We define the push statistic on permutations and multipermutations and use this to obtain various results measuring the degree to which an arbitrary permutation deviates from sorted order. We study the distribution on permutations for the statistic recording the length of the longest push and derive an explicit expression for its first moment and generating function. Several auxiliary concepts are also investigated. These include the number of cells that are not pushed; the number of cells that coincide before and after pushing (i.e., the fixed cells of a permutation); and finally the number of groups of adjacent columns of the same height that must be reordered at some point during the pushing process.

Let \( \mathcal{K} \) be a family of sets in \( \mathbb{R}^d \). For each countable subfamily \( \{K_m : m \geq 1\} \) of \( \mathcal{K} \), assume that \( \bigcap \{K_m : m \geq 1\} \) is consistent relative to staircase paths and starshaped via staircase paths, with a staircase kernel that contains a convex set of dimension \( d \). Then \( \bigcap \{K : K \in \mathcal{K} \} \) has these properties as well. For \( n \) fixed, \( n \geq 1 \), an analogous result holds for sets starshaped via staircase \( n \)-paths.

We introduce a new bivariate polynomial which we call the defensive alliance polynomial and denote it by \( da(G; x, y) \). It is a generalization of the alliance polynomial [Carballosa et al., 2014] and the strong alliance polynomial [Carballosa et al., 2016]. We show the relation between \( da(G; x, y) \) and the alliance, the strong alliance, and the induced connected subgraph [Tittmann et al., 2011] polynomials. Then, we investigate information encoded in \( da(G; x, y) \) about \( G \). We discuss the defensive alliance polynomial for the path graphs, the cycle graphs, the star graphs, the double star graphs, the complete graphs, the complete bipartite graphs, the regular graphs, the wheel graphs, the open wheel graphs, the friendship graphs, the triangular book graphs, and the quadrilateral book graphs. Also, we prove that the above classes of graphs are characterized by its defensive alliance polynomial. A relation between induced subgraphs with order three and both subgraphs with order three and size three and two respectively, is proved to characterize the complete bipartite graphs. Finally, we present the defensive alliance polynomial of the graph formed by attaching a vertex to a complete graph. We show two pairs of graphs which are not characterized by the alliance polynomial but characterized by the defensive alliance polynomial.

A cancellable number (CN) is a fraction in which a decimal digit can be removed (“canceled”) in the numerator and denominator without changing the value of the number; examples include \( \frac{64}{16} \) where the 6 can be canceled and \( \frac{98}{49} \) where the 9 can be canceled. We present a few limit theorems and provide several generalizations.

Linear codes from neighborhood designs of strongly regular graphs such as triangular, lattice, and Paley graphs have been extensively studied over the past decade. Most of these families of graphs are line graphs of a much larger class known as circulant graphs, \( \Gamma_n(S) \). In this article, we extend earlier results to obtain properties and parameters of binary codes \( C_n(S) \) from neighborhood designs of line graphs of circulant graphs.

Necessary conditions for the existence of a \( 3 \)-GDD\((n, 3, k; \lambda_1, \lambda_2)\) are obtained along with some non-existence results. We also prove that these necessary conditions are sufficient for the existence of a \( 3 \)-GDD\((n, 3, 4; \lambda_1, \lambda_2)\) for \( n \) even.

The paired domination subdivision number \( \text{sd}_{pr}(G) \) of a graph \( G \) is the minimum number of edges that must be subdivided (where an edge can be subdivided at most once) in order to increase the paired domination number of \( G \). We prove that the decision problem of the paired domination subdivision number is NP-complete even for bipartite graphs. For this reason, we define the \textit{paired domination multisubdivision number} of a nonempty graph \( G \), denoted by \( \text{msd}_{pr}(G) \), as the minimum positive integer \( k \) such that there exists an edge which must be subdivided \( k \) times to increase the paired domination number of \( G \). We show that \( \text{msd}_{pr}(G) \leq 4 \) for any graph \( G \) with at least one edge. We also determine paired domination multisubdivision numbers for some classes of graphs. Moreover, we give a constructive characterization of all trees with paired domination multisubdivision number equal to 4.