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.

Only the addition of the next number, n, will change anything.

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

However, for $n=2$ , there's an exception: there's only 2 places the 2 can go (before or after the 1).

For $n=2$ , there are $2$ permutations. Now all we need to do is simple multiplication! For 1 through 7: 1, 2, 6, 18, 54, 162, 162*3=486.

Thus for $n=7$ there are $2*3^5=\boxed{486}$ permutations.

Note that recurrence is rather common when you see something of an ascending sequence BUT it has a condition such as "the next number can be 2 smaller"; this same idea appeared on another AIME with 8 boxes.

2015 AIME II (Problems • Answer Key • Resources)
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2015 AIME II Problems/Problem 10

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