

A193404


Number of matchings (independent edge subsets) in the rooted tree with MatulaGoebel number n.


0



1, 2, 3, 3, 5, 5, 4, 4, 8, 8, 8, 7, 7, 7, 13, 5, 7, 12, 5, 11, 11, 13, 12, 9, 21, 12, 20, 10, 11, 19, 13, 6, 21, 11, 18, 16, 9, 9, 19, 14, 12, 17, 10, 18, 32, 20, 19, 11, 15, 30, 18, 17, 6, 28, 34, 13, 14, 19, 11, 25, 16, 21, 28, 7, 31, 31, 9, 15, 32, 27, 14
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OFFSET

1,2


COMMENTS

A matching in a graph is a set of edges, no two of which have a vertex in common. The empty set is considered to be a matching.
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.


REFERENCES

F. Goebel, On a 11correspondence 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 YeongNan 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.
É. Czabarka, L. Székely, and S. Wagner, The inverse problem for certain tree parameters, Discrete Appl. Math., 157, 2009, 33143319.


LINKS

Table of n, a(n) for n=1..71.
E. Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288.
Index entries for sequences related to MatulaGoebel numbers


FORMULA

Define b(n) (c(n)) to be the number of matchings of the rooted tree with MatulaGoebel number n that contain (do not contain) the root. We have the following recurrence for the pair A(n)=[b(n),c(n)]. A(1)=[0,1]; if n=p(t) (=the tth prime), then A(n)=[c(t),b(t)+c(t)]; if n=rs (r,s,>=2), then A(n)=[b(r)c(s)+c(r)b(s), c(r)c(s)]. Clearly, a(n)=b(n)+c(n). See the Czabarka et al. reference (p. 3315, (2)). The Maple program is based on this recursive formula.


EXAMPLE

a(3)=3 because the rooted tree with MatulaGoebel number 3 is the path ABC on 3 vertices; it has 3 matchings: empty, {AB}, {BC}.


MAPLE

with(numtheory): 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 [0, 1] elif bigomega(n) = 1 then [A(pi(n))[2], A(pi(n))[1]+A(pi(n))[2]] else [A(r(n))[1]*A(s(n))[2]+A(s(n))[1]*A(r(n))[2], A(r(n))[2]*A(s(n))[2]] end if end proc: a := proc (n) options operator, arrow: A(n)[1]+A(n)[2] end proc: seq(a(n), n = 1 .. 80);


CROSSREFS

Cf. A184165.
Sequence in context: A046146 A081768 A273493 * A072923 A257003 A131922
Adjacent sequences: A193401 A193402 A193403 * A193405 A193406 A193407


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Feb 11 2012


STATUS

approved



