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 ⋀⁴

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).

Pascal's 1-simplex

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

Pascal's 2-simplex

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

Pascal's 3-simplex

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

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

More information , ...

	${\displaystyle \wedge ^{1}}$	${\displaystyle \wedge ^{2}}$	${\displaystyle \wedge ^{3}}$	${\displaystyle \wedge ^{4}}$
${\displaystyle \wedge _{0}^{m}}$	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 2 2 1	1 2 1 2 2 1 2 2 2 1
${\displaystyle \wedge _{3}^{m}}$	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

Close

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:

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

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        .

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:

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

(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 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.

Symmetry

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

Geometry

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

Numeric construction

Wrapped nth power of a big number gives instantly the nth 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 ${\displaystyle \textstyle b^{dp}=(b^{dp_{1}},\cdots ,b^{dp_{m}})\in \mathbb {N} ^{m},\ p\cdot k={\sum _{i=1}^{m}{p_{i}k_{i}}}\in \mathbb {N} _{0}}$.