The multiplicity of the smallest part of a partition. This counts the number of occurrences of the smallest part $spt(\lambda)$ of a partition $\lambda$. The sum $spt(n) = \sum_{\lambda \vdash n} spt(\lambda)$ satisfies the congruences \begin{align*} spt(5n+4) &\equiv 0\quad \pmod{5}\\\ spt(7n+5) &\equiv 0\quad \pmod{7}\\\ spt(13n+6) &\equiv 0\quad \pmod{13}, \end{align*} analogous to those of the counting function of partitions, see [1] and [2].

The number of parts equal to 1 in a partition.

The largest multiplicity of a part in an integer partition.

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.

The length of the trunk of an ordered tree. This is the length of the path from the root to the first vertex which has not exactly one child.

The number of maximal antichains of minimal length in a poset.

The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. In other words, this is the lowest height of a valley of a Dyck path, or its semilength in case of the unique path without valleys.

The number of strong fixed points of a permutation. $i$ is called a strong fixed point of $\pi$ if 1. $j < i$ implies $\pi_j < \pi_i$, and 2. $j > i$ implies $\pi_j > \pi_i$ This can be described as an occurrence of the mesh pattern ([1], {(0,1),(1,0)}), i.e., the upper left and the lower right quadrants are shaded, see [3]. The generating function for the joint-distribution (RLmin, LRmax, strong fixed points) has a continued fraction expression as given in [4, Lemma 3.2], for LRmax see [[St000314]].

The number of isolated vertices of a graph.

The rix statistic of a permutation. This statistic is defined recursively as follows: $rix([]) = 0$, and if $w_i = \max\{w_1, w_2,\dots, w_k\}$, then $rix(w) := 0$ if $i = 1 < k$, $rix(w) := 1 + rix(w_1,w_2,\dots,w_{k−1})$ if $i = k$ and $rix(w) := rix(w_{i+1},w_{i+2},\dots,w_k)$ if $1 < i < k$.