An alternating permutation of the integers is a permutation of these integers that alternately increases and decreases. For example, and are alternating permutations of , but is not. Note that alternating permutations are completely different than members of the alternating group, whose elements are even permutations.
We may distinguish between "up-down" and "down-up" alternating permutations: is an up-down permutation, while is a down-up permutation. Up-down permutations have descent set while down-up permutations have descent set .
There is no general consensus about whether the phrase "alternating permutation" should refer to up-down permutations, down-up permutations, or both.
Counting alternating permutations
The number of up-down alternating permutations of length is the same as the number of down-up alternating permutations of length , and this number is denoted (though this convention is not universal). These numbers have various names, including Euler numbers, up-down numbers, and secant and tangent numbers. These latter names are the result of the remarkable generating function (or equivalently Taylor series) identity
One can compute these numbers via the recursion
Proof of these formulas
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