Columns of Pascal's Triangle contain Simplicial Polytopic Numbers

From ProofWiki
Jump to navigation Jump to search

Theorem

The columns of Pascal's triangle contain the simplicial polytopic numbers:

Column $0$: repeated instances of number $1$
Column $1$: the (strictly) positive integers
Column $2$: the triangular numbers
Column $3$: the tetrahedral numbers
Column $4$: the pentatope numbers

and so on.


Proof


This theorem requires a proof.
In particular: Need to define the simplicial polytopic numbers and demonstrate that the $n$th simplicial polytopic number of dimension $m$ is $\dbinom m n$ or whatever the formula is.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Columns_of_Pascal%27s_Triangle_contain_Simplicial_Polytopic_Numbers&oldid=382657"