

A202853


Triangle read by rows: T(n,k) is the number of kmatchings of the rooted tree having MatulaGoebel number n (n>=1, k>=0).


2



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

1,5


COMMENTS

The entries in row n are the coefficients of the matchinggenerating polynomial of the rooted tree having MatulaGoebel number n (see the MathWorld link).
A kmatching in a graph is a set of k edges, no two of which have a vertex in common.
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, 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..106.
E. Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288.
Eric Weisstein's World of Mathematics, MatchingGenerating Polynomial


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). 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.]


EXAMPLE

T(11,2)=3 because the rooted tree corresponding to n=11 is a path abcde on 5 vertices. We have three 2matchings: (ab,cd), (ab,de), and (bc,de).
Triangle starts:
1;
1,1;
1,2;
1,2;
1,3,1;
1,3,1;


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: for n to 35 do seq(coeff(m(n), x, j), j = 0 .. degree(m(n))) end do; # yields sequence in triangular form


CROSSREFS

Cf. A202854.
Sequence in context: A048220 A182593 A201167 * A228572 A334675 A078380
Adjacent sequences: A202850 A202851 A202852 * A202854 A202855 A202856


KEYWORD

nonn,tabf


AUTHOR

Emeric Deutsch, Feb 14 2012


STATUS

approved



