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

n^th component

$\underset{n}{\overset{m}{\wedge }}=\underset{n}{\overset{m-1}{△}}$ consists of the coefficients of multinomial expansion of a polynomial with m terms raised to the power of n:

$|x{|}^n=\sum _{|k|=n}(\genfrac{}{}{0ex}{}{n}{k})x^k;x\in {\mathbb{ℝ}}^m,k\in {\mathbb{\mathbb{N}}}_0^m,n\in {\mathbb{\mathbb{N}}}_0,m\in \mathbb{\mathbb{N}}$

where $|x|={\sum }_{i=1}^m{x}_i,|k|={\sum }_{i=1}^m{k}_i,x^k={\prod }_{i=1}^m{x}_i^{k_i}$.

Example for ${\wedge }^4$

Pascal's 4-simplex (sequence A189225 in the 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).

Specific Pascal's simplices

Pascal's 1-simplex

${\wedge }^1$ is not known by any special name.

n^th component

$\underset{n}{\overset{1}{\wedge }}=\underset{n}{\overset{0}{△}}$ (a point) is the coefficient of multinomial expansion of a polynomial with 1 term raised to the power of n:

$(x_1{)}^n=\sum _{k_1=n}(\genfrac{}{}{0ex}{}{n}{k_1})x_1^{k_1};k_1,n\in {\mathbb{\mathbb{N}}}_0$

Arrangement of $\underset{n}{\overset{0}{△}}$

$(\genfrac{}{}{0ex}{}{n}{n})$

which equals 1 for all n.

Pascal's 2-simplex

${\wedge }^2$ is known as Pascal's triangle (sequence A007318 in the OEIS).

n^th component

$\underset{n}{\overset{2}{\wedge }}=\underset{n}{\overset{1}{△}}$ (a line) consists of the coefficients of binomial expansion of a polynomial with 2 terms raised to the power of n:

$(x_1+x_2{)}^n=\sum _{k_1+k_2=n}(\genfrac{}{}{0ex}{}{n}{k_1,k_2})x_1^{k_1}{x}_2^{k_2};k_1,k_2,n\in {\mathbb{\mathbb{N}}}_0$

Arrangement of $\underset{n}{\overset{1}{△}}$

$(\genfrac{}{}{0ex}{}{n}{n,0}),(\genfrac{}{}{0ex}{}{n}{n-1,1}),\cdots ,(\genfrac{}{}{0ex}{}{n}{1,n-1}),(\genfrac{}{}{0ex}{}{n}{0,n})$

Pascal's 3-simplex

${\wedge }^3$ is known as Pascal's tetrahedron (sequence A046816 in the OEIS).

n^th component

$\underset{n}{\overset{3}{\wedge }}=\underset{n}{\overset{2}{△}}$ (a triangle) consists of the coefficients of trinomial expansion of a polynomial with 3 terms raised to the power of n:

$(x_1+x_2+x_3{)}^n=\sum _{k_1+k_2+k_3=n}(\genfrac{}{}{0ex}{}{n}{k_1,k_2,k_3})x_1^{k_1}{x}_2^{k_2}{x}_3^{k_3};k_1,k_2,k_3,n\in {\mathbb{\mathbb{N}}}_0$

Arrangement of $\underset{n}{\overset{2}{△}}$

$\begin{array}{rl}(\genfrac{}{}{0ex}{}{n}{n,0,0})& ,(\genfrac{}{}{0ex}{}{n}{n-1,1,0}),\cdots \cdots ,(\genfrac{}{}{0ex}{}{n}{1,n-1,0}),(\genfrac{}{}{0ex}{}{n}{0,n,0})\\ (\genfrac{}{}{0ex}{}{n}{n-1,0,1})& ,(\genfrac{}{}{0ex}{}{n}{n-2,1,1}),\cdots \cdots ,(\genfrac{}{}{0ex}{}{n}{0,n-1,1})\\ & ⋮\\ (\genfrac{}{}{0ex}{}{n}{1,0,n-1})& ,(\genfrac{}{}{0ex}{}{n}{0,1,n-1})\\ (\genfrac{}{}{0ex}{}{n}{0,0,n})\end{array}$

Properties

Inheritance of components

$\underset{n}{\overset{m}{\wedge }}=\underset{n}{\overset{m-1}{△}}$ is numerically equal to each (m − 1)-face (there is m + 1 of them) of $\underset{n}{\overset{m}{△}}=\underset{n}{\overset{m+1}{\wedge }}$, or:

$\underset{n}{\overset{m}{\wedge }}=\underset{n}{\overset{m-1}{△}}\subset \underset{n}{\overset{m}{△}}=\underset{n}{\overset{m+1}{\wedge }}$

