Related papers: Some Observations about the "Generalized Abundancy…

We generalize to the relations $(\lambda, \mu) \stackrel{\kappa}{\Rightarrow} (\lambda', \mu')$ and $\alm (\lambda, \mu) \stackrel{\kappa}{\Rightarrow} \alm (\lambda', \mu')$ some results obtained in Parts II and IV. We also present a…

We expand on the remark by Andrews on the importance of infinite sums and products in combinatorics. Let $\{g_d(n)\}_{d\geq 0,n \geq 1}$ be the double sequences $\sigma_d(n)= \sum_{\ell \mid n} \ell^d$ or $\psi_d(n)= n^d$. We associate…

Let $S_n$ denote the symmetric group. We consider \begin{equation*} N_{\ell}(n) := \frac{\left\vert Hom\left( \mathbb{Z}^{\ell},S_n\right) \right\vert}{n!} \end{equation*} which also counts the number of $\ell$-tuples $\pi=\left( \pi_1,…

Let $S_n$ denote the symmetric group of permutations acting on $n$ elements. We investigate the double sequence $\{N_{\ell}(n)\}$ counting the number of $\ell$ tuples of elements of the symmetric group $S_n$, where the components commute,…

In this short note, the second dual of generalized group algebra $(\ell^1(G,\mathcal A),\ast)$ equipped with both Arens products is investigated, where $G$ is any discrete group and $\mathcal A$ is a Banach algebra containing a complemented…

Let $s_1,s_2,\ldots s_n$ be states of a general probability theory, and $\mathcal A$ be the set of all subsets of indices $H \subset [n]\equiv\{1,2,\ldots n\}$ such that the states $(s_j)_{j\in H}$ are jointly perfectly distinguishable. All…

Grothendieck's theorem asserts that every continuous linear operator from $\ell_1$ to $\ell_2$ is absolutely $(1,1)$-summing. This kind of result is commonly called coincidence result. In this paper we investigate coincidence results in the…

The symmetric group $\mathfrak{S}_n$ acts on the polynomial ring $\mathbb{Q}[\mathbf{x}_n] = \mathbb{Q}[x_1, \dots, x_n]$ by variable permutation. The invariant ideal $I_n$ is the ideal generated by all $\mathfrak{S}_n$-invariant…

In this paper, we obtain a characterization of short normal sequences over a finite Abelian p-group, thus answering positively a conjecture of Gao for a variety of such groups. Our main result is deduced from a theorem of Alon, Friedland…

Fix a finite set $S \subset {GL}(k,\mathbb{Z})$. Denote by $a_n$ the number of products of matrices in $S$ of length $n$ that are equal to 1. We show that the sequence $\{a_n\}$ is not always P-recursive. This answers a question of…

We show that the combinatorial numbers known as {\em Bell numbers} are generic in quantum physics. This is because they arise in the procedure known as {\em Normal ordering} of bosons, a procedure which is involved in the evaluation of…

Let $\alpha$ and $\beta$ be two nonnegative integers such that $\beta < \alpha$. For an arbitrary sequence $\{a_n\}_{n\geqslant 1}$ of complex numbers, we consider the generalized Lambert series in order to investigate linear combinations…

We describe the subgroups of the group $\Z_m \times \Z_n \times \Z_r$ and derive a simple formula for the total number $s(m,n,r)$ of the subgroups, where $m,n,r$ are arbitrary positive integers. An asymptotic formula for the function…

We show that for any two sets of reals numbers $A=\{a_1,\dots,a_n\}$ and $B=\{b_1,\dots,b_n\}$, the sums of the form $\sum_{i=1}^n a_i\,b_{\pi(i)}$ always take on $\Omega(n^{3})$ distinct values, as we range over all permutations $\pi \in…

A lattice-ordered group (an $\ell$-group) $G(\oplus, \vee, \wedge)$ can be naturally viewed as a semiring $G(\vee,\oplus)$. We give a full classification of (abelian) $\ell$-groups which are finitely generated as semirings, by first showing…

By conformal welding, there is a pair of univalent functions $(f,g)$ associated to every point of the complex K\"ahler manifold $\Mob(S^1)\bk\Diff_+(S^1)$. For every integer $n\geq 1$, we generalize the definition of Faber polynomials to…

In this article we prove that $E(n)$-coactions over a finite-dimensional algebra $A$ are classified by tuples $(\varphi, d_1, ... , d_n)$ consisting of an involution $\varphi$ and a family $(d_i)_{i=1,...,n}$ of $\varphi$-derivations…

We conjecture that every infinite group $G$ can be partitioned into countably many cells $G=\bigcup_{n\in\omega}A_n$ such that $cov(A_nA_n^{-1})=|G|$ for each $n\in\omega$. Here $cov(A)=\min\{|X|:X\subseteq G, G=XA\}$. We confirm this…

A new family of generalized Pell numbers was recently introduced and studied by Br\'od \cite{Dorota}. These number possess, as Fibonacci numbers, a Binet formula. Using this, partial sums of arbitrary powers of generalized Pell numbers can…

The focus of this paper is the study of generalized Fibonacci polynomials and Fibonomial coefficients. The former are polynomials {n} in variables s and t given by {0} = 0, {1} = 1, and {n} = s{n-1}+t{n-2} for n ge 2. The latter are defined…