On the order sequence of a group

This paper provides a bridge between two active areas of research, the spectrum (set of element orders) and the power graph of a finite group. The order sequence of a finite group $G$ is the list of orders of elements of the group, arranged in non-decreasing order. Order sequences of groups of order $n$ are ordered by elementwise domination, forming a partially ordered set. We prove a number of results about this poset, among them the following. Abelian groups are uniquely determined by their order sequences, and the poset of order sequences of abelian groups of order $p^n$ is naturally isomorphic to the (well-studied) poset of partitions of $n$ with its natural partial order. If there exists a non-nilpotent group of order $n$, then there exists such a group whose order sequence is dominated by the order sequence of any nilpotent group of order $n$. There is a product operation on finite ordered sequences, defined by forming all products and sorting them into non-decreasing order. The product of order sequences of groups $G$ and $H$ is the order sequence of a group if and only if $|G|$ and $|H|$ are coprime. The paper concludes with a number of open problems.Comment: 22 pages, Comments are most welcom

Distance matrix of enhanced power graphs of finite groups

The enhanced power graph of a group $G$ is the graph $\mathcal{G}_E(G)$ with vertex set $G$ and edge set \{(u,v): u, v \in \langle w \rangle,~\mbox{for some}~ w \in G\}. In this paper, we compute the spectrum of the distance matrix of the enhanced power graph of non-abelian groups of order $pq$, dihedral groups, dicyclic groups, elementary abelian groups \El(p^n) and the non-cyclic abelian groups \El(p^n)\times \El(q^m) and \El(p^n)\times \mathbb{Z}_m, where $p$ and $q$ are distinct primes. For the non-cyclic abelian group \El(p^n)\times \El(q^m), we also compute the spectrum of the adjacency matrix of its enhanced power graph and the spectrum of the adjacency and the distance matrix of its power graph

qq-enumeration of type B and D Eulerian polynomials based on parity of descents

Carlitz and Scoville in 1973 considered four-variable polynomials enumerating the permutations according to the parity of both descents and ascents. In a recent work, Pan and Zeng proved a $q$-analogue of Carlitz-Scoville's generating function by counting the inversion number. Moreover, they also proved a type B analogue by enumerating the signed permutations with respect to the parity of descent and ascent position. In this work we prove a $q$-analogue of the type B result of Pan and Zeng by counting the type B inversion number. We also obtain a $q$-analogue of the generating functions for the bivariate alternating descent polynomials. Similar results are also obtained for type D Coxeter groups. As a by-product of our proofs, we get $q$-analogues of Hyatt's recurrences for the type B and type D Eulerian polynomials.Comment: 23 pages. Comments are welcom