On my second point, sorry, you're right that you can take any number of variables which is at least $n$. If we think in terms of fillings of semi-standard Young tableaux, then we are looking at semi-standard Young tableaux whose maximal entry is at most $n$. Because of the shape of $\lambda$, that's going to be very constraining. (In each of the first $m$ columns, there is exactly one number from 1 to $n$ that won't appear.) But, okay, there's nothing wrong with that.

On my first point, if $K$ is a constant, wouldn't you have to prove, in particular, the case $K=1$, where there is no condition on the power sums? So wouldn't that case include all cases with larger $K$ as well? (Or, if you rule out $K=1$, still, $K=2$ would include all larger cases.)

Do you really want $K$ to be an absolute constant? And $s_\lambda$ is naturally a polynomial in $|\lambda|$ variables, not $n$ variables.

The subject of the paper by Christian (and Michael Cuntz) that I referred to in my previous comment is exactly the question of whether it is possible to find a poset associated to $H_4$ "whose ideal structure returns the notion and basic properties of non-nesting partitions".

There is a different paper of Christian's which is relevant to the motivation for the question: arxiv.org/abs/1212.2876. According to this paper, there is no reasonable candidate for a "poset of positive roots" in $H_4$ (though note that there are reasonable candidates for dihedral groups and for $H_3$, due to Drew Armstrong). Thus, the hope that one could carry out the procedure for non-crystallographic Coxeter groups seems doomed. (I still think it's an interesting question on its own merits, though.)

Being the ouside corners of a Young diagram just means that you have some set of pairs $(i,\lambda_i)$, no two having the same values in the first coordinate or in the second coordinate, and such that if $i_1<i_2$, then $\lambda_{i_1}>\lambda_{i_2}$ (note: the inequalities go in opposite directions).

I am sorry if my comment struck you as overly negative. I am certainly happy that it did not deter you from continuing with MO! Your broader point ("one should refrain from discouraging opinions of the sort above, esp for newbies") is, I think, only half the story. Part of how MO works is by discouraging certain kinds of questions. While this question is not terrible, you have certainly asked other questions that are much more interesting! But I acknowledge that I may not have struck the balance particularly well in this case.

This paper arxiv.org/abs/1306.1593 by Ringel describes the posets for all the classical types, and has nice pictures for the exceptionals.

Dyckerhoff's paper arxiv.org/abs/1505.06940 says that finite $\mathbb F_1[[t]]$-modules should be considered as an $\mathbb F_1$ vector space (i.e. finite set with distinguished element $*$) together with a nilpotent endomorphism. (I guess this is a different paper from the one @darijgrinberg was looking at; the parts of the partition arise here as lengths of maximal paths to $*$.)