Pascal's simplex

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

Let {{tmath|\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 {{tmath|\wedge}}{{su|lh=0.8em|p=m|b=n}} denote its nth component, itself a finite {{nowrap|(m − 1)}}-simplex with the edge length n, with a notational equivalent $\backslash vartriangle^\{m-1\}\_n$.

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

: $|x|^n=\backslash sum\_\{|k|=n\}\{\backslash binom\{n\}\{k\}x^k\};\backslash \; \backslash \; x\backslash in\backslash mathbb\{R\}^m,\backslash \; k\backslash in\backslash mathbb\{N\}^m\_0,\backslash \; n\backslash in\backslash mathbb\{N\}\_0,\backslash \; m\backslash in\backslash mathbb\{N\}$

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

Pascal's 4-simplex {{OEIS|id=A189225}}, sliced along the k₄. All points of the same color belong to the same nth component, from red (for {{nowrap|1=n = 0}}) to blue (for {{nowrap|1=n = 3}}).

File:Simplex-4.svg

{{tmath|\wedge}}¹ is not known by any special name.

File:Simplex-1.svg

$\backslash wedge^1\_n\; =\; \backslash vartriangle^0\_n$ (a point) is the coefficient of multinomial expansion of a polynomial with 1 term raised to the power of n:

: $(x\_1)^n\; =\; \backslash sum\_\{k\_1=n\}\; \{n\; \backslash choose\; k\_1\}\; x\_1^\{k\_1\};\backslash \; \backslash \; k\_1,\; n\; \backslash in\; \backslash mathbb\{N\}\_0$

: $\backslash textstyle\; \{n\; \backslash choose\; n\}$

which equals 1 for all n.

$\backslash wedge^2$ is known as Pascal's triangle {{OEIS|id=A007318}}.

File:Simplex-2.svg

$\backslash wedge^2\_n\; =\; \backslash vartriangle^1\_n$ (a line) consists of the coefficients of binomial expansion of a polynomial with 2 terms raised to the power of n:

: $(x\_1\; +\; x\_2)^n\; =\; \backslash sum\_\{k\_1+k\_2=n\}\; \{n\; \backslash choose\; k\_1,\; k\_2\}\; x\_1^\{k\_1\}\; x\_2^\{k\_2\};\backslash \; \backslash \; k\_1,\; k\_2,\; n\; \backslash in\; \backslash mathbb\{N\}\_0$

: $\backslash textstyle\; \{n\; \backslash choose\; n,\; 0\},\; \{n\; \backslash choose\; n\; -\; 1,\; 1\},\; \backslash ldots,\; \{n\; \backslash choose\; 1,\; n\; -\; 1\},\; \{n\; \backslash choose\; 0,\; n\}$

$\backslash wedge^3$ is known as Pascal's tetrahedron {{OEIS|id=A046816}}.

File:Simplex-3.svg

$\backslash wedge^3\_n\; =\; \backslash vartriangle^2\_n$ (a triangle) consists of the coefficients of trinomial expansion of a polynomial with 3 terms raised to the power of n:

: $(x\_1\; +\; x\_2\; +\; x\_3)^n\; =\; \backslash sum\_\{k\_1+k\_2+k\_3=n\}\; \{n\; \backslash choose\; k\_1,\; k\_2,\; k\_3\}\; x\_1^\{k\_1\}\; x\_2^\{k\_2\}\; x\_3^\{k\_3\};\backslash \; \backslash \; k\_1,\; k\_2,\; k\_3,\; n\; \backslash in\; \backslash mathbb\{N\}\_0$

: $$

\begin{align}

\textstyle {n \choose n, 0, 0} &, \textstyle {n \choose n - 1, 1, 0}, \ldots\ldots, {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}, \ldots\ldots, {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{align}

