

A206494


Number of ways to take apart the rooted tree corresponding to the MatulaGoebel 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
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OFFSET

1,4


COMMENTS

The MatulaGoebel number of a rooted tree can be defined in the following recursive manner: to the onevertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the tth prime number, where t is the MatulaGoebel 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 MatulaGoebel numbers of the m branches of T.
Number of ways to label the vertices of the rooted tree corresponding to the MatulaGoebel 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 MatulaGoebel 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

F. Goebel, On a 11 correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131142.
I. Gutman and YN. Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 1722.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.
J. Fulman, Mixing time for a random walk on rooted trees, The Electronic Journal of Combinatorics, 16, 2009, #R139.
D. E. Knuth, The Art of Computer Programming, Vol.3, 2nd edition, AddisonWesley, Reading, MA, 1998.
B. E. Sagan and Y.N. Yeh, Probabilistic algorithms for trees, The Fibonacci Quarterly, 27, 1989, 201208.  Emeric Deutsch, April 28, 2015.


LINKS

Table of n, a(n) for n=1..72.
E. Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288.
Index entries for sequences related to MatulaGoebel numbers


FORMULA

a(the mth prime) = a(m); a(r*s) = a(r)*a(s)*binom(E(r*s),E(r)), where E(k) is the number of edges of the rooted tree with MatulaGoebel 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 MatulaGoebel 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 MatulaGoebel 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: A164768 A006208 A026805 * A022477 A238944 A144238
Adjacent sequences: A206491 A206492 A206493 * A206495 A206496 A206497


KEYWORD

nonn


AUTHOR

Emeric Deutsch, May 10 2012


STATUS

approved



