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%I #19 Mar 07 2017 06:19:43
%S 1,2,5,6,18,23,26,41,54,78,103,122,162,167,202,234,283,338,366,419,
%T 486,502,547,606,643,702,794,1009,1014,1093,1098,1346,1458,1506,1543,
%U 1586,1597,1818,1906,1999,2106,2371,2382,2462,2626,2719,2962
%N The Matula-Göbel numbers of rooted trees T for which the sequence formed by the number of k-matchings of T (k=0,1,2,...) is palindromic.
%C Alternatively, the Matula-Göbel numbers of rooted trees for which the matching-generating polynomial is palindromic.
%C A k-matching in a graph is a set of k edges, no two of which have a vertex in common.
%C The Matula-Göbel 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-Göbel numbers of the m branches of T.
%C After activating the Maple program, the command m(n) will yield the matching-generating polynomial of the rooted tree having Matula-Göbel number n.
%C The given Maple program gives the required Matula-Göbel numbers up to L=200 (adjustable).
%D C. D. Godsil, Algebraic Combinatorics, Chapman & Hall, New York, 1993.
%H É. Czabarka, L. Székely, and S. Wagner, <a href="http://dx.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 E. Deutsch, <a href="http://arxiv.org/abs/1111.4288"> Rooted tree statistics from Matula numbers</a>, arXiv:1111.4288 [math.CO], 2011.
%H F. Göbel, <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 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 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 D. Matula, <a href="http://www.jstor.org/stable/2027327">A natural rooted tree enumeration by prime factorization</a>, SIAM Rev. 10 (1968) 273.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Matching-Generating Polynomial.html">Matching-Generating Polynomial</a>
%H <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F Define b(n) (c(n)) to be the generating polynomials of the matchings of the rooted tree with Matula-Göbel number n that contain (do not contain) the root, with respect to the size of the matching. We have the following recurrence for the pair M(n)=[b(n),c(n)]. M(1)=[0,1]; if n=p(t) (=the t-th 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 matching-generating polynomial). [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.]
%e 5 is in the sequence because the corresponding rooted tree is a path abcd on 4 vertices. We have 1 0-matching (the empty set), 3 1-matchings (ab), (bc), (cd), and 1 2-matchings (ab, cd). The sequence 1,3,1 is palindromic.
%p L := 200: 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: PAL := {}: for n to L do if m(n) = numer(subs(x = 1/x, m(n))) then PAL := `union`(PAL, {n}) else end if end do: PAL;
%Y Cf. A202853.
%K nonn
%O 1,2
%A _Emeric Deutsch_, Feb 14 2012
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