The number of exclusive right-to-left minima of a permutation. This is the number of right-to-left minima that are not left-to-right maxima. This is also the number of non weak exceedences of a permutation that are also not mid-points of a decreasing subsequence of length 3. Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j < j$ and there do not exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$. See also [[St000213]] and [[St000119]].

The number of pure excedances of a permutation. A pure excedance of a permutation $\pi$ is a position $i < \pi_i$ such that there is no $j < i$ with $i\leq \pi_j < \pi_i$.

The number of exclusive left-to-right maxima of a permutation. This is the number of left-to-right maxima that are not right-to-left minima.

The number of descents of type 2 in a permutation. A position $i\in[1,n-1]$ is a descent of type 2 of a permutation $\pi$ of $n$ letters, if it is a descent and if $\pi(j) < \pi(i)$ for all $j < i$.

The number of nontrivial cycles in the cycle decomposition of a permutation. This statistic is equal to the difference of the number of cycles of $\pi$ (see [[St000031]]) and the number of fixed points of $\pi$ (see [[St000022]]).

The number of repeated entries in the Lehmer code of a permutation. The Lehmer code of a permutation $\pi$ is the sequence $(v_1,\dots,v_n)$, with $v_i=|\{j > i: \pi(j) < \pi(i)\}$. This statistic counts the number of distinct elements in this sequence.

The least common multiple of the parts of the partition.

The number of colourings of a cycle such that the multiplicities of colours are given by a partition. Two colourings are considered equal, if they are obtained by an action of the cyclic group. This statistic is only defined for partitions of size at least 3, to avoid ambiguity.