Let D be a set of positive integers. Let $m( n )$ be the number of n edged rooted maps on the sphere all of whose vertex degrees (or, dually, face degrees) lie in D. Using Brown’s technique, the generating function for $m( n )$ implicitly is obtained. It is used to prove that, when $\gcd ( D )$ is even, \[ m ( n ) \sim C ( D )n^{ - 5 /2} \gamma ( D )^n . \] It also yields known formulas for various special D.

Upper and lower bounds are shown for the average number of product terms required in the minimal realization, as a function of the number of nonzero output values. The variance, in addition to the bounds, allows conclusions to be drawn about how PLA size determines the set of realizable functions. Although the bounds are most accurate when there are few nonzero values, they are adequate for analyzing commercially available PLAs. The analysis shows that, for all but one commercially available PLA, the number of nonzero values is a statistically meaningful criterion for determining whether or not a given function is likely to be realized.>

An analysis of structured flowcharts is presented, where size is measured by the number, n , of decision nodes (IF-THEN-ELSE and DO-WHILE nodes). For all classes of structured flowcharts considered, the number of charts is approximately, cn -3/2 γ n , for large n , where c >and γ are parameters that depend on the class. It is also shown that most large flowcharts consist of a short sequence of basic charts (IF-THEN-ELSE and DO-WHILE charts). The average length of such sequences is 2.5.

Generating functions for the number of fanout-free and cascade networks built from an arbitrary set of symmetric gates are studied Recurslons and asymptotic estimates are obtained.The average number of gates in n-input networks .sstudied It grows hnearly with n, in contrast to the situation when a much larger set of gates is allowed KEY WORDS AND PHRASES switching functions, fanout-free networks, symmetric gates, cascades, generating functions, recurstons, asymptotic approximation CR CATEGORIES 6 1 Many authors have counted cascade and fanout-free functions.Sklansky, Korenjak, and Stone [11] determined the number of symmetry types of cascades built from two-input gates Hayes [7, 8], Kodandapani and Seth [9], Butler [5], and Bender and Butler [2] have studied various cascade and fanout-free networks where the gates are chosen from among NOT, OR, AND, Exclusive OR, and the three-input majority gate.Using a result in Bender and Canfield [3], we are able to study fanout-free and cascade networks built from any subset of the set of all symmetric gates.We first obtain a generating function for the number of n-variable functions that can be realized in this way.The generating function leads to a recursion and to asymptotic results.Special cases include the previously known results, summarized in Bender and Butler [2]; the relative number of functions for large and small n is sometimes quite different.In a typical network made from symmetric gates, the number of occurrences of a pamcular gate is proportional to n and, except for inverters, over half the gates are OR's and AND's.This contrasts sharply with the situation for arbitrary functions: The typical n-variable function reqmres an n-input gate for its fanoutfree implementation.In Section 2 we introduce some necessary terminology and recall some known results and techniques.In Sections 3, 4, and 5 we study fanout-free networks of symmetric gates, obtaining functional equations for generating functions, recursions, and asymptotlcs.

Expressions are derived for the approximate number of functions realized by various n-variable fanout-free networks. Six recently studied networks are considered. It is shown that the relative number of functions realized by two networks for small and large n is quite different in certain cases.