A193404 Number of matchings (independent edge subsets) in the rooted tree with Matula-Goebel number n.

1, 2, 3, 3, 5, 5, 4, 4, 8, 8, 8, 7, 7, 7, 13, 5, 7, 12, 5, 11, 11, 13, 12, 9, 21, 12, 20, 10, 11, 19, 13, 6, 21, 11, 18, 16, 9, 9, 19, 14, 12, 17, 10, 18, 32, 20, 19, 11, 15, 30, 18, 17, 6, 28, 34, 13, 14, 19, 11, 25, 16, 21, 28, 7, 31, 31, 9, 15, 32, 27, 14

Offset: 1

Author

Emeric Deutsch, Feb 11 2012

Keywords

• nonn

Comments

A matching in a graph is a set of edges, no two of which have a vertex in common. The empty set is considered to be a matching.

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.

Examples

			a(3)=3 because the rooted tree with Matula-Goebel number 3 is the path ABC on 3 vertices; it has 3 matchings: empty, {AB}, {BC}.
		

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.
• Index entries for sequences related to Matula-Goebel numbers

Crossrefs

Cf. A202853 (by size), A347966 (maximal), A347967 (maximum).

Cf. A184165 (independent vertex sets).

Programs

Maple

with(numtheory): A := 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 [A(pi(n))[2], A(pi(n))[1]+A(pi(n))[2]] else [A(r(n))[1]*A(s(n))[2]+A(s(n))[1]*A(r(n))[2], A(r(n))[2]*A(s(n))[2]] end if end proc: a := proc (n) options operator, arrow: A(n)[1]+A(n)[2] end proc: seq(a(n), n = 1 .. 80);

Mathematica

r[n_] := FactorInteger[n][[1, 1]];
s[n_] := n/r[n];
A[n_] := Which[n == 1, {0, 1}, PrimeOmega[n] == 1, {A[PrimePi[n]][[2]], A[PrimePi[n]][[1]] + A[PrimePi[n]][[2]]}, True, {A[r[n]][[1]]* A[s[n]][[2]] + A[s[n]][[1]]*A[r[n]][[2]], A[r[n]][[2]]*A[s[n]][[2]]}];
a[n_] := Total[A[n]];
Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Jun 25 2024, after Maple code *)

Formula

Define b(n) (c(n)) to be the number of matchings of the rooted tree with Matula-Goebel number n that contain (do not contain) the root. We have the following recurrence for the pair A(n)=[b(n),c(n)]. A(1)=[0,1]; if n=prime(t), then A(n)=[c(t),b(t)+c(t)]; if n=r*s (r,s,>=2), then A(n)=[b(r)*c(s)+c(r)*b(s), c(r)c(s)]. Clearly, a(n)=b(n)+c(n). See the Czabarka et al. reference (p. 3315, (2)). The Maple program is based on this recursive formula.