THE NUMBER OF DEGREE RESTRICTED MAPS ON GENERAL SURFACES

Let D be a finite set of positive integers with maximum bigger than two and ($) over tilde$$ m(g,n)(($) over tilde$$ m(g,n)) be the number of n-edged rooted maps on the orientable (nonorientable) surface of type g whose face degrees (or, dually, vertex degrees) all lie in D. Define ($) over tilde$$ m(g)(x)=Sigma(n greater than or equal to 0) ($) over tilde$$ m(g,n)x(n), ($) over tilde$$ m(g)(x)=Sigma(n greater than or equal to 0) ($) over tilde$$ m(g,n)x(n). We shaw that ($) over tilde$$ m(g)(x) and ($) over tilde$$ m(g),(x) are algebraic functions of a certain form. Asymptotic expressions for ($) over tilde$$ m(g,n) and ($) over tilde$$ m(g,n) are also derived for some special sets D.