The number of non-isomorphic subgroups of the Weyl group of a finite Cartan type. This statistic returns the number of non-isomorphic abstract groups. See [[St001155]] for the number of conjugacy classes of subgroups.

The number of two-component spanning forests of a graph. A '''spanning subgraph''' is a subgraph which contains all vertices of the ambient graph. A '''forest''' is a graph which contains no cycles, and has any number of connected components. A '''two-component spanning forest''' is a spanning subgraph which contains no cycles and has two connected components.

Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. It returns zero in case there is no such k.

The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. A graph is a forest if and only if in any linear ordering of its vertices, there are no three vertices $a < b < c$ such that $(a,c)$ and $(b,c)$ are edges. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.

The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. Let $n(G)$ be the number of vertices of a graph $G$, $m(G)$ be its number of edges and let $\alpha(G)$ be its independence number, [[St000093]]. Turán's theorem is that $m(G) \geq m(T^c(n(G), \alpha(G)))$ where $T^c(n, r)$ is the complement of the Turán graph. This statistic records the difference.

The size of the largest orbit of antichains under Panyushev complementation.

The product of the hook lengths of the diagonal cells in an integer partition. For a cell in the Ferrers diagram of a partition, the hook length is given by the number of boxes to its right plus the number of boxes below + 1. This statistic is the product of the hook lengths of the diagonal cells $(i,i)$ of a partition.

The number of characters of the symmetric group whose value on the partition is positive.

The number of standard desarrangement tableaux of shape equal to the given partition. A '''standard desarrangement tableau''' is a standard tableau whose first ascent is even. Here, an ascent of a standard tableau is an entry $i$ such that $i+1$ appears to the right or above $i$ in the tableau (with respect to English tableau notation). This is also the nullity of the random-to-random operator (and the random-to-top) operator acting on the simple module of the symmetric group indexed by the given partition. See also: * [[St000046]]: The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition * [[St000500]]: Eigenvalues of the random-to-random operator acting on the regular representation.

The hook length of the base cell of a partition. This is also known as the perimeter of a partition. In particular, the perimeter of the empty partition is zero.