Let's say a **[species][2]** 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][3]**.  This is the number of permutations $\sigma$ of the set $\{1, \dots, n\}$ that are **[alternating][4]**, 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][5]:

![The fourth Euler zigzag number][1]

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][6].  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][7]

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

rather than the [exponential generating function][8]

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

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?


  [1]: https://i.sstatic.net/Iqig1.gif
 [2]: https://en.wikipedia.org/wiki/Combinatorial_species
 [3]: https://oeis.org/A000111
 [4]: https://en.wikipedia.org/wiki/Alternating_permutation
 [5]: https://web.archive.org/web/20180110063213/http://mathforum.org/advanced/robertd/zigzag.html
 [6]: https://arxiv.org/abs/0912.4240
 [7]: https://en.wikipedia.org/wiki/Generating_function#Ordinary_generating_function
 [8]: https://en.wikipedia.org/wiki/Generating_function#Exponential_generating_function