Exercise · ZW-STIRLING-NUMBER-OF-THE-FIRST-KIND-010 · ★★★ Advanced
A claim says \(c(n,k)=c(n-1,k-1)+c(n-1,k)\) always holds. For which pair \((n,k)\) does the equality actually hold?
Choose one option.
Show answer
- B. \((2,1)\)
Explanation
For \((2,1)\) both sides equal 1. For \((4,1)\), \((3,1)\) and \((3,2)\) the right side gives 2, 1 and 2, while \(c(n,k)\) equals 6, 2 and 3.
Review the concept: Stirling number of the first kind
The Stirling number of the first kind \(c(n,k)\) counts the permutations of \(n\) elements that have exactly \(k\) cycles.
\[x(x+1)\cdots(x+n-1)=\sum_{k=0}^{n} c(n,k)\,x^{k}\]
More practice problems: stirling number of the first kind
- ZW-STIRLING-NUMBER-OF-THE-FIRST-KIND-001 What does the number c(n,k) count?
- ZW-STIRLING-NUMBER-OF-THE-FIRST-KIND-002 The permutations of \1,2,3\ with a single cycle are the 3-cycles. What is c(3,1)?
- ZW-STIRLING-NUMBER-OF-THE-FIRST-KIND-003 What is c(n,n) for every n≥ 1?
- ZW-STIRLING-NUMBER-OF-THE-FIRST-KIND-004 What is c(n,0) for n≥ 1?
- ZW-STIRLING-NUMBER-OF-THE-FIRST-KIND-005 What is c(n,1)?
- ZW-STIRLING-NUMBER-OF-THE-FIRST-KIND-006 Use c(4,1)=6, c(4,2)=11, c(4,3)=6 and c(4,4)=1. What is the sum of the row n=4?
- ZW-STIRLING-NUMBER-OF-THE-FIRST-KIND-007 With c(4,2)=11 and c(4,1)=6, use c(n,k)=(n-1)c(n-1,k)+c(n-1,k-1) to find c(5,2).
- ZW-STIRLING-NUMBER-OF-THE-FIRST-KIND-008 Expand x(x+1)(x+2)=∑_k c(3,k) x^k. What is c(3,2)?