The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group.
Following Stembridge [1, cor.4.7], the highest weight words indexing the irreducibles in $\mathfrak{gl_n}^{\otimes r}$ are staircase tableaux of length $2r$: sequences $(\gamma^{(0)},\dots,\gamma^{(2r)})$ of vectors in $\mathbb Z^n$ with decreasing entries, such that $\gamma^{(2i+1)}$ is obtained from $\gamma^{(2i)}$ by adding a unit vector and $\gamma^{(2i)}$ is obtained from $\gamma^{(2i-1)}$ by subtracting a unit vector.
For $n=2$, the staircase tableaux whose final element is the zero vector are in natural correspondence with Dyck paths: adding the first or subtracting the second unit vector is translated to an up step, whereas adding the second or subtracting the first unit vector is translated to a down step.
A Dyck path can be transformed into a bicoloured Motzkin path by replacing double up steps (double down, up-down, down-up steps) with up steps (down, coloured level, level steps). Note that the resulting path cannot have coloured level steps at height zero.
In this context, say that a bicoloured Motzkin path has a $\mathfrak{gl}_2$-descent between the following pairs of steps:

• an up step followed by a level step
• an up step followed by a down step, if the final height is not zero
• a coloured level step followed by any non-coloured step.

Then, conjecturally, the quasisymmetric expansion of the Frobenius character of the symmetric group $\mathfrak S_r$ acting on $\mathfrak{gl}_2^{\otimes r}$, is
$$ \sum_M F_{Des(M)}, $$
where the sum is over all length $r$ prefixes of bicoloured Motzkin paths, $Des(M)$ is the set of indices of descents of the path $M$ and $F_D$ is Gessel's fundamental quasisymmetric function.
The statistic recorded here is the number of $\mathfrak{gl}_2$-descents in the bicoloured Motzkin path corresponding to the Dyck path.
Restricting to Motzkin paths without coloured steps one obtains the quasisymmetric expansion for the Frobenius character of $\mathfrak S_r$ acting on $\mathfrak{sl}_2^{\otimes r}$. In this case, the conjecture was shown by Braunsteiner [2].

The bounce path determined by an integer composition.