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.

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

⁠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 ⁠

⁠mn denote its nth component, itself a finite (m − 1)-simplex with the edge length n, with a notational equivalent

{\displaystyle \vartriangle _{n}^{m-1}}

△

consists of the coefficients of multinomial expansion of a polynomial with m terms raised to the power of n: where

Pascal's 4-simplex (sequence A189225 in the OEIS), sliced along the k4.

All points of the same color belong to the same nth component, from red (for n = 0) to blue (for n = 3).

⁠1 is not known by any special name.

(a point) is the coefficient of multinomial expansion of a polynomial with 1 term raised to the power of n: which equals 1 for all n.

is known as Pascal's triangle (sequence A007318 in the OEIS).

(a line) consists of the coefficients of binomial expansion of a polynomial with 2 terms raised to the power of n:

is known as Pascal's tetrahedron (sequence A046816 in the OEIS).

(a triangle) consists of the coefficients of trinomial expansion of a polynomial with 3 terms raised to the power of n:

is numerically equal to each (m − 1)-face (there is m + 1 of them) of

is (m + 1)-times included in

, or: For more terms in the above array refer to (sequence A191358 in the OEIS) Conversely,

is (m + 1)-times bounded by

, or: From this follows, that for given n, all i-faces are numerically equal in nth components of all Pascal's (m > i)-simplices, or: The 3rd component (2-simplex) of Pascal's 3-simplex is bounded by 3 equal 1-faces (lines).

Each 1-face (line) is bounded by 2 equal 0-faces (vertices): Also, for all m and all n: For the nth component ((m − 1)-simplex) of Pascal's m-simplex, the number of the coefficients of multinomial expansion it consists of is given by: (where the latter is the multichoose notation).

We can see this either as a sum of the number of coefficients of an (n − 1)th component ((m − 1)-simplex) of Pascal's m-simplex with the number of coefficients of an nth component ((m − 2)-simplex) of Pascal's (m − 1)-simplex, or by a number of all possible partitions of an nth power among m exponents.

The terms of this table comprise a Pascal triangle in the format of a symmetric Pascal matrix.

An nth component ((m − 1)-simplex) of Pascal's m-simplex has the (m!

)-fold spatial symmetry.

Orthogonal axes k1, ..., km in m-dimensional space, vertices of component at n on each axis, the tip at [0, ..., 0] for n = 0.

Wrapped nth power of a big number gives instantly the nth component of a Pascal's simplex.

{\displaystyle \textstyle b^{dp}=(b^{dp_{1}},\cdots ,b^{dp_{m}})\in \mathbb {N} ^{m},\ p\cdot k={\sum _{i=1}^{m}{p_{i}k_{i}}}\in \mathbb {N} _{0}}