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Abstract If G is permutation group acting on a finite set $\Omega $ , then this action induces a natural action of G on the power set $\mathscr{P}(\Omega )$ . The number $s(G)$ of orbits in this action is an important parameter that has been used in bounding numbers of conjugacy classes in finite groups. In this context, $\inf ({\log _2 s(G)}/{\log _2 |G|})$ plays a role, but the precise value of this constant was unknown. We determine it where G runs over all permutation groups not containing any ${{\textrm {A}}}_l, l> 4$ , as a composition factor. 

Abstract Background DNA methylation is an epigenetic event involving the addition of a methyl-group to a cytosine-guanine base pair (i.e., CpG site). It is associated with different cancers. Our research focuses on studying non-small cell lung cancer hemimethylation, which refers to methylation occurring on only one of the two DNA strands. Many studies often assume that methylation occurs on both DNA strands at a CpG site. However, recent publications show the existence of hemimethylation and its significant impact. Therefore, it is important to identify cancer hemimethylation patterns. Methods In this paper, we use the Wilcoxon signed rank test to identify hemimethylated CpG sites based on publicly available non-small cell lung cancer methylation sequencing data. We then identify two types of hemimethylated CpG clusters, regular and polarity clusters, and genes with large numbers of hemimethylated sites. Highly hemimethylated genes are then studied for their biological interactions using available bioinformatics tools. Results In this paper, we have conducted the first-ever investigation of hemimethylation in lung cancer. Our results show that hemimethylation does exist in lung cells either as singletons or clusters. Most clusters contain only two or three CpG sites. Polarity clusters are much shorter than regular clusters and appear less frequently. The majority of clusters found in tumor samples have no overlap with clusters found in normal samples, and vice versa. Several genes that are known to be associated with cancer are hemimethylated differently between the cancerous and normal samples. Furthermore, highly hemimethylated genes exhibit many different interactions with other genes that may be associated with cancer. Hemimethylation has diverse patterns and frequencies that are comparable between normal and tumorous cells. Therefore, hemimethylation may be related to both normal and tumor cell development. Conclusions Our research has identified CpG clusters and genes that are hemimethylated in normal and lung tumor samples. Due to the potential impact of hemimethylation on gene expression and cell function, these clusters and genes may be important to advance our understanding of the development and progression of non-small cell lung cancer. 

A well-known conjecture of Richard Stanley posits that the $h$-vector of the independence complex of a matroid is a pure ${\mathcal O}$-sequence. The conjecture has been established for various classes but is open for graphic matroids. A biconed graph is a graph with two specified 'coning vertices', such that every vertex of the graph is connected to at least one coning vertex. The class of biconed graphs includes coned graphs, Ferrers graphs, and complete multipartite graphs.  We study the $h$-vectors of graphic matroids arising from biconed graphs, providing a combinatorial interpretation of their entries in terms of '$2$-weighted forests' of the underlying graph. This generalizes constructions of Kook and Lee who studied the Möbius coinvariant (the last nonzero entry of the $h$-vector) of graphic matroids of complete bipartite graphs. We show that allowing for partially $2$-weighted forests gives rise to a pure multicomplex whose face count recovers the $h$-vector, establishing Stanley's conjecture for this class of matroids.  We also discuss how our constructions relate to a combinatorial strengthening of Stanley's Conjecture (due to Klee and Samper) for this class of matroids. 

Abstract In this paper, we study the product of orders of composition factors of odd order in a composition series of a finite linear group.First we generalize a result by Manz and Wolf about the order of solvable linear groups of odd order.Then we use this result to find bounds for the product of orders of composition factors of odd order in a composition series of a finite linear group.