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The crossing number of a graph is a significant measure that indicates the complexity of the graph and the difficulty of visualizing it. In this paper, we examine the crossing numbers of join products involving the complete graph K 5 with discrete graphs, paths, and cycles. We analyze optimal drawings of K 5 , identify all five non-isomorphic drawings, and address previously hypothesized crossing numbers for K 5 + P n , and K 5 + C n . Through a simplified approach, we first establish cr (K 5 + D n) and then extend our method to prove the crossing numbers cr (K 5 + P n) and cr (K 5 + C n) . These results also lead to new hypotheses for cr (W m + S n) and cr (W m + W n) involving wheels and stars. Our findings correct previous inaccuracies in the literature and provide a foundation for future research. [ABSTRACT FROM AUTHOR]

Recently Alexandersson and Getachew proved some multivariate generalizations of a formula for enumerating signed excedances in derangements. In this paper we first relate their work to a recent continued fraction for permutations and confirm some of their observations. Our second main result is two refinements of their multivariate identities, which clearly explain the meaning of each term in their main formulas. We also explore some similar formulas for permutations of type B. [ABSTRACT FROM AUTHOR]