Enumerative Combinatorics · Integer partitions and Ferrers diagrams
An integer partition of n is a finite weakly decreasing sequence of positive integers…
Problem
An integer partition of \(n\) is a finite weakly decreasing sequence of positive integers with sum \(n\). Its entries are its parts, and its length is its number of parts. The empty sequence is the unique partition of zero. List every partition of 6, group the list by length, and explain a systematic procedure that makes the list exhaustive without repetitions. Why are \(4+2\) and \(2+4\) not two different partitions? Why must zero entries be excluded from the definition of a part?
Hint
Begin with the possible largest parts.
Check your work
Work the problem yourself first. Then open it in Training to check your answer and read the full worked solution.
The answer check and full solution for this problem come with ProofAnvil Practice membership ($19 USD monthly). See membership. Or start with the free Enumerative Combinatorics sample problem: Try the free sample problem.
More Enumerative Combinatorics practice problems
- A group has six distinct volunteersCounting principles, bijections, and the pigeonhole principle
- Among n distinct volunteers, a specified set of r volunteers is qualified to serve as…Counting principles, bijections, and the pigeonhole principle
- Let a,b ≥ 0 and let k be an integerBinomial and multinomial coefficients, identities, and lattice paths
- Let a,b ≥ 0 and h ≥ 1 be integers with b-a<hBinomial and multinomial coefficients, identities, and lattice paths
- For n ≥ 1, a positive composition of n is an ordered list of positive integers summing…Compositions, multisets, and distributing objects into boxes
- Among thirty labeled projects, eighteen use method A, fourteen use method B, and twelve…Inclusion-exclusion, derangements, and surjections
- Let S(n,k) count partitions of [n] into k nonempty unordered blocksSet partitions, Stirling numbers, and the cycle structure of permutations
- Let a₀=2, a₁=5, and aₙ=5aₙ₋₁-6aₙ₋₂ for n ≥ 2Ordinary generating functions, recurrences, and Catalan numbers
- If aₙ counts structures on a specified set of n distinct labels, its exponential…Exponential generating functions and labeled structures