Competition problem — Combinatorics

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Combinatorics Difficulty 5.1 AIME, harder Prove it United States

Problem:

Count the number of permutations $a_{1} a_{2} \ldots a_{7}$ of $1234567$ with longest decreasing subsequence of length at most two (i.e. there does not exist $i<j<k$ such that $a_{i}>a_{j}>a_{k}$ ).

Solution

Solution:

$C(7) = 429$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.