A202854 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.

1, 2, 5, 6, 18, 23, 26, 41, 54, 78, 103, 122, 162, 167, 202, 234, 283, 338, 366, 419, 486, 502, 547, 606, 643, 702, 794, 1009, 1014, 1093, 1098, 1346, 1458, 1506, 1543, 1586, 1597, 1818, 1906, 1999, 2106, 2371, 2382, 2462, 2626, 2719, 2962

Offset: 1

Author

Emeric Deutsch, Feb 14 2012

Keywords

• nonn

Comments

Alternatively, the Matula-Göbel numbers of rooted trees for which the matching-generating polynomial is palindromic.

A k-matching in a graph is a set of k edges, no two of which have a vertex in common.

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.

After activating the Maple program, the command m(n) will yield the matching-generating polynomial of the rooted tree having Matula-Göbel number n.

The given Maple program gives the required Matula-Göbel numbers up to L=200 (adjustable).

Examples

			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.
		

References

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

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. Göbel, 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. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
• Eric Weisstein's World of Mathematics, Matching-Generating Polynomial
• Index entries for sequences related to Matula-Goebel numbers

Crossrefs

Cf. A202853.

Programs

Maple

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;

Mathematica

L = 3000;
r[n_] := FactorInteger[n][[1, 1]];
s[n_] := n/r[n];
V[n_] := Which[n == 1, 1, PrimeOmega[n] == 1, 1 + V[PrimePi[n]], True, V[r[n]] + V[s[n]] - 1];
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]]}];
m[n_] := Total[M[n]] // Expand;
PAL = {};
Do[If [m[n] == Numerator[Together[m[n] /. x -> 1/x]], PAL = Union[PAL, {n}]], {n, 1, L}];
PAL (* Jean-François Alcover, Jun 24 2024, after Maple code *)

Formula

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=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 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.]