The number of saliances of the permutation. A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].

The size of the preimage of the map 'to poset' from Ordered trees to Posets.

The position of the first one in a binary word after appending a 1 at the end. Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.

The first ascent of a permutation. For a permutation $\pi$, this is the smallest index such that $\pi(i) < \pi(i+1)$. For the first descent, see [[St000654]].

The number of small weak excedances. A small weak excedance is an index $i$ such that $\pi_i \in \{i,i+1\}$. This is the sum of [[St000022]] and [[St000237]].

The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$.

The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset.

The second Elser number of a connected graph. For a connected graph $G$ the $k$-th Elser number is $$ els_k(G) = (-1)^{|V(G)|+1} \sum_N (-1)^{|E(N)|} |V(N)|^k $$ where the sum is over all nuclei of $G$, that is, the connected subgraphs of $G$ whose vertex set is a vertex cover of $G$. It is clear that this number is even. It was shown in [1] that it is non-negative.

The number of terminal closers of a set partition. A closer of a set partition is a number that is maximal in its block. In particular, a singleton is a closer. This statistic counts the number of terminal closers. In other words, this is the number of closers such that all larger elements are also closers.

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.