1

Show that there are $\sum_{k=1}^{n}S_{n, k}$ equivalence relations on an n-element set. The numbers Sn,k are Stirling numbers of the second kind.

I am learning discrete mathematics from different books and I have come across the problem above and I couldn't find a way to prove it.

Renzo Leonardo Gallegos Vilca
  • 11
  • 1
    There is a bijection bijection between partitions of an $n$-element set and equivalence relations on that set; this should be a familiar fact, but you can review it at this answer if necessary. $S_{n,k}$ is the number of partitions of an $n$-element set into $k$ parts; every partition of that set must have $k$ parts for some $k\in{1,\ldots,n}$, so summing $S_{n,k}$ over those values of $k$ gives you the total number of partitions of the set. – Brian M. Scott Mar 31 '21 at 05:01

1 Answers1

0

$S_{n, k}$ is the number of surjections from the set of size $n$ to one of size $k.$ Such a surjection corresponds to an equivalence relation with $k$ classes. I think you can take it from there.

Igor Rivin
  • 26,372