Permutations Of Order Research Articles

Remark on Algorithm 323: Generation of Permutations in Lexicographic Order [G6

Remark on Algorithm 323: Generation of Permutations in Lexicographic Order [G6

Point-symmetric graphs and digraphs of prime order and transitive permutation groups of prime degree

In this paper the following two problems are solved: Given any point-symmetric graph or digraph Γ of prime order the automorphism group of Γ is explicitly determined and given any transitive permutation group G of prime degree p the number of digraphs and graphs of order p having G as their automorphism group is determined.

Sequential Permutation Networks

Two sequential permutation networks which generate all nǃ permutations without duplication are presented. Each permutation can be derived from its predecessor autonomously in one clock step. Based on the principle of recursive execution of a product of transpositions, the first network is constructed in multiple cascades of ½(n²-n) switching elements each of which permutes its input pair to its output pair according to its internal state, and is controlled sequentially by the same principle. Next, the network whose structure and control sequence are complementary to those of the first is presented. The first is named a cyclic permutation network and is an explicit representation of a mapping function The second is named a lexicographic permutation network and is an explicit representation of a factorial counting function where ah is a h-nary factorial digit 0 ≤ ah ≤ (n-1). The characteristics of the networks are investigated by establishing a one-to-one correspondence among nI integers, n! permutations, and n! states of the networks. For the lexicographic permutation network which can generate all permutations in lexicographic order, three examples of applications are given.

Application of a normalized process to the solution of linear algebraic systems

AN iteration process for the solution of the linear algebraic system Ax = ‖ with the singular matrix A, based on the normalized expansion of the generalized matrix B = ∥ A ⋮ − ‖∥, is presented. The process makes it possible to analyze the rounding errors directly, to calculate and correct the numerical solution, and to analyze the system for conditionality. A direct method of solving linear algebraic systems, based on the application of a normalized process, using elementary rotations or reflections with a specified order of permutation of the rows of the matrix and of the components of the free term is presented. The system enables the solution of the system to be found without working backwards and in this sense it is an orthogonal analogue of the well-known Jordan method. Under fairly general assumptions the process makes it possible, without having recourse to additional investigations, to analyze the system to be solved for conditionality and also to analyze the rounding errors directly. The process may be used to calculate the normal solution of an incompletely defined system. Two methods of refining the solution of the system are proposed: the first essentially uses the results of the calculations of the proposed process; the second may be recommended for refining the solution of a system obtained by other methods. It is recommended that in the first instance the proposed process be used to solve ill-conditioned systems, and also for the construction of a normal solution of incompletely defined systems whose dimensions permit a calculation to be carried out using only the operative memory of the computer.

Generation of permutations in lexicographical order

When the permutations are ordered lexicographically there is an ordering number corresponding to each permutation. A relation between the ordering numbers of complementary permutations is shown which can be useful in a computer generation of permutations.

Certification of algorithm 323 [G6]: generation of permutations in lexicographic order

Certification of algorithm 323 [G6]: generation of permutations in lexicographic order

Closed systems of functions and predicates

In this paper we show that there is a one to one correspondence between systems of functions defined on a finite set A and systems of predicates defined on A. This result implies that a complete set of invariants for a universal algebra on A is given by predicates defined on A. Conversely functions on A provide a complete system of invariants for sets of predicates closed under conjunction, change of variable and application of the existential quantifier. We begin in § 2 by giving a definition of closure for systems of functions and predicates. This is followed by a definition of commutivity of a function and a predicate which gives a correspondence between the two types of systems. In Theorems 1 and 2 of § 3 we show that the correspondence is a Galois connection. In Theorem 3 we consider sets of predicates closed under the existential quantifier and show that the corresponding systems are determined by functions defined for all values of the arguments. In Theorems 4 and 5 we include disjunction and then negation in the definition of closure of a set of predicates. We also require that equality be among the predicates. The corresponding systems consist of essentially first order functions and essentially first order permutations respectively. We conclude in § 4 with some comments on the infinite case and some general comments on these results.

