The sum of the minimal distances to a greater element. Set $\pi_0 = \pi_{n+1} = n+1$, then this statistic is $$ \sum_{i=1}^n \min_d(\pi_{i-d}>\pi_i\text{ or }\pi_{i+d}>\pi_i) $$ This statistic appears in [1]. The generating function for the sequence of maximal values attained on $\mathfrak S_r$, $r\geq 0$ apparently coincides with [2], which satisfies the functional equation $$ (x-1)^2 (x+1)^3 f(x^2) - (x-1)^2 (x+1) f(x) + x = 0. $$

The number of occurrences of one of the patterns 132, 213 or 321 in a permutation. According to [1], this statistic was studied by Doron Gepner in the context of conformal field theory.

The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.

The number of nonempty primitive factors of a binary word. A word $u$ is a factor of a word $w$ if $w = p u s$ for words $p$ and $s$. A word is primitive, if it is not of the form $u^k$ for a word $u$ and an integer $k\geq 2$. Apparently, the maximal number of nonempty primitive factors a binary word of length $n$ can have is given by [[oeis:A131673]].

The number of facets in the order polytope of this poset.

The number of facets in the chain polytope of the poset.

The number of occurrences of the pattern 132 or of the pattern 213 in a permutation.

The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. The Kronecker coefficient is the multiplicity $g_{\mu,\nu}^\lambda$ of the Specht module $S^\lambda$ in $S^\mu\otimes S^\nu$: $$ S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda $$ This statistic records the Kronecker coefficient $g_{\lambda,\lambda}^{21^{n-2}}$, for $\lambda\vdash n$.

The girth of a graph, which is not a tree. This is the length of the shortest cycle in the graph.

The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles.