A206493 -id:A206493 - cp's OEIS frontend

A206494 Number of ways to take apart the rooted tree corresponding to the Matula-Goebel number n by sequentially removing terminal edges.

1, 1, 1, 2, 1, 3, 2, 6, 6, 4, 1, 12, 3, 8, 10, 24, 2, 30, 6, 20, 20, 5, 6, 60, 20, 15, 90, 40, 4, 60, 1, 120, 15, 10, 40, 180, 12, 30, 45, 120, 3, 120, 8, 30, 210, 36, 10, 360, 80, 140, 30, 90, 24, 630, 35, 240, 90, 24, 2, 420, 30, 6, 420, 720, 105, 105, 6, 60, 126, 280, 20, 1260

Offset: 1

Emeric Deutsch, May 10 2012

nonn

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.

Number of ways to label the vertices of the rooted tree corresponding to the Matula-Goebel number n with the elements of {1,2,...,n} so that the label of each vertex is less than that of its descendants. Example: a(8)=6 because the rooted tree with Matula-Goebel number 8 is the star \|/; the root has label 1 and the 3 leaves are labeled with any permutation of {2,3,4}. See the Knuth reference, p. 67, Exercise 20. There is a simple bijection between the ways of the described labeling of a rooted tree and the ways of taking it apart by sequentially removing terminal edges: remove the edges in the inverse order of the labeling.

			a(7)=2 because the rooted tree with Matula-Goebel number 7 is Y; denoting the edges in preorder by 1,2,3, it can be taken apart either in the order 231 or 321. a(11) = 1 because the rooted tree with Matula-Goebel number 11 is the path tree with 5 vertices; any path tree can be taken apart in only one way.

D. E. Knuth, The Art of Computer Programming, Vol.3, 2nd edition, Addison-Wesley, Reading, MA, 1998.

Kevin Ryde, Table of n, a(n) for n = 1..10000
Emeric Deutsch, Tree statistics from Matula numbers, arXiv:1111.4288 [math.CO], 2011.
J. Fulman, Mixing time for a random walk on rooted trees, The Electronic J. of Combinatorics, 16, 2009, R139.
F. Goebel, 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. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
Kevin Ryde, PARI/GP Code
B. E. Sagan and Y.-N. Yeh, Probabilistic algorithms for trees, The Fibonacci Quarterly, 27, 1989, 201-208. [_Emeric Deutsch_, Apr 28 2015]
Index entries for sequences related to Matula-Goebel numbers

Cf. A061775, A206493.

with(numtheory): E := 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 elif bigomega(n) = 1 then 1+E(pi(n)) else E(r(n))+E(s(n)) end if end proc: a := 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 a(pi(n)) else a(r(n))*a(s(n))*binomial(E(r(n))+E(s(n)), E(r(n))) end if end proc: seq(a(n), n = 1 .. 72);

r[n_] := FactorInteger[n][[1, 1]];
s[n_] := n/r[n];
e[n_] := e[n] = Which[n == 1, 0, PrimeOmega[n] == 1, 1+e[PrimePi[n]], True, e[r[n]] + e[s[n]]];
a[n_] := a[n] = Which[n == 1, 1, PrimeOmega[n] == 1, a[PrimePi[n]], True, a[r[n]]*a[s[n]]*Binomial[e[r[n]] + e[s[n]], e[r[n]]] ];
Table[a[n], {n, 1, 72}] (* Jean-François Alcover, Aug 06 2024, after Maple program, replacing E(n) with e[n] *)

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a(prime(m)) = a(m); a(r*s) = a(r)*a(s)*binomial(E(r*s),E(r)), where E(k) is the number of edges of the rooted tree with Matula-Goebel number k. The Maple program is based on these recurrence relations.

a(n) = V(n)!/TF(n), where V denotes "number of vertices" (A061775) and TF denotes "tree factorial" (A206493) (see Eq. (3) in the Fulman reference).

