Let’s count – Partitions

Do your best in trying to solve the following problems. It is normal that this step will take some time, we recommend to perseverate and wait before looking at the solution of Exercise 1 in the video or of both in the pdf below.

Exercise 1.

Write how many ways there are to subdivide 4 people into 3 (nonempty) distinguishable groups (A, B, C) in two different ways:
a) by means of a known formula;
b) Count first partitions into 3 nonempty sets and then “label’’ the sets.

Exercise 2.

Recurrence formula for the Stirling numbers of II kind (solved in the pdf).

Let (nge 1, kge n+1). Prove that [Biglbracebegin{matrix}k newline nend{matrix}Bigrbrace=Biglbracebegin{matrix}k-1 newline n-1end{matrix}Bigrbrace+nBiglbracebegin{matrix}k – 1newline nend{matrix}Bigrbrace.] Deduce the values of the Stirling numbers of II type (Biglbracebegin{matrix}k newline nend{matrix}Bigrbrace) for (1le nle kle 4).