arXiv subjects
Publications and source records attributed to Alnour Altoum.
In this paper, we will introduce the Cauchy numbers of both kinds in type B and produce their corresponding exponential generating functions. Then we will provide some identities involving Cauchy, Lah, and Stirling numbers in type B through combinatorial methods.
In this paper, we introduce an inversion statistic on the hyperoctahedral group $B_n$ by using an decomposition of a positive root system of this reflection group. Then we prove some combinatorial properties for the inversion statistic. We establish an enumeration system on the group $B_n$ and give an efficient method to uniquely derive any group element known its enumeration order with the help of the inversion table. In addition, we prove that the \textit{flag-major index} is equi-distributed with this inversion statistic on $B_n$.
In this paper, we construct a mixed-base number system over the generalized symmetric group $G(m,1,n)$, which is a complex reflection group with a root system of type $B_n^{(m)}$. We also establish one-to-one correspondence between all positive integers in the set $\{1,\cdots,m^nn!\}$ and the elements of $G(m,1,n)$ by constructing the subexceedant function in relation to this group. In addition, we provide a new enumeration system for $G(m,1,n)$ by defining the inversion statistic on $G(m,1,n)$. Finally, we prove that the \textit{flag-major index} is equi-distributed with this inversion statistic on $G(m,1,n)$. Therefore, the flag-major index is Mahonian on $G(m,1,n)$ with respect to the length function $L$.
In this paper, we define a mixed-base number system over a Weyl group of type $D$, the group even-signed permutations. We introduce one-to-one correspondence between positive integers and elements of Weyl groups of type $D$ after constructed the subexceedant function associating to the group. Thus, the integer representations of all classical Weyl groups are now completed.