A206494 Number of ways to take apart the rooted tree corresponding to the Matula-Goebel number n by sequentially removing terminal edges.

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

Offset: 1

Author

Emeric Deutsch, May 10 2012

Keywords

• nonn

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.

Examples

			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.
		

References

• D. E. Knuth, The Art of Computer Programming, Vol.3, 2nd edition, Addison-Wesley, Reading, MA, 1998.

Links

• Kevin Ryde, Table of n, a(n) for n = 1..10000
• Emeric 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.
• Kevin Ryde, PARI/GP Code
• B. E. Sagan and Y.-N. Yeh, Probabilistic algorithms for trees, The Fibonacci Quarterly, 27, 1989, 201-208. [_Emeric Deutsch_, Apr 28 2015]
• Index entries for sequences related to Matula-Goebel numbers

Crossrefs

Cf. A061775, A206493.

Programs

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);

Mathematica

r[n_] := FactorInteger[n][[1, 1]];
s[n_] := n/r[n];
e[n_] := e[n] = Which[n == 1, 0, PrimeOmega[n] == 1, 1+e[PrimePi[n]], True, e[r[n]] + e[s[n]]];
a[n_] := a[n] = Which[n == 1, 1, PrimeOmega[n] == 1, a[PrimePi[n]], True, a[r[n]]*a[s[n]]*Binomial[e[r[n]] + e[s[n]], e[r[n]]] ];
Table[a[n], {n, 1, 72}] (* Jean-François Alcover, Aug 06 2024, after Maple program, replacing E(n) with e[n] *)

PARI

\\ See links.

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).