

A206483


The matching number of the rooted tree having MatulaGoebel number n.


1



0, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 3, 1, 2, 3, 1, 2, 2, 3, 3, 2, 3, 3, 3, 2, 2, 3, 3, 1, 3, 2, 3, 3, 2, 2, 3, 2, 3, 3, 2, 3, 4, 3, 3, 2, 2, 3, 3, 3, 1, 4, 4, 2, 2, 3, 2, 3, 3, 3, 3, 1, 4, 4, 2, 2, 4, 3, 2, 3, 3, 3, 4, 2, 3, 4, 3, 2, 4, 3, 3, 3, 3, 3, 3, 3, 2, 4, 3, 3, 4, 4, 3, 2, 3, 3, 4, 3, 3, 3, 4, 3, 4, 2, 2, 4, 3, 4
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OFFSET

1,5


COMMENTS

A matching in a graph is a set of edges, no two of which have a vertex in common. The matching number of a graph is the maximum of the cardinalities of all the matchings in the graph. Consequently, the matching number of a graph is the degree of the matchinggenerating polynomial of the graph (see the MathWorld link).
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.
After activating the Maple program, which yields the sequence, the command m(n) will yield the matchinggenerating polynomial of the rooted tree corresponding to the MatulaGoebel number n.


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.
C. D. Godsil, Algebraic Combinatorics, Chapman & Hall, New York, 1993.


LINKS

Table of n, a(n) for n=1..110.
E. Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288.
Eric Weisstein's World of Mathematics, MatchingGenerating Polynomial
Index entries for sequences related to MatulaGoebel numbers


FORMULA

Define b(n) (c(n)) to be the generating polynomials of the matchings of the rooted tree with MatulaGoebel number n that contain (do not contain) the root, with respect to the size of the matching (a kmatching has size k). We have the following recurrence for the pair M(n)=[b(n),c(n)]. M(1)=[0,1]; if n=p(t) (=the tth prime), then M(n)=[xc(t),b(t)+c(t)]; if n=rs (r,s,>=2), then M(n)=[b(r)c(s)+c(r)b(s), c(r)c(s)]. Then m(n)=b(n)+c(n) is the generating polynomial of the matchings of the rooted tree with respect to the size of the matchings (called the matchinggenerating polynomial). The matching number is the degree of this polynomial.


EXAMPLE

a(11)=2 because the rooted tree corresponding to n=11 is a path abcde on 5 vertices. We have matchings with 2 edges (for example, (ab, cd)) but not with 3 edges.


MAPLE

with(numtheory): N := 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 1+N(pi(n)) else N(r(n))+N(s(n))1 end if end proc: M := 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 [x*M(pi(n))[2], M(pi(n))[1]+M(pi(n))[2]] else [M(r(n))[1]*M(s(n))[2]+M(r(n))[2]*M(s(n))[1], M(r(n))[2]*M(s(n))[2]] end if end proc: m := proc (n) options operator, arrow: sort(expand(M(n)[1]+M(n)[2])) end proc: seq(degree(m(n)), n = 1 .. 110);


CROSSREFS

Cf. A202853.
Sequence in context: A212632 A025885 A198337 * A087011 A294602 A000174
Adjacent sequences: A206480 A206481 A206482 * A206484 A206485 A206486


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Feb 14 2012


STATUS

approved