Algorithm 323:Generation of permutations in lexicographic order

Algorithm 323:Generation of permutations in lexicographic order

Algorithm 308: Generation of the permutations in pseudo-lexicographic order

Algorithm 308: Generation of the permutations in pseudo-lexicographic order

Remarks on algorithm 87, 102, 130, 202: generation of permutations in lexicographical order

Remarks on algorithm 87, 102, 130, 202: generation of permutations in lexicographical order

Certification of algorithm 202 [G6] generation of permutations in lexicographical order

Certification of algorithm 202 [G6] generation of permutations in lexicographical order

Algorithm 202: generation of permutations in lexicographical order

Algorithm 202: generation of permutations in lexicographical order

Special Orthogonal Latin Squares of Order 10

The orthogonal latin squares displayed in [1] and [2] have the property that their row permutations are transformed amongst themselves by a permutation of order 7. In this note I present three examples of orthogonal latin squares of order 10 whose row permutations are transformed amongst themselves by a permutation of order 9.

On the generation of permutations and combinations

In this paper a new method of generating permutations in lexicographical order is developed. Analysis shows that the method is superior to the known method of generation by addition. A simple method of generating combinations is also described.

Algorithm 102: Permutation in lexicographical order

Algorithm 102: Permutation in lexicographical order

Permutation Ordering and Identification

(1961). Permutation Ordering and Identification. Mathematics Magazine: Vol. 34, No. 6, pp. 353-358.

Permutation Ordering and Identification

Permutation Ordering and Identification

Erzeugung lexikographisch geordneter Permutationen in Rechenautomaten

A convenient method of programming digital computers to generate permutations of n elements, in the same order as a dictionary, is introduced and described in detail. The mathematical basis of the method consists of the introduction of permutation numbers, one number being provided for each permutation. These permutation numbers provide a theoretically simple method of generating the permutation with any dictionary number selected at will, or conversely determining the dictionary number and sign of a given permutation. Moreover the use of permutation numbers permits the generation of permutations following a given permutation in dictionary order, including their signs, with the least possible number of transpositions. It is shown that, with continuous generation of permutations in dictionary order in accordance with the practical optimum programming method here developed, the average number of transfers required is only 1.24 or 1.31 times, according to the computer employed, what would be needed for continuous generation by successive transposition. With the latter method, moreover, the principle of dictionary order must be abandoned. The programming is illustrated by flow diagrams.

New Approach to the Quantum-Mechanical Analysis of the Electronic Structures of Molecules. The Method of Deformed Atoms in Molecules

A modification of Moffit's method of atoms in molecules is here proposed. In this treatment, the modified atomic functions X are introduced, which are made up from the exact eigenfunctions of atoms modified by correction factors so as to take account of the deformation of atoms in molecules. The advantage of these basic functions X is as follows. Firstly, the difficulty caused by the correlation energy in atoms can be removed by the use of the experimental atomic term values just as in Moffitt's method. Secondly, the common effective charges for neutral and ionic structures can be applied for the convenience of calculating the interaction operators, which sometimes produces serious errors in Moffitt's method. Moreover the improvement on the original Heitler-London treatment—corresponding to Wang's or Weinbaum's work for the hydrogen molecule—can be expected. Then, it is discussed how to determine these modified atomic functions in order that the results appertain to the good range in spite of the rather simple calculation. The energy loss of atoms in molecules, induced by the deformation, is corrected, and the interatomic interaction energy is calculated by the use of the approximate functions composed of the orbital functions, where the molecular orbital method can be utilized for saving the difficulty of the higher order permutations. Then the unsatisfactory results, obtained in the calculation of Li2 by utilizing the common effective charges in Moffitt's method, are improved.

Use of Non-orthogonal Wave Functions in the Treatment of Solids, with Applications to Ferromagnetism

It is proved by a rearrangement of L\"owdin's solution to the many-body problem that a vector model type energy expression is valid for a solid provided (in the simplest case of one electron per atom) the number of nearest neighbors times the overlap integral between them is small compared with unity.The approximation is also carried to include third order permutation terms, and the spin coupling via closed shells these "triple exchange" terms can produce is discussed.General expressions are given for the exchange integrals to be used in a vector model solution. These under the proper conditions reduce to the conventional integrals except for a slight modification, which helps to make the integrals negative.