Integer partitions of $n$ are viewed as bargraphs (i.e., Ferrers diagrams rotated anticlockwise by 90 degrees) in which the $i$-th part of the partition $x_i$ is given by the $i$-th column of the bargraph with $x_i$ cells. The sun is at infinity in the northwest of our two-dimensional model, and each partition casts a shadow in accordance with the rules of physics. The number of unit squares in this shadow but not being part of the partition is found through a bivariate generating function in $q$ tracking partition size and $u$ tracking shadow. To do this, we define triangular $q$-binomial coefficients which are analogous to standard $q$-binomial coefficients, and we obtain a formula for these. This is used to obtain a generating function for the total number of shaded cells in (weakly decreasing)
partitions of $n$.

We consider a scalar-valued implicit function of many variables, and provide two closed formulae for all of its partial derivatives. One formula is based on products of partial derivatives of the defining function, the other one involves fewer products of building blocks of multinomial type, and we study the combinatorics of the coefficients showing up in both formulae.

Tensors, or multi-linear forms, are important objects in a variety of areas from analytics, to combinatorics, to computational complexity theory. Notions of tensor rank aim to quantify the “complexity” of these forms, and are thus also important. While there is one single definition of rank that completely captures the complexity of matrices (and thus linear transformations), there is no definitive analog for tensors. Rather, many notions of tensor rank have been defined over the years, each with their own set of uses.

In this paper we survey the popular notions of tensor rank. We give a brief history of their introduction, motivating their existence, and discuss some of their applications in computer science. We also give proof sketches of recent results by Lovett, and Cohen and Moshkovitz, which prove asymptotic equivalence between three key notions of tensor rank over finite fields with at least three elements.

We prove two conjectures due to Sun concerning binomial-harmonic sums. First, we introduce a proof of a formula for Catalan’s constant that had been conjectured by Sun in 2014. Then, using a similar approach as in our first proof, we solve an open problem due to Sun involving the sequence of alternating odd harmonic numbers. Our methods, more broadly, allow us to reduce difficult binomial-harmonic sums to finite combinations of dilogarithms that are evaluable using previously known algorithms.

The aim of this work is to establish congruences $(\mathrm{mod}{p}^2)$ involving the trinomial coefficients ${(\genfrac{}{}{0ex}{}{np-1}{p-1})}_2$ and ${(\genfrac{}{}{0ex}{}{np-1}{(p-1)/2})}_2$ arising from the expansion of the powers of the polynomial $1+x+x^2$. In main results, we extend some known congruences involving the binomial coefficients $(\genfrac{}{}{0ex}{}{np-1}{p-1})$ and $(\genfrac{}{}{0ex}{}{np-1}{(p-1)/2})$, and establish congruences linking binomial coefficients and harmonic numbers.

In analogy with the semi-Fibonacci partitions studied recently by Andrews, we define semi-$m$-Pell compositions. We find that these are in bijection with certain weakly unimodal $m$-ary compositions. We give generating functions, bijective proofs, and a number of unexpected congruences for these objects. In the special case of $m=2$, we have a new combinatorial interpretation of the semi-Pell sequence and connections to other objects.

The aim of this paper is to introduce and study a new class of analytic functions which generalize the classes of $\lambda$-Spirallike Janowski functions. In particular, we gave the representation theorem, the right side of the covering theorem, starlikeness estimates and some properties related to the functions in the class $S_{\lambda }(T,H,F)$.

criteria to verify log-convexity of sequences is presented. Iterating this criteria produces infinitely log-convex sequences. As an application, several classical examples of sequences arising in Combinatorics and Special Functions are presented. The paper concludes with a conjecture regarding coefficients of chromatic polynomials.

We discuss the VC-dimension of a class of multiples of integers and primes (equivalently indicator functions) and demonstrate connections to prime counting functions. Additionally, we prove limit theorems for the behavior of an empirical risk minimization rule as well as the weights assigned to the output hypothesis in AdaBoost for these “prime-identifying” indicator functions, when we sample $mn$ i.i.d. points uniformly from the integers $\{2,\dots ,n\}$.

Integer compositions and related counting problems are a rich and ubiquitous topic in enumerative combinatorics. In this paper we explore the definition of symmetric and asymmetric peaks and valleys over compositions. In particular, we compute an explicit formula for the generating function for the number of integer compositions according to the number of parts, symmetric, and asymmetric peaks and valleys.

In this paper we show some identities come from the $q$-identities of Euler, Jacobi, Gauss, and Rogers-Ramanujan. Some of these identities relate the function sum of divisors of a positive integer $n$ and the number of integer partitions of $n$. One of the most intriguing results found here is given by the next equation, for $n>0$,
$$\sum _{l=1}^n\frac{1}{l!}\sum _{w_1+w_2+\cdots +w_l\in C(n)}\frac{{\sigma }_1(w_1){\sigma }_1(w_2)\cdots {\sigma }_1(w_l)}{w_1{w}_2\cdots w_l}=p_1(n),$$
where ${\sigma }_1(n)$ is the sum of all positive divisors of $n$, $p_1(n)$ is the number of integer partitions of $n$, and $C(n)$ is the set of integer compositions of $n$. In the last section, we show seven applications, one of them is a series expansion for
$$\frac{(q^{a_1};q^{b_1}{)}_{\mathrm{\infty }}(q^{a_2};q^{b_2}{)}_{\mathrm{\infty }}\cdots (q^{a_k};q^{b_k}{)}_{\mathrm{\infty }}}{(q^{c_1};q^{d_1}{)}_{\mathrm{\infty }}(q^{c_2};q^{d_2}{)}_{\mathrm{\infty }}\cdots (q^{c_r};q^{d_r}{)}_{\mathrm{\infty }}},$$
where $a_1,\dots ,a_k,b_1,\dots ,b_k,c_1,\dots ,c_r,d_1,\dots ,d_r$ are positive integers, and $|q|<1$.

