Partitions Of Positive Integer Research Articles

A novel approach to find partitions of $ Z_{m} $ with equal sum subsets via complete graphs

<abstract> In mathematics and computer sciences, the partitioning of a set into two or more disjoint subsets of equal sums is a well-known NP-complete problem, also referred to as partition problem. There are various approaches to overcome this problem for some particular choice of integers. Here, we use quadratic residue graph to determine the possible partitions of positive integers $ m = 2^{\beta}, q^{\beta}, 2^{\beta}q, $ $ 2q^{\beta}, qp, $ where $ p $, $ q $ are odd primes and $ \beta $ is any positive integer. The quadratic residue graph is defined on the set $ Z_{m} = \{\overline{0}, \overline{1}, \cdots, \overline{m-1}\}, $ where $ Z_{m} $ is the ring of residue classes of $ m $, i.e., there is an edge between $ \overline{x}, $ $ \overline{y}\in Z_{m} $ if and only if $ \overline{x}^{2}\equiv \overline{y}^{2}\; (\text{mod}\; m) $. We characterize these graphs in terms of complete graph for some particular classes of $ m $. </abstract>

Counting short cycles of (c,d)-regular bipartite graphs

Recently, Tanner graphs which represented low density parity check (LDPC) codes have become an interesting research topic. Finding the number of short cycles of Tanner graphs motivated Blake and Lin to investigate the multiplicity of cycles of length equal to the girth of bi-regular bipartite graphs by using the spectrum and degree distribution of the graph. While there were many algorithms to find the number of cycles, they chose to take a computational approach. Dehghan and Banihashemi counted the number of cycles of length [Formula: see text] and [Formula: see text] where [Formula: see text] is a bi-regular bipartite graph and [Formula: see text] is the girth of [Formula: see text] But for the cycles of length smaller than [Formula: see text] in bi-regular bipartite graphs, they only proposed a descriptive technique. In this paper, we find the number of cycles of length less than [Formula: see text] by using the spectrum and the degree distribution of bi-regular bipartite graphs such that the formula depends only on the partitions of positive integers and the number of closed cycle-free walks from any vertex of [Formula: see text] and [Formula: see text] which are known.

A recursive enumeration of connected Feynman diagrams with an arbitrary number of external legs in the fermionic non-relativistic interacting gas

In this work, we generalize a recursive enumerative formula for connected Feynman diagrams with two external legs. The Feynman diagrams are defined from a fermionic gas with a two-body interaction. The generalized recurrence is valid for connected Feynman diagrams with an arbitrary number of external legs and an arbitrary order. The recurrence formula terms are expressed in function of weak compositions of non-negative integers and partitions of positive integers in such a way that to each term of the recurrence correspond a partition and a weak composition. The foundation of this enumeration is the Wick theorem, permitting an easy generalization to any quantum field theory. The iterative enumeration is constructive and enables a fast computation of the number of connected Feynman diagrams for a large amount of cases. In particular, the recurrence is solved exactly for two and four external legs, leading to the asymptotic expansion of the number of different connected Feynman diagrams.

Lower bounds for generalized Hausdorff matrices and lower triangular matrices on the block weighted sequence space ℓp(w, F)

Let and be a non-negative matrix. Denote by , the supremum of those , satisfying the following inequalitywhere is a partition of positive integers, , is a non-negative sequence in and is a monotone and non-negative sequence of real number. In this paper a Hardy-type formula is obtained for , where is the generalized Hausdorff matrix, and Another purpose of this paper is to establish a general upper estimate for the exact value of , for which recently a lower estimate was established in Lashkaripour and Talebi [Lashkaripour R, Talebi G. Bull. Iran. Math. Soc. 2011;37:115–126], where is a non-negative lower triangular matrix and . We also derive the corresponding result for with In particular, we apply our results to summability matrices, weighted mean matrices, Nörlund matrices. Our results also generalize some results in Chen and Wang [Chen C-P, Wang K-Z. Linear Multilinear Algebra, March 2011;59:321–337] and Lashkaripour and Talebi [Lashkaripour R, Talebi G. Czech. Math. J. 2012; 62: 293–04.].

Compositions of n as alternating sequences of weakly increasing and strictly decreasing partitions

Abstract Compositions and partitions of positive integers are often studied in separate frameworks where partitions are given by q-series generating functions and compositions exhibiting specific patterns are designated by generating functions for these patterns. Here, we view compositions as alternating sequences of weakly increasing and strictly decreasing partitions (i.e. alternating blocks). We obtain generating functions for the number of such partitions in terms of the size of the composition, the number of parts and the total number of “valleys” and “peaks”. From this, we find the total number of “peaks” and “valleys” in the composition of n which have the mentioned pattern. We also obtain the generating function for compositions which split into just two partition blocks. Finally, we obtain the two generating functions for compositions of n that start either with a weakly increasing partition or a strictly decreasing partition.

