# Pascal's simplex

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.

## To the subject

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). ${\ displaystyle d}$ ${\ displaystyle d \ geq 1}$ ${\ displaystyle {\ binom {n} {k_ {1}, \ ldots, k_ {d}}}}$ ${\ displaystyle n, k_ {1}, \ ldots, k_ {d-1}}$ ${\ displaystyle k_ {d}}$ ${\ displaystyle n-k_ {1} - \ ldots -k_ {d-1}}$ ${\ displaystyle d}$ ${\ displaystyle n}$ ## properties

• 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 .${\ displaystyle n}$ ${\ displaystyle n}$ ${\ displaystyle n> 0}$ ${\ displaystyle n-1}$ ${\ displaystyle {\ binom {n} {k_ {1}, \ ldots, k_ {d}}} = {\ binom {n-1} {k_ {1} -1, \ ldots, k_ {d}}} + {\ binom {n-1} {k_ {1}, k_ {2} -1, \ ldots, k_ {d}}} + \ ldots + {\ binom {n-1} {k_ {1}, \ ldots, k_ {d} -1}}}$ ${\ displaystyle 0}$ ${\ displaystyle 1}$ • The sum of all the numbers in the nth (d-1) -sub-simplex is .${\ displaystyle d ^ {n-1}}$ • The limiting (d-1) simplex is equal to Pascal's (d-1) simplex. This can be expressed by .${\ displaystyle {\ binom {n} {k_ {1}, ..., k_ {d-1}, 0}} = {\ binom {n} {k_ {1}, ..., k_ {d- 1}}}}$ ## Individual evidence

1. 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.