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.