Let's say a **species** is a functor

$$F: \mathrm{FinSet}_0 \to \mathrm{FinSet}_0$$

from the groupoid of finite sets and bijections to itself. Let $F(n)$ be its value on your favorite $n$-element set; then its **generating function** is the formal power series

$$ |F|(z) = \sum_{n = 0}^\infty \frac{|F(n)| z^n}{n!} $$

where the absolute value denotes cardinality.

In plain English: $F$ is a way of putting structures on finite sets, and the generating function is a power series whose $n$th coefficient is the number of ways of putting this structure on an $n$-element set, divided by $n!$.

Is there an interesting species whose generating function is $\sec z + \tan z$?

There's an answer that comes frustratingly close to being good. We have

$$ \sec z + \tan z = \sum_{n = 0}^\infty \frac{A_n z^n}{n!} $$

where $A_n$ is the $n$th **Euler zigzag number**. This is the number of permutations $\sigma$ of the set $\{1, \dots, n\}$ that are **alternating**, by which I mean that

$$ \sigma(1) < \sigma(2) > \sigma(3) < \sigma(4) > \cdots $$

For example, here is a picture that shows $A_4 = 5$, drawn by Robert M. Dickau:

This seems nice and combinatorial. However, to define an alternating permutation of a finite set, we need to equip it with a total ordering. There is a species that assigns to any finite set its collection of total orderings *together with* alternating permutations... but for an $n$-element set, there are $A_{n}$ times $n!$ of these, so the generating function of this species is

$$ \sum_{n = 0}^\infty A_{n} z^{n}, $$

not what I want.

I believe we could fix this by creating a species $F$ that assigns to each finite set the collection of *isomorphism classes* of total orderings and alternating permutations, where two are considered isomorphic if they differ by the action of a permutation. However, the resulting species, if indeed it's well-defined, will be 'uninteresting' in that now

$$F: \mathrm{FinSet}_0 \to \mathrm{FinSet}_0$$

maps every permutation of a finite set to an identity morphism.

Richard Stanley has many other interpretations of the Euler zigzag numbers in A survey of alternating permutations. However, I believe they all suffer from the same problem: they count structures on *totally ordered* finite sets. In this situation we expect to get the ordinary generating function

$$ \sum_{n = 0}^\infty A_{n} z^{n} $$

rather than the exponential generating function

$$ \sum_{n = 0}^\infty \frac{A_n z^n}{n!} $$

If this is inevitable, I'd like to know why the function $\sec z + \tan z$ comes so close to being the generating function of an interesting species, yet fails! Could we get it using a species valued in some other groupoid, like the groupoid of finite-dimensional vector spaces? Or maybe some other trick?

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