# 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 ${\displaystyle \wedge ^{m}}$ 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 ${\displaystyle \wedge _{n}^{m}}$ denote its nth component, itself a finite (m − 1)-simplex with the edge length n, with a notational equivalent ${\displaystyle \vartriangle _{n}^{m-1}}$.

### nth component

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

${\displaystyle |x|^{n}=\sum _{|k|=n}{{\binom {n}{k}}x^{k}};\ \ x\in \mathbb {R} ^{m},\ k\in \mathbb {N} _{0}^{m},\ n\in \mathbb {N} _{0},\ m\in \mathbb {N} }$

where ${\displaystyle \textstyle |x|=\sum _{i=1}^{m}{x_{i}},\ |k|=\sum _{i=1}^{m}{k_{i}},\ x^{k}=\prod _{i=1}^{m}{x_{i}^{k_{i}}}}$.

### Example for ${\displaystyle \wedge ^{4}}$

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

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

#### nth component

${\displaystyle \wedge _{n}^{1}=\vartriangle _{n}^{0}}$ (a point) is the coefficient of multinomial expansion of a polynomial with 1 term raised to the power of n:

${\displaystyle (x_{1})^{n}=\sum _{k_{1}=n}{n \choose k_{1}}x_{1}^{k_{1}};\ \ k_{1},n\in \mathbb {N} _{0}}$
##### Arrangement of ${\displaystyle \vartriangle _{n}^{0}}$
${\displaystyle \textstyle {n \choose n}}$

which equals 1 for all n.

### Pascal's 2-simplex

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

#### nth component

${\displaystyle \wedge _{n}^{2}=\vartriangle _{n}^{1}}$ (a line) consists of the coefficients of binomial expansion of a polynomial with 2 terms raised to the power of n:

${\displaystyle (x_{1}+x_{2})^{n}=\sum _{k_{1}+k_{2}=n}{n \choose k_{1},k_{2}}x_{1}^{k_{1}}x_{2}^{k_{2}};\ \ k_{1},k_{2},n\in \mathbb {N} _{0}}$
##### Arrangement of ${\displaystyle \vartriangle _{n}^{1}}$
${\displaystyle \textstyle {n \choose n,0},{n \choose n-1,1},\cdots ,{n \choose 1,n-1},{n \choose 0,n}}$

### Pascal's 3-simplex

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

#### nth component

${\displaystyle \wedge _{n}^{3}=\vartriangle _{n}^{2}}$ (a triangle) consists of the coefficients of trinomial expansion of a polynomial with 3 terms raised to the power of n:

${\displaystyle (x_{1}+x_{2}+x_{3})^{n}=\sum _{k_{1}+k_{2}+k_{3}=n}{n \choose k_{1},k_{2},k_{3}}x_{1}^{k_{1}}x_{2}^{k_{2}}x_{3}^{k_{3}};\ \ k_{1},k_{2},k_{3},n\in \mathbb {N} _{0}}$
##### Arrangement of ${\displaystyle \vartriangle _{n}^{2}}$
{\displaystyle {\begin{aligned}\textstyle {n \choose n,0,0}&,\textstyle {n \choose n-1,1,0},\cdots \cdots ,{n \choose 1,n-1,0},{n \choose 0,n,0}\\\textstyle {n \choose n-1,0,1}&,\textstyle {n \choose n-2,1,1},\cdots \cdots ,{n \choose 0,n-1,1}\\&\vdots \\\textstyle {n \choose 1,0,n-1}&,\textstyle {n \choose 0,1,n-1}\\\textstyle {n \choose 0,0,n}\end{aligned}}}

## Properties

### Inheritance of components

${\displaystyle \wedge _{n}^{m}=\vartriangle _{n}^{m-1}}$ is numerically equal to each (m − 1)-face (there is m + 1 of them) of ${\displaystyle \vartriangle _{n}^{m}=\wedge _{n}^{m+1}}$, or:

${\displaystyle \wedge _{n}^{m}=\vartriangle _{n}^{m-1}\subset \ \vartriangle _{n}^{m}=\wedge _{n}^{m+1}}$

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

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

#### Example

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

${\displaystyle \wedge _{0}^{m}}$     1          1          1          1

${\displaystyle \wedge _{1}^{m}}$     1         1 1        1 1        1 1  1
1          1

${\displaystyle \wedge _{2}^{m}}$     1        1 2 1      1 2 1      1 2 1  2 2  1
2 2        2 2    2
1          1

${\displaystyle \wedge _{3}^{m}}$     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, ${\displaystyle \wedge _{n}^{m+1}=\vartriangle _{n}^{m}}$ is (m + 1)-times bounded by ${\displaystyle \vartriangle _{n}^{m-1}=\wedge _{n}^{m}}$, or:

${\displaystyle \wedge _{n}^{m+1}=\vartriangle _{n}^{m}\supset \vartriangle _{n}^{m-1}=\wedge _{n}^{m}}$

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

${\displaystyle \wedge _{n}^{i+1}=\vartriangle _{n}^{i}\subset \vartriangle _{n}^{m>i}=\wedge _{n}^{m>i+1}}$

#### 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:

${\displaystyle 1=\wedge _{n}^{1}=\vartriangle _{n}^{0}\subset \vartriangle _{n}^{m-1}=\wedge _{n}^{m}}$

### Number of coefficients

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:

${\displaystyle {(n-1)+(m-1) \choose (m-1)}+{n+(m-2) \choose (m-2)}={n+(m-1) \choose (m-1)},}$

that is, either by 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.

#### Example

Number of coefficients of nth 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

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

### Symmetry

(An nth 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.)

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

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