A206496 The Connes-Moscovici weight of the rooted tree with Matula-Goebel number n. It is defined as the number of ways to build up the rooted tree from the one-vertex tree by adding successively edges to the existing vertices.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 3, 4, 1, 6, 3, 4, 10, 1, 1, 15, 1, 10, 10, 5, 3, 10, 10, 15, 15, 10, 4, 60, 1, 1, 15, 5, 20, 45, 6, 5, 45, 20, 3, 60, 4, 15, 105, 18, 10, 15, 10, 70, 15, 45, 1, 105, 35, 20, 15, 24, 1, 210, 15, 6, 105, 1, 105, 105, 1, 15, 63, 140

Offset: 1

Emeric Deutsch, Jul 20 2012

nonn

See A206494 for the number of ways to take apart the rooted tree corresponding to the Matula-Goebel number n by sequentially removing terminal edges.

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.

			a(6)=3 because the rooted tree with Matula-Goebel number 6 is the path ARBC with root at R; starting with R we can obtain the tree ARBC by adding successively edges at the vertices  (i) R, R, A, or at (ii) R, R, B, or at (iii) R, A, R.
a(8)=1 because the rooted tree with Matula-Goebel number 8 is the star tree with 3 edges emanating from the root; obviously, there is only 1 way to build up this tree from the root.

J. C. Butcher, The Numerical Analysis of Ordinary Differential Equations, 1987, Wiley, Chichester.

Ch. Brouder, Runge-Kutta methods and renormalization, Eur. Phys. J. C 12, 2000, 521-534.
D. J. Broadhurst and D. Kreimer, Renormalization automated by Hopf algebra, J. Symbolic Computation, 27, 1999, 581-600.
Emeric Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288 [math.CO], 2011.
J. Fulman, Mixing time for a random walk on rooted trees, The Electronic J. of Combinatorics, 16, 2009, R139.
F. Goebel, 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 Y.-N. Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.
Index entries for sequences related to Matula-Goebel numbers

Cf. A061775, A206493, A206497, A206494.

A139002 is a permutation of this sequence.

with(numtheory): 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: TF := 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 V(n)*TF(pi(n)) else TF(r(n))*TF(s(n))*V(n)/(V(r(n))*V(s(n))) end if end proc: SF := proc (n) if n = 1 then 1 elif nops(factorset(n)) = 1 then factorial(log[factorset(n)[1]](n))*SF(pi(factorset(n)[1]))^log[factorset(n)[1]](n) else SF(expand(op(1, ifactor(n))))*SF(expand(n/op(1, ifactor(n)))) end if end proc: a := proc (n) options operator, arrow: factorial(V(n))/(TF(n)*SF(n)) end proc: seq(a(n), n = 1 .. 120);

V[n_] := Module[{u, v}, u = FactorInteger[#][[1, 1]]&; v = #/u[#]&; If[n == 1, 1, If[PrimeQ[n], 1 + V[PrimePi[n]], V[u[n]] + V[v[n]] - 1]]];
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];
TF[n_] := Which[n == 1, 1, PrimeOmega[n] == 1, V[n]*TF[PrimePi[n]], True, TF[r[n]]*TF[s[n]]*v[n]/(v[r[n]]*v[s[n]])];
SF[n_] := SF[n] = If[n == 1, 1, If[PrimeQ[n], SF[PrimePi[n]], Module[{p, e}, Product[{p, e} = pe; e! * SF[p]^e, {pe, FactorInteger[n]}]]]];
a[n_] := V[n]!/(TF[n]*SF[n]);
Table[a[n], {n, 1, 120}] (* Jean-François Alcover, Jun 24 2024, after Emeric Deutsch in A061775, A206493 and A206497 *)

a(n) = V(n)!/[TF(n)*SF(n)], where V denotes "number of vertices" (A061775), TF denotes "tree factorial" (A206493), and SF denotes "symmetry factor" (A206497).

cp's OEIS Frontend

A206494 Number of ways to take apart the rooted tree corresponding to the Matula-Goebel number n by sequentially removing terminal edges.

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