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A202853
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Triangle read by rows: T(n,k) is the number of k-matchings of the rooted tree having Matula-Goebel number n (n>=1, k>=0).
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2
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1, 1, 1, 1, 2, 1, 2, 1, 3, 1, 1, 3, 1, 1, 3, 1, 3, 1, 4, 3, 1, 4, 3, 1, 4, 3, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 5, 6, 1, 1, 4, 1, 4, 2, 1, 5, 5, 1, 1, 4, 1, 5, 5, 1, 5, 5, 1, 5, 6, 1, 1, 5, 5, 1, 1, 5, 3, 1, 6, 10, 4, 1, 5, 5, 1, 1, 6, 9, 4, 1, 5, 4, 1, 5, 5, 1, 6, 9, 3, 1, 5, 6, 1, 1, 5, 1, 6, 10, 4, 1, 5, 5, 1, 6, 9, 2
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OFFSET
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1,5
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COMMENTS
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The entries in row n are the coefficients of the matching-generating polynomial of the rooted tree having Matula-Goebel number n (see the MathWorld link).
A k-matching in a graph is a set of k edges, no two of which have a vertex in common.
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, 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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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 Review, 10, 1968, 273.
É. Czabarka, L. Székely, and S. Wagner, The inverse problem for certain tree parameters, Discrete Appl. Math., 157, 2009, 3314-3319.
C. D. Godsil, Algebraic Combinatorics, Chapman & Hall, New York, 1993.
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LINKS
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Table of n, a(n) for n=1..106.
E. Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288.
Eric Weisstein's World of Mathematics, Matching-Generating Polynomial
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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). T(n,k) is the coefficient of x^k in the polynomial m(n). [The actual matching polynomial is obtained by the substitution x = -1/x^2, followed by multiplication by x^N(n), where N(n) is the number of vertices of the rooted tree.]
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EXAMPLE
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T(11,2)=3 because the rooted tree corresponding to n=11 is a path abcde on 5 vertices. We have three 2-matchings: (ab,cd), (ab,de), and (bc,de).
Triangle starts:
1;
1,1;
1,2;
1,2;
1,3,1;
1,3,1;
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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: for n to 35 do seq(coeff(m(n), x, j), j = 0 .. degree(m(n))) end do; # yields sequence in triangular form
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CROSSREFS
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Cf. A202854.
Sequence in context: A048220 A182593 A201167 * A228572 A334675 A078380
Adjacent sequences: A202850 A202851 A202852 * A202854 A202855 A202856
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KEYWORD
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nonn,tabf
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AUTHOR
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Emeric Deutsch, Feb 14 2012
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STATUS
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approved
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