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Number of permutation satisfying a summation relation

Source: INAMO 2023 P6 (OSN 2023)

August 30, 2023
combinatoricscountingpermutationIndonesiaIndonesia MO

Problem Statement

Determine the number of permutations a1,a2,…,ana_1, a_2, \dots, a_n of 1,2,…,n1, 2, \dots, n such that for every positive integer kk with 1≤k≤n1 \le k \le n, there exists an integer rr with 0≤r≤n−k0 \le r \le n - k which satisfies 1+2+⋯+k=ar+1+ar+2+⋯+ar+k. 1 + 2 + \dots + k = a_{r+1} + a_{r+2} + \dots + a_{r+k}.
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