Exercise · ZW-BELL-NUMBER-002 · ★★☆ Intermediate
Which of these collections is a partition of \(\{1,2,3\}\)?
Choose one option.
Show answer
- B. \(\{\{1\},\{2\},\{3\}\}\)
Explanation
A partition requires nonempty, pairwise disjoint blocks whose union is the whole set; \(\{\{1\},\{2\},\{3\}\}\) satisfies all three. The others fail by overlapping at 2, containing an empty block, or leaving out an element.
Review the concept: Bell number
The Bell number \(B_n\) is the number of partitions of a set with \(n\) elements, that is, of ways to split it into nonempty disjoint subsets.
\[B_n=\sum_{k=0}^{n-1}\binom{n-1}{k}B_k\]
More practice problems: bell number
- ZW-BELL-NUMBER-001 What is the value of B_2?
- ZW-BELL-NUMBER-003 What is the value of B_0?
- ZW-BELL-NUMBER-004 How many partitions does a 3-element set have in total?
- ZW-BELL-NUMBER-005 Which expression relates B_n to the Stirling numbers of the second kind?
- ZW-BELL-NUMBER-006 Compute B_4 knowing that S(4,1)=1, S(4,2)=7, S(4,3)=6 and S(4,4)=1.
- ZW-BELL-NUMBER-007 Using B_0=1, B_1=1 and B_2=2, compute B_3 with Bell's recurrence.
- ZW-BELL-NUMBER-008 How many partitions of \1,2,3,4\ have exactly two blocks?
- ZW-BELL-NUMBER-009 Show that B_5=52 by summing S(5,k) for k=1,dots,5, knowing that S(5,1)=1, S(5,2)=15, S(5,3)=25, S(5…