This paper introduces a variation on the binomial coefficient that depends on a poset and interpolates between q-binomials and 1-binomials: a total order gives the usual q-binomial, and a poset with no relations gives the usual binomial coefficient. These coefficients arise naturally in the study of supercharacters of the finite groups of unipotent upper-triangular matrices, whose representatio...

We present a new approach to evaluating combinatorial sums by using finite differences. Let {ak}k=0 and {bk}k=0 be sequences with the property that ∆bk = ak for k ≥ 0. Let gn = ∑n k=0 ( n k ) ak, and let hn = ∑n k=0 ( n k ) bk. We derive expressions for gn in terms of hn and for hn in terms of gn. We then extend our approach to handle binomial sums of the form ∑n k=0 ( n k ) (−1)ak, ∑ k ( n 2k ...

Let n = ∏ p p νp(n) denote the canonical factorization of n ∈ N. The binomial convolution of arithmetical functions f and g is defined as (f ◦g)(n) = ∑ d|n (∏ p (νp(n) νp(d) )) f(d)g(n/d), where ( a b ) is the binomial coefficient. We provide properties of the binomial convolution. We study the Calgebra (A,+, ◦,C), characterizations of completely multiplicative functions, Selberg multiplicative...

Let N(t) denote the number of ways of writing t as a binomial coefficient. We show that N(t) = O ( (log t)(log log log t) (log log t)3 ) .

We first pose a Sudoku-type puzzle, involving lattice points in Pascal’s Triangle, and lines passing through them. Then, through the use of an expanded notation for the binomial coefficient (

For non-negative integers $k\leq n$, we prove a combinatorial identity for the $p$-binomial coefficient $\binom{n}{k}_p$ based on abelian p-groups. A purely proof of this is not known. While proving identity, $r\in \mathbb{N}\cup\{0\},s\in \mathbb{N}$ and $p$ prime, present formula number subgroups $\mathbb{Z}^s$ finite index $p^r$ with quotient isomorphic to $p$-group type $\underline{\lambda}...

INTRODUCTION With the renewed drive towards malaria elimination, there is a need for improved surveillance tools. While time series analysis is an important tool for surveillance, prediction and for measuring interventions' impact, approximations by commonly used Gaussian methods are prone to inaccuracies when case counts are low. Therefore, statistical methods appropriate for count data are re...