The multiplicity of the sign 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 $g_{\lambda,\lambda}^{1^n}$, for $\lambda\vdash n$. It equals $1$ if and only if $\lambda$ is self-conjugate.

The size of the automorphism group of a poset. A poset automorphism is a permutation of the elements of the poset preserving the order relation.

The distinguishing number of a poset. This is the minimal number of colours needed to colour the vertices of a poset, such that only the trivial automorphism of the poset preserves the colouring. See also [[St000469]], which is the same concept for graphs.

The number of occurrences of hills of size 3 in a Dyck path. A hill of size three is a subpath beginning at height zero, consisting of three up steps followed by three down steps.

The number of 1-rises at odd height of a Dyck path.

Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra.

The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module.

The index of the maximal parabolic seaweed algebra associated with the composition. Let $a_1,\dots,a_m$ and $b_1,\dots,b_t$ be a pair of compositions of $n$. The meander associated to this pair is obtained as follows: * place $n$ dots on a horizontal line * subdivide the dots into $m$ blocks of sizes $a_1, a_2,\dots$ * within each block, connect the first and the last dot, the second and the next to last, and so on, with an arc above the line * subdivide the dots into $t$ blocks of sizes $b_1, b_2,\dots$ * within each block, connect the first and the last dot, the second and the next to last, and so on, with an arc below the line By [1, thm.5.1], the index of the seaweed algebra associated to the pair of compositions is $$\operatorname{ind}\displaystyle\frac{b_1|b_2|...|b_t}{a_1|a_2|...|a_m} = 2C+P-1,$$ where $C$ is the number of cycles (of length at least $2$) and P is the number of paths in the meander. This statistic is $\operatorname{ind}\displaystyle\frac{b_1|b_2|...|b_t}{n}$.

The number of occurrences of the Hertzsprung pattern 132 in a permutation. A Hertzsprung occurrence of the pattern $\tau=(\tau_1,\dots,\tau_k)$ in a permutation $\pi$ is a factor $\pi_i, \pi_{i+1}, \dots,\pi_{i+k-1}$ of $\pi$ such that $\pi_{i+j-1} - \tau_j$ is constant for $1\leq j\leq k$.

The length of the minimal rise of a Dyck path. For the length of a maximal rise, see [[St000444]].