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Number of partitions (IMO SL 1987-P11)

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August 19, 2010
combinatoricspartitionscountingExtremal combinatoricsIMO Shortlist

Problem Statement

Find the number of partitions of the set {1,2,⋯ ,n}\{1, 2, \cdots, n\} into three subsets A1,A2,A3A_1,A_2,A_3, some of which may be empty, such that the following conditions are satisfied:
(i)(i) After the elements of every subset have been put in ascending order, every two consecutive elements of any subset have different parity.
(ii)(ii) If A1,A2,A3A_1,A_2,A_3 are all nonempty, then in exactly one of them the minimal number is even .
Proposed by Poland.
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