Interpretations for the q-binomial coefficient evaluated at −q are discussed. A (q, t)-version is established, including an instance of a cyclic sieving phenomenon involving unitary spaces. 1. The q-binomial The q-binomial coefficient is defined for integers k and n, with 0 ≤ k ≤ n, and an indeterminate q by

In this section we will discuss the Inclusion-Exclusion principle, with a few applications (including a formula for the chromatic polynomial of a graph), and then consider a wide generalisation of it due to Gian-Carlo Rota, involving the Möbius function of a partially ordered set. The q-binomial theorem gives a simple formula for the Möbius function of the lattice of subspaces of a vector space.

In this paper, binomial convolution in the frame of quantum calculus is studied for set Aq q-Appell sequences. It has been shown that sequences forms an Abelian group under operation convolution. Several properties structure have studied. A new definition polynomials associated with a random variable proposed. Scale transformation as well based on expectation respect to used present determinant...

Abstract We study the following $$(q-1)$$ ( q - 1 ) th convex ordering relation for q convolution power of difference probability distributions $$\mu $$ ? and $$\nu ? $$\begin{aligned} ...

The aim of this paper is to prove a Turán type theorem for random graphs. For 0 < γ ≤ 1 and graphs G and H, write G→γ H if any γ-proportion of the edges of G spans at least one copy of H in G. We show that for every l ≥ 2 and every fixed real 1/(l − 1) > δ > 0 almost every graph G in the binomial random graph model G(n, q), with q = q(n) ((logn)4/n)1/(l−1), satisfies G→(l−2)/(l−1)+δ Kl, where K...

In this article, we demonstrate how the Binomial Theorem in turn arises from a one-parameter generalization of the Sierpinski triangle. The connection between them is given by the sum-of-digits function, s(k), defined as the sum of the digits in the binary representation of k (see [1]). For example, s(3) = s(1·2+1·2) = 2. Towards this end, we begin with a well-known matrix formulation of Sierpi...