The number of full-support reflections in the Weyl group of a finite Cartan type. A reflection has full support if any (or all) reduced words for it in simple reflections use all simple reflections. This number is given by $\frac{nh}{|W|}d_1^*\cdots d_{n-1}^*$ where $n$ is the rank, $h$ is the Coxeter number, $W$ is the Weyl group, and $d_1^* \geq \ldots \geq d_{n-1}^* \geq d_n^* = 0$ are the codegrees of the Weyl group of a Cartan type.

The Dyson rank of a partition. This rank is defined as the largest part minus the number of parts. It was introduced by Dyson [1] in connection to Ramanujan's partition congruences $$p(5n+4) \equiv 0 \pmod 5$$ and $$p(7n+6) \equiv 0 \pmod 7.$$

The number of double up and double down steps of a Dyck path. In other words, this is the number of double rises (and, equivalently, the number of double falls) of a Dyck path.

The maximal area to the right of an up step of a Dyck path.

The position of the last double rise in a Dyck path. If the Dyck path has no double rises, this statistic is $0$.

The depth of the binary word interpreted as a path. This is the maximal value of the number of zeros minus the number of ones occurring in a prefix of the binary word, see [1, sec.9.1.2]. The number of binary words of length $n$ with depth $k$ is $\binom{n}{\lfloor\frac{(n+1) - (-1)^{n-k}(k+1)}{2}\rfloor}$, see [2].

The number of critical steps in the Catalan decomposition of a binary word. Every binary word can be written in a unique way as $(\mathcal D 0)^\ell \mathcal D (1 \mathcal D)^m$, where $\mathcal D$ is the set of Dyck words. This is the Catalan factorisation, see [1, sec.9.1.2]. This statistic records the number of critical steps $\ell + m$ in the Catalan factorisation. The distribution of this statistic on words of length $n$ is $$ (n+1)q^n+\sum_{\substack{k=0\\\text{k even}}}^{n-2} \frac{(n-1-k)^2}{1+k/2}\binom{n}{k/2}q^{n-2-k}. $$

Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path.

Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra.

The number of parts in an integer partition whose next smaller part has the same size. In other words, this is the number of distinct parts subtracted from the number of all parts.