$\backslash wedge^m\_n\; =\; \backslash vartriangle^\{m-1\}\_n$ is numerically equal to each {{nowrap|(m − 1)}}-face (there is {{nowrap|m + 1}} of them) of $\backslash vartriangle^m\_n\; =\; \backslash wedge^\{m+1\}\_n$, or:

: $\backslash wedge^m\_n\; =\; \backslash vartriangle^\{m-1\}\_n\; \backslash subset\backslash \; \backslash vartriangle^m\_n\; =\; \backslash wedge^\{m+1\}\_n$

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

: $\backslash wedge^m\; \backslash subset\; \backslash wedge^\{m+1\}$

 1 

   1

   1

   1

 1 

  1 1

  1 1

1

  1 1        1

1

 1 

 1 2 1

 1 2 1

2 2
1

 1 2 1      2 2      1

2 2        2
1

 1 

1 3 3 1

1 3 3 1

3 6 3
3 3
1

1 3 3 1    3 6 3    3 3    1

3 6 3      6 6      3
3 3        3
1

For more terms in the above array refer to {{OEIS|id=A191358}}

Conversely, $\backslash wedge^\{m+1\}\_n\; =\; \backslash vartriangle^m\_n$ is ({{nowrap|m + 1)}}-times bounded by $\backslash vartriangle^\{m-1\}\_n\; =\; \backslash wedge^m\_n$, or:

: $\backslash wedge^\{m+1\}\_n\; =\; \backslash vartriangle^m\_n\; \backslash supset\; \backslash vartriangle^\{m-1\}\_n\; =\; \backslash wedge^m\_n$

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

: $\backslash wedge^\{i+1\}\_n\; =\; \backslash vartriangle^i\_n\; \backslash subset\; \backslash vartriangle^\{m>i\}\_n\; =\; \backslash wedge^\{m>i+1\}\_n$

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:

: $1\; =\; \backslash wedge^1\_n\; =\; \backslash vartriangle^0\_n\; \backslash subset\; \backslash vartriangle^\{m-1\}\_n\; =\; \backslash wedge^m\_n$

For the nth component ({{nowrap|(m − 1)}}-simplex) of Pascal's m-simplex, the number of the coefficients of multinomial expansion it consists of is given by:

: $\{(n-1)\; +\; (m-1)\; \backslash choose\; (m-1)\}\; +\; \{n\; +\; (m\; -\; 2)\; \backslash choose\; (m\; -\; 2)\}\; =\; \{n\; +\; (m\; -\; 1)\; \backslash choose\; (m\; -\; 1)\}\; =\; \backslash left(\backslash !\backslash !\; \backslash binom\{m\}\{n\}\; \backslash !\backslash !\backslash right),$

(where the latter is the multichoose notation). We can see this either as a sum of the number of coefficients of an {{nowrap|(n − 1)}}th component ({{nowrap|(m − 1)}}-simplex) of Pascal's m-simplex with the number of coefficients of an nth component ({{nowrap|(m − 2)}}-simplex) of Pascal's {{nowrap|(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 ({{nowrap|(m − 1)}}-simplex) of Pascal's m-simplex has the (m!)-fold spatial symmetry.

Orthogonal axes k₁, ..., k_m in m-dimensional space, vertices of component at n on each axis, the tip at {{nowrap|[0, ..., 0]}} for {{nowrap|1=n = 0}}.

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

: $\backslash left|b^\{dp\}\backslash right|^n=\backslash sum\_\{|k|=n\}\{\backslash binom\{n\}\{k\}b^\{dp\backslash cdot\; k\}\};\backslash \; \backslash \; b,d\backslash in\backslash mathbb\{N\},\backslash \; n\backslash in\backslash mathbb\{N\}\_0,\backslash \; k,p\backslash in\backslash mathbb\{N\}\_0^m,\backslash \; p:\backslash \; p\_1=0,\; p\_i=(n+1)^\{i-2\}$

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

Category:Factorial and binomial topics

Category:Triangles of numbers