2015 AIME II Problems/Problem 10

Problem

Call a permutation $a_1, a_2, \ldots, a_n$ of the integers $1, 2, \ldots, n$ quasi-increasing if $a_k \leq a_{k+1} + 2$ for each $1 \leq k \leq n-1$ . For example, 53421 and 14253 are quasi-increasing permutations of the integers $1, 2, 3, 4, 5$ , but 45123 is not. Find the number of quasi-increasing permutations of the integers $1, 2, \ldots, 7$ .

Solution

The simple recurrence can be found.

When inserting an integer n into a string with n-1 integers, we notice that the integer n has 3 spots where it can go: before n-1, before n-2, and at the very end.

EXAMPLE: putting 4 into the string 123: 4 can go before the 2 : 1423, before the 3: 1243, and at the very end: 1234.

Thus the number of permutations with n elements is three times the number of permutations with n-1 elements.

For n=3, there are 6 permutations. Thus for n=7 there are 6*3^4 = 486 permutations.

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2015 AIME II Problems/Problem 10

Problem

Solution