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

%I A193404 #34 Jun 26 2024 09:32:47
%S A193404 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,
%T A193404 19,13,6,21,11,18,16,9,9,19,14,12,17,10,18,32,20,19,11,15,30,18,17,6,
%U A193404 28,34,13,14,19,11,25,16,21,28,7,31,31,9,15,32,27,14
%N A193404 Number of matchings (independent edge subsets) in the rooted tree with Matula-Goebel number n.
%C A193404 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.
%C A193404 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.
%H A193404 É. Czabarka, L. Székely, and S. Wagner, <a href="https://doi.org/10.1016/j.dam.2009.07.004">The inverse problem for certain tree parameters</a>, Discrete Appl. Math., 157, 2009, 3314-3319.
%H A193404 Emeric Deutsch, <a href="http://arxiv.org/abs/1111.4288"> Rooted tree statistics from Matula numbers</a>, arXiv:1111.4288 [math.CO], 2011.
%H A193404 F. Goebel, <a href="http://dx.doi.org/10.1016/0095-8956(80)90049-0">On a 1-1-correspondence between rooted trees and natural numbers</a>, J. Combin. Theory, B 29 (1980), 141-143.
%H A193404 I. Gutman and A. Ivic, <a href="http://dx.doi.org/10.1016/0012-365X(95)00182-V">On Matula numbers</a>, Discrete Math., 150, 1996, 131-142.
%H A193404 I. Gutman and Yeong-Nan Yeh, <a href="http://www.emis.de/journals/PIMB/067/3.html">Deducing properties of trees from their Matula numbers</a>, Publ. Inst. Math., 53 (67), 1993, 17-22.
%H A193404 D. W. Matula, <a href="http://www.jstor.org/stable/2027327">A natural rooted tree enumeration by prime factorization</a>, SIAM Rev. 10 (1968) 273.
%H A193404 <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F A193404 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.
%e A193404 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}.
%p A193404 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);
%t A193404 r[n_] := FactorInteger[n][[1, 1]];
%t A193404 s[n_] := n/r[n];
%t A193404 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]]}];
%t A193404 a[n_] := Total[A[n]];
%t A193404 Table[a[n], {n, 1, 80}] (* _Jean-François Alcover_, Jun 25 2024, after Maple code *)
%Y A193404 Cf. A202853 (by size), A347966 (maximal), A347967 (maximum).
%Y A193404 Cf. A184165 (independent vertex sets).
%K A193404 nonn
%O A193404 1,2
%A A193404 _Emeric Deutsch_, Feb 11 2012