Between Bernoulli/Euler polynomials and Pell/Lucas polynomials, convolution sums are evaluated in closed form via the generating function method. Several interesting identities involving Fibonacci and Lucas numbers are shown as consequences including those due to Byrd $(1975)$ and Frontczak $(2020)$.

The notion of length spectrum for natural numbers was introduced by Pong in $[5]$. In this article, we answer the question of how often one can recover a random number from its length spectrum. We also include a quick deduction of a result of LeVeque in $[4]$ on the average order of the size of length spectra.

This paper uses exponential sum methods to show that if $E\subset M_2(\mathbb{Z}/p^r)$ is a finite set of $2\times 2$ matrices with sufficiently large density and $j$ is any unit in the finite ring $\mathbb{Z}/p^r$, then there exist at least two elements of $E$ whose difference has determinant $j$.

In this paper, we introduce a generalized family of numbers and polynomials of one or more variables attached to the formal composition $f\cdot (g\circ h)$ of generating functions $f$, $g$, and $h$. We give explicit formulae and apply the obtained result to two special families of polynomials; the first concerns the generalization of some polynomials applied to the theory of hyperbolic differential equations recently introduced and studied by $M.Mihoubi$ and $M.Sahari$. The second concerns two-variable Laguerre-based generalized Hermite-Euler polynomials introduced and should be updated to studied recently by $N.U.Khan\mathit{\text{et al.}}$.

In this paper, we show that the generalized exponential polynomials and the generalized Fubini polynomials satisfy certain binomial identities and that these identities characterize the mentioned polynomials (up to an affine transformation of the variable) among the class of the normalized Sheffer sequences.

Let $A$ be a subset of a finite field $\mathbb{F}$. When $\mathbb{F}$ has prime order, we show that there is an absolute constant $c>0$ such that, if $A$ is both sum-free and equal to the set of its multiplicative inverses, then $|A|<(0.25–c)|\mathbb{F}|+o(|\mathbb{F}|)$ as $|\mathbb{F}|\to \mathrm{\infty }$. We contrast this with the result that such sets exist with size at least $0.25|\mathbb{F}|–o(|\mathbb{F}|)$ when $\mathbb{F}$ has characteristic 2.

In this paper, we will recover the generating functions of Tribonacci numbers and Chebychev polynomials of first and second kind. By making use of the operator defined in this paper, we give some new generating functions for the binary products of Tribonacci with some remarkable numbers and polynomials. The technique used here is based on the theory of the so-called symmetric functions.

It is shown that if $V\subseteq {\mathbb{F}}_p^{n\times \cdots \times n}$ is a subspace of $d$-tensors with dimension at least $t{n}^{d-1}$, then there is a subspace $W\subseteq V$ of dimension at least $t/(dr)–1$ whose nonzero elements all have analytic rank ${\mathrm{\Omega }}_{d,p}(r)$. As an application, we generalize a result of Altman on Szemerédi’s theorem with random differences.

Extensions of a set partition obtained by imposing bounds on the size of the parts and the coloring of some of the elements are examined. Combinatorial properties and the generating functions of some counting sequences associated with these partitions are established. Connections with Riordan arrays are presented.

Every set of natural numbers determines a generating function convergent for $q\in (-1,1)$ whose behavior as $q\to 1^{-}$ determines a germ. These germs admit a natural partial ordering that can be used to compare sets of natural numbers in a manner that generalizes both cardinality of finite sets and density of infinite sets. For any finite set $D$ of positive integers, call a set $S$ “$D$-avoiding” if no two elements of $S$ differ by an element of $D$. We study the problem of determining, for fixed $D$, all $D$- avoiding sets that are maximal in the germ order. In many cases, we can show that there is exactly one such set. We apply this to the study of one-dimensional packing problems.

We prove some combinatorial identities by an analytic method. We use the property that singular integrals of particular functions include binomial coefficients. In this paper, we prove combinatorial identities from the fact that two results of the particular function calculated as two ways using the residue theorem in the complex function theory are the same. These combinatorial identities are the generalization of a combinatorial identity that has been already obtained

Bargraphs are column convex polyominoes, where the lower edge lies on a horizontal axis. We consider the inner site-perimeter, which is the total number of cells inside the bargraph that have at least one edge in common with an outside cell and obtain the generating function that counts this statistic. From this we find the average inner perimeter and the asymptotic expression for this average as the semi-perimeter tends to infinity. We finally consider those bargraphs where the inner site-perimeter is exactly equal to the area of the bargraph.

Let $G$ be a graph with $n$ vertices, then the $c$-dominating matrix of $G$ is the square matrix of order $n$ whose $(i,j)$-entry is equal to 1 if the $i$-th and $j$-th vertex of $G$ are adjacent, is also equal to 1 if $i=j$, $v_i\in D_c$, and zero otherwise.

The $c$-dominating energy of a graph $G$, is defined as the sum of the absolute values of all eigenvalues of the $c$-dominating matrix.

The main purposes of this paper are to introduce the $c$-dominating Estrada index of a graph. Moreover, to obtain upper and lower bounds for the $c$-dominating Estrada index and investigate the relations between the $c$-dominating Estrada in