10 Jan
2017
10 Jan
'17
4:58 a.m.
* The formula is easily proved by induction, since e.g. the nth tetrahedral number is just a stack of the first n triangular numbers. But is there a geometrical reason that it should be the number of size-k subsets of a set of n+k-1 objects?
Yes, the nth k-simplicial number counts the number of (k+1)-tuples: (a_0, a_1, a_2, ..., a_k) where each a_i is in {0, 1, 2, ...} and the sum is exactly n-1. [This is using the standard embedding of a regular k-simplex as the convex span of (k+1) orthonormal vectors.] Now, for instance, map the tuple (3, 2, 7, 4) to the string: ooo|oo|ooooooo|oooo which contains k copies of '|' and n-1 copies of 'o'. The result follows. Best wishes, Adam P. Goucher