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Maximums and minimums in permutation

Source: Iberoamerican 2018 Problem 5

September 26, 2018
combinatorics

Problem Statement

Let nn be a positive integer. For a permutation a1,a2,…,ana_1, a_2, \dots, a_n of the numbers 1,2,…,n1, 2, \dots, n we define
bk=min⁡1≤i≤kai+max⁡1≤j≤kajb_k = \min_{1 \leq i \leq k} a_i + \max_{1 \leq j \leq k} a_j
We say that the permutation a1,a2,…,ana_1, a_2, \dots, a_n is guadiana if the sequence b1,b2,…,bnb_1, b_2, \dots, b_n does not contain two consecutive equal terms. How many guadiana permutations exist?
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