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

nth component

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

where .

Example for

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

is not known by any special name.

nth component

(a point) is the coefficient of multinomial expansion of a polynomial with 1 term raised to the power of n:

Arrangement of

which equals 1 for all n.

Pascal's 2-simplex

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

nth component

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

Arrangement of

Pascal's 3-simplex

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

nth component

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

Arrangement of

Properties

Inheritance of components

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

From this follows, that the whole is (m + 1)-times included in , or:

Example

                                  

     1          1          1          1

     1         1 1        1 1        1 1  1
                             1          1

     1        1 2 1      1 2 1      1 2 1  2 2  1
                            2 2        2 2    2
                             1          1

     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, 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:

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:

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:

(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.

Example

Number of coefficients of nth component ((m − 1)-simplex) of Pascal's m-simplex
m-simplexnth componentn = 0n = 1n = 2n = 3n = 4n = 5
1-simplex 0-simplex 111111
2-simplex 1-simplex 123456
3-simplex 2-simplex 136101521
4-simplex 3-simplex 1410203556
5-simplex 4-simplex 15153570126
6-simplex 5-simplex 162156126252

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 in m-dimensional space, vertices of component at n on each axe, the tip at [0,...,0] for .

Numeric construction

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

where .

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