**Generating Function**

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A system with more than two states is called a multistate system (MSS), and such systems have already become a general trend in the arena of complex industrial products and/or systems. Fault-tolerant technology often plays a very important role in improving the reliability of an MSS. However, the existence of imperfect coverage failure (ICF) in a work-sharing group (WSG) decreases the reliability of MSS. A method is proposed to assess the reliability and sensitivity of an MSS with ICF. The components in a WSG can cooperate so as to improve overall efficiency by increasing performance levels. Using the technique of the universal generating function (UGF), a component’s UGF expression with ICF can be incorporated in two steps. During the computation of the system’s UGF, an algorithm based on matrix (ABM) is developed to reduce the computational complexity. Consequently, indices of reliability can be easily calculated based on the UGF expression of an MSS. Sensitivity analysis can help engineers judge which WSG should be eliminated first under various resource limitations. Examples illustrate and validate this method.

Network reliability is one of the most important concepts in this modern era. Reliability characteristics, component significance measures, such as the Birnbaum importance measure, critical importance measure, the risk growth factor and average risk growth factor, and network reliability stability of the communication network system have been discussed in this paper to identify the critical components in the network, and also to quantify the impact of component failures. The study also proposes an efficient algorithm to compute the reliability indices of the network. The authors explore how the universal generating function can work to solve the problems related to the network using the exponentially distributed failure rate. To illustrate the proposed algorithm, a numerical example has been taken.

For a graph $\Gamma$ , let $K(H_{\Gamma },\,1)$ denote the Eilenberg–Mac Lane space associated with the right-angled Artin (RAA) group $H_{\Gamma }$ defined by $\Gamma$ . We use the relationship between the combinatorics of $\Gamma$ and the topological complexity of $K(H_{\Gamma },\,1)$ to explain, and generalize to the higher TC realm, Dranishnikov's observation that the topological complexity of a covering space can be larger than that of the base space. In the process, for any positive integer $n$ , we construct a graph $\mathcal {O}_n$ whose TC-generating function has polynomial numerator of degree $n$ . Additionally, motivated by the fact that $K(H_{\Gamma },\,1)$ can be realized as a polyhedral product, we study the LS category and topological complexity of more general polyhedral product spaces. In particular, we use the concept of a strong axial map in order to give an estimate, sharp in a number of cases, of the topological complexity of a polyhedral product whose factors are real projective spaces. Our estimate exhibits a mixed cat-TC phenomenon not present in the case of RAA groups.

Shuffle Exchange Networks (SENs) are considered as an appropriate interconnection network because they consist of switching elements of small size and possess a straight forward and simple configuration. In this paper, we have proposed a method for analyzing reliability of 4×4 SEN, 4×4 SEN+1 and 4×4 SEN+2. The reliability has been obtained on the basis of three indices, namely, terminal reliability, broadcast reliability and network reliability by using universal generating function (UGF) method. This study also examines effect of adding the additional stages in 4×4 shuffle exchange networks (SENs).

Abstract Gireesh and Mahadeva Naika [‘On 3-regular partitions in 3-colors’, Indian J. Pure Appl. Math.50 (2019), 137–148] proved an infinite family of congruences modulo powers of 3 for the function $p_{\{3,3\}}(n)$ , the number of 3-regular partitions in three colours. In this paper, using elementary generating function manipulations and classical techniques, we significantly extend the list of proven arithmetic properties satisfied by $p_{\{3,3\}}(n).$

In this article, a study is carried out around the Perrin sequence, these numbers marked by their applicability and similarity with Padovan’s numbers. With that, we will present the recurrence for Perrin’s polynomials and also the definition of Perrin’s complex bivariate polynomials. From this, the recurrence of these numbers, their generating function, generating matrix and Binet formula are defined.

In this paper, we construct the Lax operator of the multi-component Boussinesq hierarchy. Based on the Sato theory and the dressing structure of the multi-component Boussinesq hierarchy, the adjoint wave function and the Orlov–Schulman’s operator are introduced, which are useful for constructing the additional symmetry of the multi-component Boussinesq hierarchy. Besides, the additional flows can commute with the original flows, and these flows form an infinite dimensional [Formula: see text] algebra. Taking the above discussion into account, we mainly study the additional symmetry flows and the generating function for both strongly and weakly multi-component of the Boussinesq hierarchies. By the way, using the [Formula: see text] constraint of the multi-component Boussinesq hierarchy, the string equation can be derived.