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%I #17 Apr 29 2016 05:25:28
%S 0,1,-1,0,1,-4,-6,0,1,-4,-6,0,1,-12,-32,-20,0,1,-12,-32,-20,0,1,-12,
%T -28,-15,0,1,-12,-28,-15,0,1,-32,-120,-140,-50,0,1,-32,-120,-140,-50,
%U 0,1,-32,-120,-140,-50,0,1,-32,-112,-116,-38,0,1,-32,-112,-116,-38,0,1,-32,-112,-116,-38,0,1
%N Triangle read by rows: row n contains the coefficients (of the increasing powers of the variable) of the characteristic polynomial of the distance matrix of the rooted tree having Matula-Göbel number n.
%C Row n contains 1+A061775(n) entries (= 1+ number of vertices of the rooted tree). The pairs 0,1 are ends of rows.
%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-Göbel 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.
%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. W. Matula, <a href="http://www.jstor.org/stable/2027327">A natural rooted tree enumeration by prime factorization</a>, SIAM Rev. 10 (1968) 273.
%F Let T be a rooted tree with root b. If b has degree 1, then let A be the rooted tree with root c, obtained from T by deleting the edge bc emanating from the root. If b has degree >=2, then A is obtained (not necessarily in a unique way) by joining at b two trees B and C, rooted at b. It is straightforward to express the distance matrix of T in terms of the entries of the distance matrix of A (resp. of B and C). Making use of this, the Maple program (improvable!) finds recursively the distance matrices of the rooted trees with Matula-Göbel numbers 1..1000 (upper limit can be altered) and then finds the coefficients of their characteristic polynomials.
%e Row 4 is -4,-6,0,1 because the rooted tree having Matula-Göbel number 4 is V; the distance matrix is [0,1,1; 1,0,2; 1,2,0], having characteristic polynomial -4-6x+x^3.
%e Triangle starts:
%e 0,1;
%e -1,0,1;
%e -4,-6,0,1;
%e -4,-6,0,1;
%e -12,-32,-20,0,1;
%e -12,-32,-20,0,1;
%e -12,-28,-15,0,1;
%p with(numtheory): with(linalg): with(LinearAlgebra): V := 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+V(pi(n)) else V(r(n))+V(s(n))-1 end if end proc: d := proc (n) local r, s, C, a: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: C := proc (A, B) local c: c := proc (i, j) options operator, arrow: A[1, i]+B[1, j+1] end proc: Matrix(RowDimension(A), RowDimension(B)-1, c) end proc: a := proc (i, j) if i = 1 and j = 1 then 0 elif 2 <= i and 2 <= j then dd[pi(n)][i-1, j-1] elif i = 1 then 1+dd[pi(n)][1, j-1] elif j = 1 then 1+dd[pi(n)][i-1, 1] else end if end proc: if n = 1 then Matrix(1, 1, [0]) elif bigomega(n) = 1 then Matrix(V(n), V(n), a) else Matrix(blockmatrix(2, 2, [dd[r(n)], C(dd[r(n)], dd[s(n)]), Transpose(C(dd[r(n)], dd[s(n)])), SubMatrix(dd[s(n)], 2 .. RowDimension(dd[s(n)]), 2 .. RowDimension(dd[s(n)]))])) end if end proc: for n to 1000 do dd[n] := d(n) end do: for n to 14 do seq(coeff(CharacteristicPolynomial(d(n), x), x, k), k = 0 .. V(n)) end do;
%Y Cf. A061775.
%K sign,tabf
%O 1,6
%A _Emeric Deutsch_, Feb 08 2012
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