Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
I would bet that this is an empirical observation, with no special motivation. One interesting point about this self-map is that it satisfies the cyclic sieving phenomenon, as defined by Reiner, Stanton, and White in 2004.
There is an inconsistency between the value at 1 (claimed to be a binomial coefficient) and the last formula, as illustrated in the examples, which seems to imply that the value at 1 is twice a binomial coefficient. Maybe there is a factor $1-q^2$ missing in the last formula ?
Tilting modules (in the derived categories of quiver representations) can sometimes be displayed using the Auslander-Reiten quiver. This is the case in particular for coeherent sheaves on P¹. See for examples images at the end of math.jussieu.fr/~keller/publ/dct.pdf
Such a Lie algebra is also a pre-Lie algebra. The pre-Lie product is obtained from the Lie bracket by adjunction with respect to $\omega$. You may have a look at various articles of Alberto Medina.
This Legendre transform, or rather the inversion formula obtained by derivating both sides, is related to the results and conjectures of fr.arxiv.org/abs/1010.3176 Indeed, the PreLie operad (dimension $n^{n-1}$ in degree $n$) contains a subspace of dimension $(n-2)!$ (related to the cyclic Lie operad), which conjecturally generates a free sub-operad (of dimension $(n-1)^{n-1}$ in degree $n$).
