The standardized bi-alternating inversion number of a permutation. The bi-alternating inversion number $i(\pi)$ is $\sum_{1\leq y < x\leq n} (-1)^{y+x} \mathrm{sgn}(\pi(x)-\pi(y))$, see Section 3.1 of [1]. It takes values in between $-\lfloor\frac n2\rfloor^2$ and $\lfloor\frac n2\rfloor^2$, and $i(\pi)+\lfloor\frac n2\rfloor^2$ is even, so we standardize it to $(i(\pi)+\lfloor\frac n2\rfloor^2)/2$. For odd $n$ the bi-alternating inversion is invariant under rotation, that is, conjugation with the long cycle.

The pebbling number of a connected graph.

The number of simple modules with projective dimension at most 1.

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

The number of invariant oriented cycles when acting with a permutation of given cycle type.

The Grundy value of Chomp on Ferrers diagrams. Players take turns and choose a cell of the diagram, cutting off all cells below and to the right of this cell in English notation. The player who is left with the single cell partition looses. The traditional version is played on chocolate bars, see [1]. This statistic is the Grundy value of the partition, that is, the smallest non-negative integer which does not occur as value of a partition obtained by a single move.