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A206483 The matching number of the rooted tree having Matula-Goebel number n.

%I A206483 #33 Feb 16 2025 08:33:16
%S A206483 0,1,1,1,2,2,1,1,2,2,2,2,2,2,3,1,2,3,1,2,2,3,3,2,3,3,3,2,2,3,3,1,3,2,
%T A206483 3,3,2,2,3,2,3,3,2,3,4,3,3,2,2,3,3,3,1,4,4,2,2,3,2,3,3,3,3,1,4,4,2,2,
%U A206483 4,3,2,3,3,3,4,2,3,4,3,2,4,3,3,3,3,3,3,3,2,4,3,3,4,4,3,2,3,3,4,3,3,3,4,3,4,2,2,4,3,4
%N A206483 The matching number of the rooted tree having Matula-Goebel number n.
%C A206483 A matching in a graph is a set of edges, no two of which have a vertex in common. The matching number of a graph is the maximum of the cardinalities of all the matchings in the graph. Consequently, the matching number of a graph is the degree of the matching-generating polynomial of the graph (see the MathWorld link).
%C A206483 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.
%C A206483 After activating the Maple program, which yields the sequence, the command m(n) will yield the matching-generating  polynomial of the rooted tree corresponding to the Matula-Goebel number n.
%D A206483 C. D. Godsil, Algebraic Combinatorics, Chapman & Hall, New York, 1993.
%H A206483 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 A206483 Emeric Deutsch, <a href="http://arxiv.org/abs/1111.4288"> Rooted tree statistics from Matula numbers</a>, arXiv:1111.4288 [math.CO], 2011.
%H A206483 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 A206483 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 A206483 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 A206483 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 A206483 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Matching-Generating Polynomial.html">Matching-Generating Polynomial</a>
%H A206483 <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F A206483 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), 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). The matching number is the degree of this polynomial.
%e A206483 a(11)=2 because the rooted tree corresponding to n=11 is a path abcde on 5 vertices. We have matchings with 2 edges (for example, (ab, cd)) but not with 3 edges.
%p A206483 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: seq(degree(m(n)), n = 1 .. 110);
%t A206483 r[n_] := FactorInteger[n][[1, 1]];
%t A206483 s[n_] := n/r[n];
%t A206483 V[n_] := Which[n == 1, 1, PrimeOmega[n] == 1, 1 + V[PrimePi[n]], True, V[r[n]] + V[s[n]] - 1];
%t A206483 M[n_] := 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]]}];
%t A206483 m[n_] := Total[M[n]] // Expand;
%t A206483 a[n_] := Exponent[m[n], x];
%t A206483 Table[a[n], {n, 1, 110}] (* _Jean-François Alcover_, Jun 24 2024, after Maple code *)
%Y A206483 Cf. A202853.
%K A206483 nonn
%O A206483 1,5
%A A206483 _Emeric Deutsch_, Feb 14 2012