The size of the overlap set of a permutation. For a permutation $\pi\in\mathfrak S_n$ this is the number of indices $i < n$ such that the standardisation of $\pi_1\dots\pi_{n-i}$ equals the standardisation of $\pi_{i+1}\dots\pi_n$. In particular, for $n > 1$, the statistic is at least one, because the standardisations of $\pi_1$ and $\pi_n$ are both $1$. For example, for $\pi=2143$, the standardisations of $21$ and $43$ are equal, and so are the standardisations of $2$ and $3$. Thus, the statistic on $\pi$ is $2$.

The number of augmented double ascents of a permutation. An augmented double ascent of a permutation $\pi$ is a double ascent of the augmented permutation $\tilde\pi$ obtained from $\pi$ by adding an initial $0$. A double ascent of $\tilde\pi$ then is a position $i$ such that $\tilde\pi(i) < \tilde\pi(i+1) < \tilde\pi(i+2)$.

The number of descents of a permutation minus one if its first entry is not one. This statistic appears in [1, Theorem 2.3] in a gamma-positivity result, see also [2].

The number of alignments of a signed permutation. An alignment of a signed permutation $n\in\mathfrak H_n$ is either a nesting alignment, [[St001866]], an alignment of type EN, [[St001867]], or an alignment of type NE, [[St001868]]. Let $\operatorname{al}$ be the number of alignments of $\pi$, let \operatorname{cr} be the number of crossings, [[St001862]], let \operatorname{wex} be the number of weak excedances, [[St001863]], and let \operatorname{neg} be the number of negative entries, [[St001429]]. Then, $\operatorname{al}+\operatorname{cr}=(n-\operatorname{wex})(\operatorname{wex}-1+\operatorname{neg})+\binom{\operatorname{neg}{2}$.

The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.

The number of indecomposable projective-injective modules in the algebra corresponding to a subset. Let $A_n=K[x]/(x^n)$. We associate to a nonempty subset S of an (n-1)-set the module $M_S$, which is the direct sum of $A_n$-modules with indecomposable non-projective direct summands of dimension $i$ when $i$ is in $S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of $M_S$. We decode the subset as a binary word so that for example the subset $S=\{1,3 \} $ of $\{1,2,3 \}$ is decoded as 101.

The descent variation of a composition. Defined in [1].

The length of the boundary of an ordered tree. This is the sum of the number of edges to the left most and the right most leaf.

The length of the boundary minus the length of the trunk of an ordered tree. This is the size of the set of edges which are either on the left most path or on the right most path from the root.

The number of very big ascents of a permutation. A very big ascent of a permutation $\pi$ is an index $i$ such that $\pi_{i+1} - \pi_i > 2$. For the number of ascents, see [[St000245]] and for the number of big ascents, see [[St000646]]. General $r$-ascents were for example be studied in [1, Section 2].