From this follows, that the whole ${\wedge }^m$ is (m + 1)-times included in ${\wedge }^{m+1}$, or:

${\wedge }^m\subset {\wedge }^{m+1}$

Example

        ${\wedge }^1$         ${\wedge }^2$        ${\wedge }^3$         ${\wedge }^4$

$\underset{0}{\overset{m}{\wedge }}$     1          1          1          1

$\underset{1}{\overset{m}{\wedge }}$     1         1 1        1 1        1 1  1
                             1          1

$\underset{2}{\overset{m}{\wedge }}$     1        1 2 1      1 2 1      1 2 1  2 2  1
                            2 2        2 2    2
                             1          1

$\underset{3}{\overset{m}{\wedge }}$     1       1 3 3 1    1 3 3 1    1 3 3 1  3 6 3  3 3  1
                           3 6 3      3 6 3    6 6    3
                            3 3        3 3      3
                             1          1

For more terms in the above array refer to (sequence A191358 in the OEIS)

Equality of sub-faces

Conversely, $\underset{n}{\overset{m+1}{\wedge }}=\underset{n}{\overset{m}{△}}$ is (m + 1)-times bounded by $\underset{n}{\overset{m-1}{△}}=\underset{n}{\overset{m}{\wedge }}$, or:

$\underset{n}{\overset{m+1}{\wedge }}=\underset{n}{\overset{m}{△}}\supset \underset{n}{\overset{m-1}{△}}=\underset{n}{\overset{m}{\wedge }}$

From this follows, that for given n, all i-faces are numerically equal in n^th components of all Pascal's (m > i)-simplices, or:

$\underset{n}{\overset{i+1}{\wedge }}=\underset{n}{\overset{i}{△}}\subset \underset{n}{\overset{m>i}{△}}=\underset{n}{\overset{m>i+1}{\wedge }}$

Example

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):

2-simplex   1-faces of 2-simplex         0-faces of 1-face

 1 3 3 1    1 . . .  . . . 1  1 3 3 1    1 . . .   . . . 1
  3 6 3      3 . .    . . 3    . . .
   3 3        3 .      . 3      . .
    1          1        1        .

Also, for all m and all n:

$1=\underset{n}{\overset{1}{\wedge }}=\underset{n}{\overset{0}{△}}\subset \underset{n}{\overset{m-1}{△}}=\underset{n}{\overset{m}{\wedge }}$

Number of coefficients

For the n^th component ((m − 1)-simplex) of Pascal's m-simplex, the number of the coefficients of multinomial expansion it consists of is given by:

$(\genfrac{}{}{0ex}{}{(n-1)+(m-1)}{(m-1)})+(\genfrac{}{}{0ex}{}{n+(m-2)}{(m-2)})=(\genfrac{}{}{0ex}{}{n+(m-1)}{(m-1)})=\left((\genfrac{}{}{0ex}{}{m}{n})\right),$

(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 n^th component ((m − 2)-simplex) of Pascal's (m − 1)-simplex, or by a number of all possible partitions of an n^th power among m exponents.

Example

Number of coefficients of n^th component ((m − 1)-simplex) of Pascal's m-simplex

m-simplex	nth component	n = 0	n = 1	n = 2	n = 3	n = 4	n = 5
1-simplex	0-simplex	1	1	1	1	1	1
2-simplex	1-simplex	1	2	3	4	5	6
3-simplex	2-simplex	1	3	6	10	15	21
4-simplex	3-simplex	1	4	10	20	35	56
5-simplex	4-simplex	1	5	15	35	70	126
6-simplex	5-simplex	1	6	21	56	126	252

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

Symmetry

An n^th component ((m − 1)-simplex) of Pascal's m-simplex has the (m!)-fold spatial symmetry.

Geometry

Orthogonal axes $k_1...k_m$ in m-dimensional space, vertices of component at n on each axe, the tip at [0,...,0] for $n=0$.

Numeric construction

Wrapped n-th power of a big number gives instantly the n-th component of a Pascal's simplex.

${\left|b^{dp}\right|}^n=\sum _{|k|=n}(\genfrac{}{}{0ex}{}{n}{k})b^{dp\cdot k};b,d\in \mathbb{\mathbb{N}},n\in {\mathbb{\mathbb{N}}}_0,k,p\in {\mathbb{\mathbb{N}}}_0^m,p:p_1=0,p_i=(n+1{)}^{i-2}$

where $b^{dp}=(b^{d{p}_1},\cdots ,b^{d{p}_m})\in {\mathbb{\mathbb{N}}}^m,p\cdot k={\sum }_{i=1}^m{p}_i{k}_i\in {\mathbb{\mathbb{N}}}_0$.