Abstract: 0. Introduction. In this paper we continue the study, begun in [1], of some combinatorial problems related to monotonicities that occur in certain spaces of finite sequences. These spaces are equipped with standard probability measures, so that one may study the distribution of monotonicities in such spaces and, in particular, the expected lengths of maximal monotonie subsequences. These, in turn, are upper bounds on the expected lengths of monotonie subsequences obtained by applying some selection process over the space of sequences. The problem which we have called natural sorting is concerned with the maximization of these expected lengths. The scheme of the paper is as follows. In Section 1 we give definitions and review some background material. In Section 2 we describe the distribution of maximal sequences occurring in the space of permutation sequences.* These distributions have been computed exactly for spaces of permutations of length n = 2(1)36, and have been approximated by Monte Carlo computations for certain values of n ranging up to 10,000. In Section 3 we consider several selection strategies and the corresponding distribution of selected monotonie subsequences. The computations were performed on the IBM 7094 at the Computer Center of the University of California, Berkeley. We are indebted to David M. Matula for some of the calculations in Section 3. We should like to thank Geri Stephen for her assistance in the preparation of the manuscript.

Abstract: Rearrangeable switching networks are considered as permutation generators. Two-by-two network arrays may be implemented in these networks using a β reversing element. Nested tree arrays of these elements may be used to synthesize rearrangeable switching networks which appear minimal. Multistage network arrays of these elements may be implemented within a single coordinate device.

Abstract: Suppose that a (nan) and b (n->bn) are sequences in a compact metric space with distance d If d(an, bn)->O, then, clearly, a and b have the same set of cluster points If, more generally, 7r is a permutation of the set of positive integers, such that d(an, b,,n)->O, then, again, a and b have the same set of cluster points It is a remarkable result due to J von Neumann (Charakterisierung des Spektrums eines Integraloperators, Hermann, Paris, 1935, pp 11-12) that the converse is true: if two sequences a and b have the same set of cluster points, then there exists a permutation 7r of the set of positive integers suchthat d(a, b7,)>O (In fact von Neumann discussed real sequences only, but that is merely a notational specialization) If the common cluster set is a singleton, the permutation can be chosen to be the identity and the conclusion is immediate If, more generally, the common cluster set is finite, the result is less trivial but still quite easy The proof von Neumann gave for the general case is a page densely packed with subscripts The purpose of this note is to give a simpler proof The simplification is achieved by use of the best known result in infinite combinatorics, the Schroder-Bernstein theorem An a posteriori analysis of von Neumann's proof shows that the subscripts hide a re-proof of that theorem in the special case at hand If C is the common cluster set of a and b, writee, = 1e/n+d(a, C), and let U7, be the open ball with center an, and radius en, Since e > d(a, C), the ball U7, contains at least one point of C, and, consequently, U7, contains infinitely many terms of the sequence b Let o1 be the smallest positive integer such that 1 > 1 and b,l C U1 Inductively, let o(n+1) be the smallest positive integer such that o(n+1) >o(n) and b,(n+l) E U+1 (Observe that o(n) >n) Since en -30 it follows that d(a7, bn,)>O Summary: there exists a one-toone mapping oof the set of positive integers into itself such that o-n>n for all n and such that d(an, ban)->O Similarly, there exists a one-to-one mapping i of the set of positive integers into itself such that -n > n for all n and such that d (a7n, bn)->O The Schroder-Bemstein theorem says that if M and N are sets and if c: M->N and r: NM are injections, then there exists a bijection

Abstract: b e g i n p := sqrt(z); d := 0.5 X z X p; p := 1 p e n d e l s e b e g i n d : = z X z ; p : = w X z e n d ; y := 2 X w / z ; for j := b + 2 s t e p 2 u n t i l n do b e g i n d := (1 + a / ( j -2 ) ) X d X z; p := i f a = 1 t h e n p + d X y / ( j 1 ) e l s e ( p + w ) X z e n d j; y := w X z; z := 2/z; b := n 2; for i := a + 2 s t e p 2 u n t i l m do b e g i n j : = i + b; d := y X d X j / ( i 2 ) ; p := p -z X d / j e n d i; Fisher := p e n d Fisher

Abstract: An algorithm for the realization of k-threshold threshold realizable functions is presented. Instead of solving the set of linear inequalities, where the unknowns are the weights corresponding to the input variables, incremental weights are sought. The procedure reduces to that of resolving contradicting pairs of vertices by the incremental weights. The minimum number of thresholds are sought for each complementation and permutation of input variables. A definition of an optimal multithreshold weight threshold vector is derived from the reliability viewpoint. The desired solution is obtained through a search of all possible obtainable realization vectors of the function. For single-threshold realizable functions, permutation and complementation of input variables need not be considered if the input variables of the function are ordered and positivized. The procedure is systematic and has been programmed in FORTRAN IV. As a comparison with Haring and Ohori's tabulation on the 221 equivalence classes of four variable Boolean functions under the NPN1 operation, it can be seen that 42 of the 221 equivalence classes need fewer numbers of thresholds for their realization. For the same number of thresholds, 58 equivalence classes have less absolute sum of weights. Finally, with the number of thresholds and absolute sum of weights being equal, 36 equivalence classes have lower threshold values.

