The number of Hecke atoms of a permutation. For a permutation $z\in\mathfrak S_n$, this is the cardinality of the set $$ \{ w\in\mathfrak S_n | w^{-1} \star w = z\}, $$ where $\star$ denotes the Demazure product. Note that $w\mapsto w^{-1}\star w$ is a surjection onto the set of involutions.

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.