Pascal's simplex

In mathematics, Pascal's simplex is a generalisation of Pascal's triangle into arbitrary number of dimensions, based on the multinomial theorem.

Generic Pascal's m-simplex

Let m (m > 0) be a number of terms of a polynomial and n (n ≥ 0) be a power the polynomial is raised to.

Let $\wedge^m$ denote a Pascal's m-simplex. Each Pascal's m-simplex is a semi-infinite object, which consists of an infinite series of its components.

Let $\wedge^m_n$ denote its n^th component, itself a finite (m − 1)-simplex with the edge length n, with a notational equivalent $\vartriangle^{m-1}_n$ .

n^th component

$\wedge^m_n = \vartriangle^{m-1}_n$ consists of the coefficients of multinomial expansion of a polynomial with m terms raised to the power of n:

where $\textstyle|x|=\sum_{i=1}^m{x_i},\ |k|=\sum_{i=1}^m{k_i},\ x^k=\prod_{i=1}^m{x_i^{k_i}}$ .

Example for $\wedge^4$

Pascal's 4-simplex (sequence A189225 in OEIS), sliced along the k₄. All points of the same color belong to the same n-th component, from red (for n = 0) to blue (for n = 3).