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A206483
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The matching number of the rooted tree having Matula-Goebel number n.
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3
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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
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1,5
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COMMENTS
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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 matching-generating polynomial of the graph (see the MathWorld link).
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.
After activating the Maple program, which yields the sequence, the command m(n) will yield the matching-generating polynomial of the rooted tree corresponding to the Matula-Goebel number n.
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REFERENCES
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C. D. Godsil, Algebraic Combinatorics, Chapman & Hall, New York, 1993.
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LINKS
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FORMULA
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Define b(n) (c(n)) to be the generating polynomials of the matchings of the rooted tree with Matula-Goebel number n that contain (do not contain) the root, with respect to the size of the matching (a k-matching 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 t-th 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 matching-generating polynomial). The matching number is the degree of this polynomial.
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EXAMPLE
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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.
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MAPLE
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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);
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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