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A193405 The Matula numbers of the rooted trees that have a perfect matching. 3
2, 5, 6, 15, 18, 22, 23, 26, 31, 41, 45, 54, 55, 65, 66, 69, 78, 93, 94, 103, 122, 123, 135, 137, 158, 162, 165, 166, 167, 195, 198, 202, 207, 211, 234, 235, 242, 253, 254, 279, 282, 283, 286, 299, 305, 309, 338, 341, 358, 366, 369, 394, 395, 401, 403, 405, 411, 415, 419, 431, 451, 474, 486, 495, 498 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
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.
It is known that a tree has at most one perfect matching.
Complement of A193406.
REFERENCES
C. D. Godsil, Algebraic Combinatorics, Chapman & Hall, New York, 1993.
LINKS
É. Czabarka, L. Székely, and S. Wagner, The inverse problem for certain tree parameters, Discrete Appl. Math., 157, 2009, 3314-3319.
E. Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288 [math.CO], 2011.
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.
FORMULA
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. 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 (a modified matching polynomial). The tree has a perfect matching if and only if the degree of this polynomial is 1/2 of the number of vertices of the tree.
EXAMPLE
2,6,31 are in the sequence because they are the Matula numbers of the paths on 2,4,6 vertices, respectively.
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: PM := {}: for n to 500 do if N(n) = 2*degree(m(n)) then PM := `union`(PM, {n}) else end if end do: PM;
CROSSREFS
Sequence in context: A287203 A291211 A193402 * A037079 A101325 A042980
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Feb 12 2012
STATUS
approved

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Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)