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Upper bound of the number of oscillatory sequences

In the terminology of Chapter 3 of Enumerative Combinatorics, vol. 1, second ed., you are asking for the value of the strict order polynomial of the zigzag poset $Z_n$ at $q$ (see Exercise 3.66). ...
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Random covering of a set

It follows immediately from the special case (with $m=n$ and $s=t=k$) of Theorem 1 of Canfield and Mckay that the probability -- say $p_{n,k}$ -- in question is $$\sim\frac1{\sqrt e}\dfrac{\...

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