In how many ways can the set {1, 2, 3, 4, 5, 6, 7, 8, 9} be…

Question

In how many ways can the set {1, 2, 3, 4, 5, 6, 7, 8, 9} be partitioned into exactly 3 non-empty unlabeled blocks such that no block is a singleton (every block has size at least 2)?

Accepted Answer

We count partitions of a 9-element set into 3 unlabeled non-empty blocks with no singletons. The total number into exactly 3 non-empty blocks is the Stirling number S(9,3) = 3025.

Subtract partitions that have at least one singleton block. Use inclusion-exclusion on the number of singleton blocks among the 3 blocks.

Partitions with at least one block being a singleton: choose the singleton element and partition the rest into the remaining 2 unlabeled blocks. But careful with overcounting when two blocks are singletons. The possible block-size types for 9 elements in 3 blocks with a singleton: sizes (1,a,b) with a+b=8. The 'no singleton' valid size types are (2,2,5),(2,3,4),(3,3,3),(2,4,3 same),(5,2,2),(4,4,1 invalid). Enumerate valid integer partitions of 9 into 3 parts each >=2: (2,2,5),(2,3,4),(3,3,3).

Count each: (2,2,5): choose the 5-set C(9,5)=126, then split remaining 4 into two unordered pairs = 3, total 126*3 = 378. (2,3,4): choose 2 (C(9,2)=36), choose 3 of remaining 7 (C(7,3)=35), last 4 fixed; all sizes distinct so no symmetry division: 36*35 = 1260. (3,3,3): partition 9 into three unordered triples = 9!/(3!^3 · 3!) = 362880/(216·6) = 362880/1296 = 280.

Total = 378 + 1260 + 280 = 1918.

Final answer: 1918

In how many ways can the set {1, 2, 3, 4, 5, 6, 7, 8, 9} be…

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