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

The number of invariant subsets of size 3 when acting with a permutation of given cycle type.

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 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 projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1.

The number of finite solvable groups that are realised by the given partition over the complex numbers. A finite group $G$ is ''realised'' by the partition $(a_1,\dots,a_m)$ if its group algebra over the complex numbers is isomorphic to the direct product of $a_i\times a_i$ matrix rings over the complex numbers. The smallest partition which does not realise a solvable group, but does realise a finite group, is $(5,4,3,3,1)$.

The number of finite groups that are realised by the given partition over the complex numbers. A finite group $G$ is 'realised' by the partition $(a_1,...,a_m)$ if its group algebra over the complex numbers is isomorphic to the direct product of $a_i\times a_i$ matrix rings over the complex numbers.

This is the number of standard Young tableaux of the given shifted shape. For an integer partition $\lambda = (\lambda_1,\dots,\lambda_k)$, the shifted diagram is obtained by moving the $i$-th row in the diagram $i-1$ boxes to the right, i.e., $$\lambda^∗ = \{(i, j) | 1 \leq i \leq k, i \leq j \leq \lambda_i + i − 1 \}.$$ In particular, this statistic is zero if and only if $\lambda_{i+1} = \lambda_i$ for some $i$.

Another weight of a partition according to Alladi. According to Theorem 3.4 (Alladi 2012) in [1] $$\sum_{\pi\in GG_1(r)} w_1(\pi)$$ equals the number of partitions of $r$ whose odd parts are all distinct. $GG_1(r)$ is the set of partitions of $r$ where consecutive entries differ by at least $2$, and consecutive even entries differ by at least $4$.