The Pascal's simplices are - analogous to Pascal's triangle and Pascal's tetrahedron - geometric representations of multinomial . In Pascal's d- simplex , every number is the sum of d numbers above it. The properties known from Pascal's triangle and tetrahedron can be transferred to Pascal's simplices.
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A Pascal simplex can be imagined in any dimension ( natural number): the multinomial coefficient can be assigned to each point with integer coordinates ( these are the respective coordinates, results from ). The envelope of the points that are not zero, then form a -dimensional, in direction un limited , "Simplex" (usually is a simplex limited).
- The -th level of a Pascalian simplex (i.e., the non-zero entries for a solid.. ) For can be calculated from the level above (i.e., for.. Calculate) . The only entry on the level is one , from which all others can be recursively derived .
- The sum of all the numbers in the nth (d-1) -sub-simplex is .
- The limiting (d-1) simplex is equal to Pascal's (d-1) simplex. This can be expressed by .
- Peter Hilton, Derek Holton, Jean Pedersen: Mathematical Vistas. From a Room with Many Windows. Springer, New York a. a. 2002. ISBN 978-0-387-95064-8 . Pp. 188-190.