Let q be regarded as either a complex number q ∈ C or a p-adic number q ∈ Cp. If q ∈ C, then we always assume |q| < 1. If q ∈ Cp, we normally assume |1− q|p < p − 1 p−1 , which implies that q = exp(x log q) for |x|p ≤ 1. Here, | · |p is the p-adic absolute value in Cp with |p|p = 1 p . The q-basic natural number are defined by [n]q = 1−q 1−q = 1 + q + · · · + q , ( n ∈ N), and q-factorial are a...

We give new q-(1+q)-analogue of the Gaussian coefficient, also know as the q-binomial which, like the original q-binomial [ n k ] q , is symmetric in k and n− k. We show this q-(1 + q)-binomial is more compact than the one discovered by Fu, Reiner, Stanton and Thiem. Underlying our q-(1 + q)-analogue is a Boolean algebra decomposition of an associated poset. These ideas are extended to the Birk...

‎the number of factorizations of a finite abelian group as the product of two subgroups is computed in two different ways and a combinatorial identity involving gaussian binomial coefficients is presented‎.

For a prime p and nonnegative integers n, k, consider the set A (p) n,k = {x ∈ [0, 1, ..., n] : p|| ( n x ) }. Let the expansion of n + 1 in base p be n + 1 = α0p ν + α1p ν−1 + · · · + αν , where 0 ≤ αi ≤ p − 1, i = 0, . . . , ν. Then n is called a binomial coefficient predictor in base p(p-BCP), if |A (p) n,k| = αkp , k = 0, 1, . . . , ν. We give a full description of the p-BCP’s in every base p.

The coefficient of determination R2 quantifies the proportion of variance explained by a statistical model and is an important summary statistic of biological interest. However, estimating R2 for generalized linear mixed models (GLMMs) remains challenging. We have previously introduced a version of R2 that we called [Formula: see text] for Poisson and binomial GLMMs, but not for other distribut...