Abstract: The group of timelike permutations onRn+1,Tn+1, the group of spacelike permutations onRn+1,Sn+1, and the group of lightlike permutations onRn+1,Ln+1, (n≥1), are defined, whereRn+1 is the Cartesian productRn+1=Rx0×Rx1×…×Rx1 andRxi, (i=0, 1, …,n), is the set of all real numbers A permutation on the setRn+1 is a timelike permutationtn+1∈Tn+1, (n≥1), if bothtn+1 and (tn+1)−1 preserve the relationTx, y being two points whatsoever belonging toRn+1,xTy meansQ(x−y)>0, whereQ(z)=z02−z12−…−zn2,z=(z0,z1,…,zn)∈Rn+1, (n≥1) The spacelike permutationssn+1∈Sn+1 and the lightlike permutationsln+1∈Ln+1, (n≥1), are defined in a similar way by using the respective conditionsQ(x−y) 1), whereDn+1, (n≥1), is the group whose elements are all and only the transformations onRn+1 which are obtained by composing an element of the Poincare group,Pn+1, with an element of the group of the positive dilatations,R+x To prove said theorem we use the relationTn+1=Sn+1, (n≥1), previously proved, and the propertytn+1∈Tn+1⇒tn+1∈Ln+1, (n≥1) The casen=1 is then studied and one deduces thatT2=S2 is a proper subgroup ofL2 Finally the groupT2=S2 andL2 are studied in detail

Abstract: It is extremely difficult to make general statements about functional completeness. (For the main reference on the subject see Post [2].) In this paper we restrict ourselves to the case of unary functions in a finite valued logic, and prove a result concerning minimal functionally complete sets, along with a necessary and sufficient condition for completeness. A basic familiarity with group theory would be helpful. The following conventions will be followed in notation. The symbol F" will denote the set of all unary functions in an n-valued logic. If f e F" and g E FD then by the product fg is meant a function h e Fn such that for any truth value i, h(i) = f(g(i)). As usual the symbol fm denotes the product of f with itself m times, and f3 will denote the identity map. It will be assumed that the truth values are represented by integers 1 through n. We now state some preliminary definitions and a lemma. DEFINITION. A permutation is a function f e Fn such that every truth value occurs in its truth table once and only once. The symbol Sn will denote the group of all permutations which belong to Fn. DEFINITION. Let a be a truth value, and let f e F,. The horizontal class of f containing a is the set of truth values {i I f(i) = f(a)}. LEMMA. Suppose f, g e Fn, and a one-to-one correspondence can be established between the horizontal classes of f and the horizontal classes of g such that corresponding classes have an equal number of elements. Then there exist permutations x and y such that f = xgy. The proof of this lemma is very straightforward and will be omitted. DEFINITION. A function f E Fn is singly reduced iff there exist two and only two truth values i and ] such that f(i) = f(j). We are now able to state and prove a theorem on functional completeness in one variable giving sufficient and necessary conditions for a set of unary functions to be complete. THEOREM. A set C c Fn is functionally complete iff some subset of C generates Sn and C contains at least one singly reduced function. PROOF. Let C be a set which satisfies the hypothesis above. It is easy to verify that any two singly reduced functions satisfy the hypothesis of the lemma. Now define a function fk so that

Abstract: Frobenius derived an equation which is sufficient to determine all simple characters and character recursion relations for the permutation groups. A simple procedure for obtaining these results involves comparing monomials on both sides of Frobenius’ equation. To make such comparisons one needs to individually consider each permutation of each pertinent monomial. A modified form of Frobenius’ equation is derived. This form makes it possible to compare monomials without considering individual permutations. This considerably reduces the number of explicit comparisons needed to determine characters and character recursion relations.

Abstract: The paper describes a method for laying out networks by computer so that the number of crossings between the network connections is close to a minimum. The problem is relevant to the design of printed circuits, where special wiring arrangements have to be made when crossings occur. The network is expressed in the form of a permutation, which is convenient for manipulation, by deforming the network so that the node points lie on a straight line with the connections drawn as semicircles above an below the node line. Locally optimal networks are defined so that no gain can result from moving an individual node to a new position, and a 2-stage method of construction is proposed. The formulas used to calculate the number of crossings consist primarily of summations, so that the procedure is quickly performed on a computer. The method has been tested on some trial networks for which the minimum number of crossings is known, and it has also been compared with Monte Carlo methods on random networks. The results are encouraging in all cases.

Abstract: such that Ij,j (j= l(1)n) and the latter asks for the number of permutations for which there are a fixed number of j such that Ij