A202853 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).

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

Offset: 1

Author

Emeric Deutsch, Feb 14 2012

Keywords

• nonn
• tabf

Comments

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.

Examples

			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;
  ...
		

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.
• Emeric 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.
• Eric Weisstein's World of Mathematics, Matching-Generating Polynomial
• Index entries for sequences related to Matula-Goebel numbers

Crossrefs

Cf. A206483 (matching number), A193404 (row sums), A347967 (end-most each row), A193403.

Cf. A202854 (palindromic rows).

Programs

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

Mathematica

r[n_] := FactorInteger[n][[1, 1]];
s[n_] := n/r[n];
V[n_] := Which[n == 1, 1, PrimeOmega[n] == 1, 1 + V[PrimePi[n]], True, V[r[n]] + V[s[n]] - 1];
M[n_] := Which[n == 1, {0, 1}, PrimeOmega[n] == 1, {x*M[PrimePi[n]][[2]], M[PrimePi[n]][[1]] + M[PrimePi[n]][[2]]}, True, {M[r[n]][[1]]* M[s[n]][[2]] + M[r[n]][[2]]*M[s[n]][[1]], M[r[n]][[2]]*M[s[n]][[2]]}];
m[n_] := Total[M[n]];
T[n_] := CoefficientList[m[n], x];
Table[T[n], {n, 1, 35}] // Flatten (* Jean-François Alcover, Jun 24 2024, after Maple code *)

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 (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=prime(t) (=the t-th prime), then M(n)=[xc(t),b(t)+c(t)]; if n=r*s (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.]