root has label 1 and the 3 leaves are labeled with any permutation of {2,3,4}. See the Knuth reference, p. 67, Exercise 20. There is a simple bijection between the ways of the described labeling of a rooted tree and the ways of taking it apart by sequentially removing terminal edges: remove the edges in the inverse order of the labeling.
%D D. E. Knuth, The Art of Computer Programming, Vol.3, 2nd edition, Addison-Wesley, Reading, MA, 1998.
%H E. Deutsch, <a href="http://arxiv.org/abs/1111.4288">Tree statistics from Matula numbers</a>, arXiv:1111.4288 [math.CO], 2011.
%H J. Fulman, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v16i1r139">Mixing time for a random walk on rooted trees</a>, The Electronic J. of Combinatorics, 16, 2009, R139.
%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 B. E. Sagan and Y.-N. Yeh, <a href="https://www.fq.math.ca/Scanned/27-3/sagan.pdf">Probabilistic algorithms for trees</a>, The Fibonacci Quarterly, 27, 1989, 201-208. [_Emeric Deutsch_, Apr 28 2015]
%H <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F a(prime(m)) = a(m); a(r*s) = a(r)*a(s)*binomial(E(r*s),E(r)), where E(k) is the number of edges of the rooted tree with Matula-Goebel number k. The Maple program is based on these recurrence relations.
%F a(n) = V(n)!/TF(n), where V denotes "number of vertices" (A061775) and TF denotes "tree factorial" (A206493) (see Eq. (3) in the Fulman reference).
%e a(7)=2 because the rooted tree with Matula-Goebel number 7 is Y; denoting the edges in preorder by 1,2,3, it can be taken apart either in the order 231 or 321. a(11) = 1 because the rooted tree with Matula-Goebel number 11 is the path tree with 5 vertices; any path tree can be taken apart in only one way.
%p with(numtheory): E := 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 bigomega(n) = 1 then 1+E(pi(n)) else E(r(n))+E(s(n)) end if end proc: 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 1 elif bigomega(n) = 1 then a(pi(n)) else a(r(n))*a(s(n))*binomial(E(r(n))+E(s(n)), E(r(n))) end if end proc: seq(a(n), n = 1 .. 72);
%Y Cf. A061775, A206493.
%K nonn
%O 1,4
%A _Emeric Deutsch_, May 10 2012
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