On the regular semisimple elements and primary classes of [formula omitted

The present paper deals with counting the number and orders of regular semisimple elements, together with the number of primary classes of GL ( n , q ) . The approach used in this paper depends essentially on partitions of positive integers ⩽ n. We give the numbers of regular semisimple elements and primary classes of GL ( n , q ) for n ∈ { 1 , 2 , … , 6 } and see that the number of regular semisimple elements is an integral polynomial in q, while the number of primary classes is a rational polynomial in q.

Using classical groups to construct authentication codes

To design a simple and practical authentication scheme is one of the important issues in the authentication codes research.Applying partition of positive integers and Anzahl theorem of subgroups of classical groups,a class of authentication codes was obtained and all parameters of this scheme were computed.Simultaneously,supposing that the secret key was chosen according to the uniform probability distribution,security analysis of the scheme was given.The construction increased the number of the source state dramatically,which indicates that this code has better traits than some other codes obtained by matrices method before.

Two-parameter family of infinite-dimensional diffusions on the Kingman simplex

We construct a two-parameter family of diffusion processes X α,θ on the Kingman simplex, which consists of all nonincreasing infinite sequences of nonnegative numbers with sum less than or equal to one. The processes on this simplex arise as limits of finite Markov chains on partitions of positive integers. For α = 0, our process coincides with the infinitely-many-neutral-alleles diffusion model constructed by Ethier and Kurtz (1981) in population genetics. The general two-parameter case apparently lacks population-genetic interpretation. In the present paper, we extend Ethier and Kurtz’s main results to the two-parameter case. Namely, we show that the (two-parameter) Poisson-Dirichlet distribution PD(α,θ) is the unique stationary distribution for the process X α,θ and that the process is ergodic and reversible with respect to PD(α, θ). We also compute the spectrum of the generator of X α,θ . The Wright-Fisher diffusions on finite-dimensional simplices turn out to be special cases of X α,θ for certain degenerate parameter values.

Some partitions of positive integers

Research problem 231, Discrete Mathematics 140 (1995) says: Let A be a set of 2k, k ⩾ 2, distinct positive integers. It is desired to partition A into two subsets A0 and A1 each with cardinality k so that the sum of any k−1 elements of Ai is not an element of Ai+1, i = 0, 1 mod 2. It is not possible to find such a partition when A is {1, 3, 4, 5, 6, 7} or any of {1, 2, 3, 4, 5, x}, x ⩾ 7. Can it be done in all other cases? We show that the answer is affirmative for k ⩾ 3 with some exceptions for k = 3.

Exchanges of "Quanta" Among Distinguishable and Indistinguishable Objects

Beginning with a physical problem of exchange of n indistinguishable "quanta" of energy in an ensemble of k oscillators we define a new wide class of combinatorial problems, which also contains statistics intermediate between Fermi–Dirac and Bose–Einstein. One such problem is related to the number theoretic problem of computing the partitions of positive integers. After establishing such a connection, we give explicit formulas for the partitions M(n,k) of an integer n into k parts with k ≤ 4. Moreover, we derive a recursion relation for fixed n and varying k which is valid for any k. A formula which relates partitions to the cardinality of the partition set taking order into account is also derived. The leading asymptotic behavior for n large is obtained for any k. A suggestive interpretation of this formulas is proposed in terms of simplicial lattices. Recursive formulas at fixed k and varying n are then written for k ≤ 5 using the concept of factorial triangle, which is amenable for generalizations to larger k's. The problem of distinct partitions is mapped onto the probability problem of ball removal with replacement, for which we give again recursive solution formulas. Finally, the method of generalized Tartaglia triangle allows the derivation of recursive formulas for limited partitions which take order into account. This latter result is related to the problem of finding the number of distinct ways of dividing n indistinguishable objects into k distinguishable groups, for which explicit summations had been previously found.

Partitions of positive integers

Because they exhibit so many beautiful and intriguing relations, numbers are an inexhaustible source of fascination. This paper will show how to answer a variety of questions about partitions of a positive integer n using three easily constructed tables, and give a particularly simple and efficient way to calculate the number of partitions of large n. Accompanying computer programs are given in the appendix.

Partitions of Positive Integers: An Elementary Topic of Number Theory

The author presents some interesting aspects of number theory and graphing that should be appealing to students.