Primes: SeqEnum Default: [ ]
Compute the growth function of the word acceptor automaton associated with G. The growth function of a DFA, A, is the quotient of two integral polynomials in a single variable x. The coefficient of xn in the Taylor expansion (about 0) of this quotient is equal to the number of words of length n accepted by A. That is, the result is a closed form for this generating function.The algorithm is by Derek Holt. The Primes parameter is no longer used, but is kept for backward compatibility. It may be removed in future releases.
> R<x> := RationalFunctionField(Integers()); > FG<a,b> := FreeGroup(2); > Q := quo< FG | a^5, b^2, a^b = a^-1>; > G := AutomaticGroup(Q); > f := GrowthFunction(G); > R!f; 2*x^3 + 4*x^2 + 3*x + 1Now we take as example an infinite dihedral group. The group is infinite, so the result cannot be polynomial. We then extract the coefficients of the growth function for word lengths 0 to 14.
> FG2<d,e> := FreeGroup(2); > Q2 := quo<FG2| e^2, d^e = d^-1>; > G2 := AutomaticGroup(Q2); > f2 := GrowthFunction(G2); > R!f2; (-x^2 - 2*x - 1)/(x - 1) > PSR := PowerSeriesRing(Integers():Precision := 15); > Coefficients(PSR!f2); [ 1, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 ][Next][Prev] [Right] [Left] [Up] [Index] [Root]