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 A206494 Number of ways to take apart the rooted tree corresponding to the Matula-Goebel number n by sequentially removing terminal edges. 4
 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, 60, 1, 120, 15, 10, 40, 180, 12, 30, 45, 120, 3, 120, 8, 30, 210, 36, 10, 360, 80, 140, 30, 90, 24, 630, 35, 240, 90, 24, 2, 420, 30, 6, 420, 720, 105, 105, 6, 60, 126, 280, 20, 1260 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS 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. 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. REFERENCES D. E. Knuth, The Art of Computer Programming, Vol.3, 2nd edition, Addison-Wesley, Reading, MA, 1998. LINKS E. Deutsch, Tree statistics from Matula numbers, arXiv:1111.4288 [math.CO], 2011. J. Fulman, Mixing time for a random walk on rooted trees, The Electronic J. of Combinatorics, 16, 2009, R139. F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143. I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142. I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22. D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273. B. E. Sagan and Y.-N. Yeh, Probabilistic algorithms for trees, The Fibonacci Quarterly, 27, 1989, 201-208. [Emeric Deutsch, Apr 28 2015] FORMULA 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. 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). EXAMPLE 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. MAPLE 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); CROSSREFS Cf. A061775, A206493. Sequence in context: A006208 A026805 A335688 * A022477 A238944 A144238 Adjacent sequences: A206491 A206492 A206493 * A206495 A206496 A206497 KEYWORD nonn AUTHOR Emeric Deutsch, May 10 2012 STATUS approved

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