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 $S_{\lambda }(T,H,F)$.