The number of maximal proper sublattices of a lattice.

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 partitions of the same length below the given integer partition. For a partition $\lambda_1 \geq \dots \lambda_k > 0$, this number is $$\det\left( \binom{\lambda_{k+1-i}}{j-i+1} \right)_{1 \le i,j \le k}.$$

The height of a Dyck path. The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.

The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. In other words, this is the number of valleys and peaks whose first step is in odd position, the initial position equal to 1. The generating function is given in [1].

Eigenvalues of the top-to-random operator acting on a simple module. These eigenvalues are given in [1] and [3]. The simple module of the symmetric group indexed by a partition $\lambda$ has dimension equal to the number of standard tableaux of shape $\lambda$. Hence, the eigenvalues of any linear operator defined on this module can be indexed by standard tableaux of shape $\lambda$; this statistic gives all the eigenvalues of the operator acting on the module. This statistic bears different names, such as the type in [2] or eig in [3]. Similarly, the eigenvalues of the random-to-random operator acting on a simple module is [[St000508]].

The position of the first down step of a Dyck path.

The Hadwiger number of the graph. Also known as clique contraction number, this is the size of the largest complete minor.

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 $$(n+1)q^n+\sum_{\substack{k=0\\\text{k even}}}^{n-2} \frac{(n-1-k)^2}{1+k/2}\binom{n}{k/2}q^{n-2-k}.$$