number of permutations in which no two consecutive numbers are adjacent

combinatorics

Solution 1:

This is OEIS sequence A002464. The value for $n=10$ is $479306$; the entry gives several formulas for calculating the terms. It also refers to p. 373 of Analytical Combinatorics by Flajolet and Sedgewick, which you can download here; that page gives a derivation of the ordinary generating function of the sequence.

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