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Given sequences 9 , X : IN0 + C , g(0) = 1, linked by the convolution X * g = 9'(g'(n) := (n + 1)g(n + 1)) we study what can be inferred about X (n --f m) from some concrete information about the behaviour of g (n + 00).

Introduction. The problems of "factorisatio numerorum", which go back more than 65 years, are concerned principally with (i) the total number f (n) of factorizations of a natural number n > 1 into products of natural numbers larger than 1, where the order of the factors is not counted, and (ii) the corresponding total number F (n) of factorizations when the order of the factors is counted. The object of this paper is to consider similar problems and results, with emphasis on the average numbers of factorizations of each kind, within the partly analogous but also quite distinct context of additive arithmetical semigroups. Such semigroups (to be defined below) are treated in the monographs [3], [4] within an abstract setting designed to conveniently cover (under minimal assumptions) concrete cases like (i) the multiplicative semigroup G q of all monic polynomials in one indeterminate over a finite field F q , (ii) semigroups of ideals in principal orders within algebraic function fields over F q , (iii) semigroups formed under direct sum by the isomorphism classes of certain kinds of finite modules or algebras over such principal orders.Although the first main result below will be derived within a still more general framework, we shall formulate and state our second main conclusion within a context (see [4]) which conveniently covers the preceding concrete cases:An (additive) arithmetical semigroup will be understood to be a free commutative semigroup G with identity element 1, generated by a (countable) set P of "prime" elements, which admits an integer-valued "degree"

The generating power series for the number of permutations avoiding particular consecutive patterns are derived in a new and simple fashion.

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Asymptotic representations are given for the three sums

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