On some partition theorems of M. V. Subbarao

M.V. Subbarao proved that the number of partitions of $n$ in which parts occur with multiplicities 2, 3 and 5 is equal to the number of partitions of $n$ in which parts are congruent to $\pm2, \pm3, 6 \pmod{12}$, and generalized this result. In this paper, we give a new generalization of this identity and also present a new partition theorem in the spirit of Subbarao's generalization of the identity

Intriguing sets of strongly regular graphs and their related structures

In this paper we outline a technique for constructing directed strongly regular graphs by using strongly regular graphs having a "nice" family of intriguing sets. Further, we investigate such a construction method for rank three strongly regular graphs having at most $45$ vertices. Finally, several examples of intriguing sets of polar spaces are provided

On signs of certain Toeplitz-Hessenberg determinants whose elements involve Bernoulli numbers: On negativity and positivity of Hessenberg determinants

In the paper, by virtue of Wronski's formula and Kaluza's theorem related to a power series and its reciprocal, by means of Cahill and Narayan's recursive relation, and with the aid of the logarithmic convexity of the sequence of the Bernoulli numbers, the author presents the signs of certain Toeplitz-Hessenberg determinants whose elements involve the Bernoulli numbers and combinatorial numbers. Moreover, with the help of a derivative formula for the ratio of two differentiable functions, the author provides an alternative proof of Wronski's formula

A graph related to the sum of element orders of a finite group

A finite group is called $\psi$-divisible iff $\psi(H)|\psi(G)$ for any subgroup $H$ of a finite group $G$. Here, $\psi(G)$ is the sum of element orders of $G$. For now, the only known examples of such groups are the cyclic ones of square-free order. The existence of non-abelian $\psi$-divisible groups still constitutes an open question. The aim of this paper is to make a connection between the $\psi$-divisibility property and graph theory. Hence, for a finite group $G$, we introduce a simple undirected graph called the $\psi$-divisibility graph of $G$. We denote it by $\psi_G$. Its vertices are the non-trivial subgroups of $G$, while two distinct vertices $H$ and $K$ are adjacent iff $H\subset K$ and $\psi(H)|\psi(K)$ or $K\subset H$ and $\psi(K)|\psi(H)$. We prove that $G$ is $\psi$-divisible iff $\psi_G$ has a universal (dominating) vertex. Also, we study various properties of $\psi_G$, when $G$ is a finite cyclic group. The choice of restricting our study to this specific class of groups is motivated in the paper

q-Analogues of π\pi-Series by Applying Carlitz Inversions to q-Pfaff-Saalschutz Theorem

By applying multiplicate forms of the Carlitz inverse series relations to the $q$-Pfaff-Saalschtz summation theorem, we establish twenty five nonterminating $q$-series identities with several of them serving as $q$-analogues of infinite series expressions for $\pi$ and $1/\pi$, including some typical ones discovered by Ramanujan (1914) and Guillera

On some new families of kk-Mersenne and generalized kk-Gaussian Mersenne numbers and their polynomials

In this paper, we define the generalized k-Mersenne numbers and give a formula of generalized Mersenne polynomials and further, we study their properties. Moreover, we define Gaussian Mersenne numbers and obtain some identities like Binet Formula, Cassini's identity, D'Ocagne's Identity, and generating functions. The generalized Gaussian Mersenne numbers are described and their relation with the classical Mersenne numbers are explained. We also introduce a generalization of Gaussian Mersenne polynomials and establish some properties of these polynomials

Structural theory of trees II. Completeness and completions of trees

Trees are partial orderings where every element has a linearly ordered set of smaller elements. We define and study several natural notions of completeness of trees, extending Dedekind completeness of linear orders and Dedekind-MacNeille completions of partial orders. We then define constructions of tree completions that extend any tree to a minimal one satisfying the respective completeness property

On a generalized basic series and Rogers–Ramanujan type identities - II

This paper is in continuation with our recent paper “On a generalized basic series and Rogers-Ramanujan type identities”. Here, we consider two generalized basic series and interpret these basic series as the generating function of some restricted $(n + t)$-color partitions and restricted weighted lattice paths. The basic series discussed in the aforementioned paper, is now a mere particular case of one of the generalized basic series that are discussed in this paper. Besides, eight particular cases are also discussed which give combinatorial interpretations of eight Rogers–Ramanujan type identities which are combinatorially unexplored till date

Further Rogers-Ramanujan type identities for modified lattice paths

Recently, the authors introduced the modified lattice paths which generalize Agarwal-Bressoud weighted lattice paths. Using these new objects they interpreted combinatorially two basic series identities which led to two new combinatorial Rogers-Ramanujan type identities. In this paper we obtain three more Rogers-Ramanujan type identities for modified lattice paths. This also leads to three new 3-way combinatorial identities

Total dominator total coloring of a graph

Here, we initiate to study the total dominator total coloring of a graph which is a total coloring of the graph such that each object of the graph is adjacent or incident to every object of some color class. In more details, while in section 2 we present some tight lower and upper bounds for the total dominator total chromatic number of a graphs in terms of some parameters such as order, size, the total dominator chromatic and total domination numbers of the graph and its line graph, in section 3 we restrict our to trees and present a Nordhaus-Gaddum-like relation for trees. Finally in last section we show that there exist graphs that their total dominator total chromatic numbers are equal to their orders