I have come across a sequence of representations $V\_n$ of the symmetric group $S\_{n+2}$ which has the property that restricting the action $S\_n \subset S\_{n+2}$ gives the regular representation:
$$ Res^{S\_{n+2}}\_{S\_n} V\_n = \mathbb{Q}S\_n. $$
In other words, there is some natural way to give the regular rep of $S\_n$ an action of $S\_{n+2}$. This (to me) is surprising, but I imagine this has already been observed. 


For concreteness, here are the first few terms in the sequence, written as a sum of partitions (using the usual indexing of representations of $S\_{n+2}$ by partitions of $n+2$):



```
[2]
[3]
[2,2]
[3,1,1]
[3,3]+[2,2,1,1]+[4,1,1]
...

```

I have two questions:


1. Is there already work on unrestrictions of the regular representation of a symmetric group? Is my particular sequence of representations $V\_n$ well-known?
2. In general, are there circumstances under which a representation of $S\_n$ has a canonical way to extend the action to $S\_{n+1}$?



An n!-dimensional representation of the symmetric group S_{n+2}
1. Yes your series of representations looks (except for the first term - but there must be the law of small numbers lurking around) like the sign representation times the *Whitehouse module*, see, e.g. [these slides by Richard Stanley](http://www-math.mit.edu/~rstan/transparencies/whouse.ps). Basically, the Whitehouse module can be viewed as the respective component of the cyclic operad Lie.
2. The last sentence of the previous paragraph gives an operadchik's view on your second question. If your representation of $S\_n$ appears as a component of arity $n$ of a certain operad, then the most "natural" situation in which it can be extended to $S\_{n+1}$ is when your operad is cyclic or anticyclic (see, e.g. [this paper of Getzler and Kapranov](http://www.google.lu/url?sa=t&rct=j&q=getzler%20kapranov%20cyclic&source=web&cd=1&ved=0CCgQFjAA&url=http%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fdownload%3Fdoi%3D10.1.1.146.2678%26rep%3Drep1%26type%3Dpdf&ei=PVEqT6eFFsbL8QO86dWdDw&usg=AFQjCNFBQq43vG8QKOCo9ulIr5DkPRRfqA), and [this paper of Chapoton](http://arxiv.org/abs/math.RT/0411508) for some flavour of the story).


