# Stirling numbers of a second kind, from the hint in the book, look correct?

1. Nov 23, 2006

### mr_coffee

Hello everyone i was wondering if someone could check to see if this looks correct or not. The question is:

If X is a set with n elements and Y is a set with m elements, express the number of onto functions from X and Y using Stirling numbers of the second kind. Justify your answer.

note: the x in the latex generated graphics means multiply not the variable 'x'.

for more info on stirling numbers of a second kind look here:
http://mathworld.wolfram.com/StirlingNumberoftheSecondKind.html

Well the back of the book gave the following hint:
The number of onto functions from X = {x_1,x_2,x_3,x_4} to Y = {y_1,y_2,y_3} is $$S_{4,3}x3!$$ because the construction of an onto function can be thought of as a 2-step processs where
step 1: is to choose a partion of X into 3 subsets and
step 2: is to choose, for each subset of the partition, an element of Y for the elements of the subset to be sent to.

Well in the question X has n elements in the hint 4 elements.
Y has m elements in the question and in the hint it has 3 elements.
So it looks like the answer would be:
$$S_{n,m} x m!$$
does that look right to you?

thanks!

Last edited: Nov 23, 2006