A193406 - OEIS

%I #29 Jun 25 2024 13:12:00

%S 1,3,4,7,8,9,10,11,12,13,14,16,17,19,20,21,24,25,27,28,29,30,32,33,34,

%T 35,36,37,38,39,40,42,43,44,46,47,48,49,50,51,52,53,56,57,58,59,60,61,

%U 62,63,64,67,68,70,71,72,73,74,75,76,77,79,80,81,82,83,84,85,86,87,88,89,90,91,92,95,96,97,98,99,100

%N The Matula numbers of the rooted trees that have no perfect matchings.

%C 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 It is known that a tree has at most one perfect matching.

%C Complement of A193405.

%D C. D. Godsil, Algebraic Combinatorics, Chapman & Hall, New York, 1993.

%H 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 Emeric Deutsch, <a href="http://arxiv.org/abs/1111.4288">Rooted tree statistics from Matula numbers</a>, arXiv:1111.4288 [math.CO], 2011.

%H 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 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. 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 <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-Goebel 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=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 (a modified matching polynomial). The tree has a perfect matching if and only if the degree of this polynomial is 1/2 of the number of vertices of the tree.

%e 3 and 11 are in the sequence because they are the Matula numbers of paths on 3 and 5  vertices, respectively.

%p 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: NPM := {}: for n to 100 do if N(n) <> 2*degree(m(n)) then NPM := `union`(NPM, {n}) else  end if end do: NPM;

%t r[n_] := FactorInteger[n][[1, 1]];

%t s[n_] := n/r[n];

%t V[n_] := Which[n == 1, 1, PrimeOmega[n] == 1, 1 + V[PrimePi[n]], True, V[r[n]] + V[s[n]] - 1];

%t M[n_] := Which[n == 1, {0, 1}, PrimeOmega[n] == 1, {x*M[PrimePi[n]][[2]], Total[M[PrimePi[n]]]}, 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 m[n_] := M[n] // Total // Expand;

%t NPM = {};

%t Do[If[V[n] != 2 Exponent[m[n], x], NPM = Union[NPM, {n}]], {n, 1, 100}];

%t NPM (* _Jean-François Alcover_, Jun 21 2024, after Maple code *)

%Y Cf. A061775, A193405.

%K nonn

%O 1,2

%A _Emeric Deutsch_, Feb 12 2012