Pascal's simplex

The Pascal's simplices are - similar to Pascal's triangle and Pascal's tetrahedron - geometric representations of multinomial. In Pascal's d- simplex, each number is the sum of d about their standing figures. The well-known from the Pascal's triangle and tetrahedron properties can be transferred to Pascalian simplices.

The concept

Can be present in each dimension ( natural number) A pascal cal simplex: each point with integer coordinates can be assigned to this the multinomial coefficient (the respective coordinates, is given by ). The envelope of the points which are not zero, then one-dimensional, in - direction form unrestricted, " simplex " ( usually a limited simplex ).

Properties

• The -th level of Pascal's simplex (ie the non-zero entries for a solid ) for can be derived from the level above (ie for ) calculate: . At the level of single entry is a, from which then recursively all other result.
• The sum of all the numbers in the nth ( d-1) - part simplex.
• The limiting ( d-1) - simplices are equal to the Pascal's ( d-1) - simplex. This can be expressed by.