A206497 The symmetry factor of the rooted tree with Matula-Goebel number n.

1, 1, 1, 2, 1, 1, 2, 6, 2, 1, 1, 2, 1, 2, 1, 24, 2, 2, 6, 2, 2, 1, 2, 6, 2, 1, 6, 4, 1, 1, 1, 120, 1, 2, 2, 4, 2, 6, 1, 6, 1, 2, 2, 2, 2, 2, 1, 24, 8, 2, 2, 2, 24, 6, 1, 12, 6, 1, 2, 2, 2, 1, 4, 720, 1, 1, 6, 4, 2, 2, 2, 12, 2, 2, 2, 12, 2, 1, 1, 24, 24, 1, 2, 4, 2, 2, 1, 6, 6, 2, 2, 4, 1, 1, 6, 120

Offset: 1

Author

Emeric Deutsch, May 29 2012

Keywords

• nonn
• mult

Comments

The symmetry factor of a rooted tree T is defined to be the number of indistinguishable permutations of the branches of T. For example, for the rooted tree obtained by identifying the roots of two copies of Y, the symmetry factor is 8; indeed the two branches of the first Y admit 2 indistinguishable permutations, the two branches of the second Y admit 2 indistinguishable permutations, and the 2 Y's admit 2 indistinguishable permutations (2*2*2=8).

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 (rooted at the root of T).

Examples

			a(2)=1 because the rooted tree with Matula-Goebel number 2 is the 1-edge tree I; a(4)=2 because the rooted tree with Matula-Goebel number 4 is V and there are 2 indistinguishable permutations of the branches.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Ch. Brouder, Runge-Kutta methods and renormalization, arXiv:hep-th/9904014, 1999; Eur. Phys. J. C 12, 2000, 521-534.
• D. J. Broadhurst and D. Kreimer, Renormalization automated by Hopf algebra, arXiv:hep-th/9810087, 1998; J. Symbolic Computation, 27, 1999, 581-600.
• Emeric Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288 [math.CO], 2011.
• 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.
• M. E. Hoffman, An analogue of covering space theory for ranked posets, The Electronic J. of Combinatorics, 8, 2001, #R32.
• D. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
• Index entries for sequences related to Matula-Goebel numbers

Crossrefs

Cf. A049084, A027748, A124010, A000142, A276625 (indices of 1's).

Programs

Haskell

import Data.List (genericIndex)
a206497 n = genericIndex a206497_list (n - 1)
a206497_list = 1 : g 2 where
  g x = y : g (x + 1) where
    y | t > 0     = a206497 t
      | otherwise = product $ zipWith (\p e -> a000142 e * a206497 p ^ e)
                                      (a027748_row x) (a124010_row x)
      where t = a049084 x
-- Reinhard Zumkeller, Sep 03 2013

Maple

with(numtheory): a := proc (n) if n = 1 then 1 elif nops(factorset(n)) = 1 then factorial(log[factorset(n)[1]](n))*a(pi(factorset(n)[1]))^log[factorset(n)[1]](n) else a(expand(op(1, ifactor(n))))*a(expand(n/op(1, ifactor(n)))) end if end proc: seq(a(n), n = 1 .. 160);
with(numtheory): SF := proc (n) local IF, A, FIF, FP, EFP: IF := proc (n) options operator, arrow: ifactors(n) end proc: A := proc (n) options operator, arrow: op(2, IF(n)) end proc: FIF := proc (n) options operator, arrow: op(1, A(n)) end proc: FP := proc (n) options operator, arrow: op(1, FIF(n)) end proc: EFP := proc (n) options operator, arrow: op(2, FIF(n)) end proc: if n = 1 then 1 elif bigomega(n) = 1 then SF(pi(n)) elif nops(A(n)) = 1 then factorial(EFP(n))*SF(pi(FP(n)))^EFP(n) else SF(FP(n)^EFP(n))*SF(n/FP(n)^EFP(n)) end if end proc: seq(SF(n), n = 1 .. 160); # Emeric Deutsch, Apr 30 2015

Mathematica

a[n_] := a[n] = If[n==1, 1, If[PrimeQ[n], a[PrimePi[n]], Product[{p, e} = pe; e! a[p]^e, {pe, FactorInteger[n]}]]];
a /@ Range[1, 100] (* Jean-François Alcover, Sep 25 2019 *)

PARI

a(n)={my(f=factor(n)); prod(i=1, #f~, my([p,e]=f[i,]); e!*a(primepi(p))^e)} \\ Andrew Howroyd, Aug 01 2018

Formula

a(1)=1; if n=prime(t), then a(n) = a(t); if n = p^u*q^v ... is the prime factorization of n, then a(n) =(u!*a(p)^u)*(v!*a(q)^v)... . (see Eq (2) in the Brouder reference).

If m and n are relatively prime, then a(m*n) = a(m)*a(n).

a(1)=1; if n=p^u, where p is the t-th prime, then a(n) = u!*a(t)^u; if n=r*s, r,s>=2, gcd(r,s)=1, then a(n)=a(r)a(s). Both Maple programs are based on these recurrence relations.

Extensions

Keyword:mult added by Andrew Howroyd, Aug 01 2018