The geometry of classical groups over finite fields is widely used in many fields. In this paper, we study the rank-generating function, the characteristic polynomial, and the Poincaré polynomial of lattices generated by the orbits of subspaces under finite orthogonal groups of even characteristic. We also determine their expressions.

The problem of relevant enumeration with pattern-avoiding permutations is a significant topic in enumerative combinatorics and has wide applications in physics, chemistry, and computer science. This paper summarizes the relevant conclusions of the enumeration of pattern-avoiding permutations on the n-element symmetric group {\mathcal{S}}_n, alternating permutations, Dumont permutations, Ballot permutations, and inversion sequences. It also introduces relevant research results on avoiding vincular patterns and barred patterns in {\mathcal{S}}_n.

Let Γ=Cay(G, S) be the Cayley graph of a group G with respect to its subset S. The graph Γ is said to be normal edge-transitive if the normalizer of G in the automorphism group Aut(Γ) of Γ acts transitively on the edge set of Γ. In this paper, we study the structure of normal edge-transitive Cayley graphs on a class of non-abelian groups with order 2p² (p refers to an odd prime). The structure and automorphism groups of the non-abelian groups are first presented, and then the tetravalent normal edge-transitive Cayley graphs on such groups are investigated. Finally, the normal edge-transitive Cayley graphs on group G are characterized and classified.

Let N be a sufficiently large integer. In this paper, it is proved that with at most O\left(N^{\frac{17}{18}+\epsilon }\right)exceptions, all positive integers satisfying some necessary congruence conditions up to N can be represented in the form p_1^3+p_2^4+p_3^4+p_4^4+p_5^4+p_6^4+p_7^4+p_8^4+p_9^4+p_{10}^4, where p_1,p_2,\dots ,p_{10} are prime numbers.