Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path.

The number of augmented double ascents of a permutation. An augmented double ascent of a permutation $\pi$ is a double ascent of the augmented permutation $\tilde\pi$ obtained from $\pi$ by adding an initial $0$. A double ascent of $\tilde\pi$ then is a position $i$ such that $\tilde\pi(i) < \tilde\pi(i+1) < \tilde\pi(i+2)$.

Eigenvalues of the top-to-random operator acting on a simple module. These eigenvalues are given in [1] and [3]. The simple module of the symmetric group indexed by a partition $\lambda$ has dimension equal to the number of standard tableaux of shape $\lambda$. Hence, the eigenvalues of any linear operator defined on this module can be indexed by standard tableaux of shape $\lambda$; this statistic gives all the eigenvalues of the operator acting on the module. This statistic bears different names, such as the type in [2] or eig in [3]. Similarly, the eigenvalues of the random-to-random operator acting on a simple module is [[St000508]].

The least common multiple of the parts of the partition.

The major index of an integer partition when read from bottom to top. This is the sum of the positions of the corners of the shape of an integer partition when reading from bottom to top. For example, the partition $\lambda = (8,6,6,4,3,3)$ has corners at positions 3,6,9, and 13, giving a major index of 31.

The number of positive values of the symmetric group character corresponding to the partition. For example, the character values of the irreducible representation $S^{(2,2)}$ are $2$ on the conjugacy classes $(4)$ and $(2,2)$, $0$ on the conjugacy classes $(3,1)$ and $(1,1,1,1)$, and $-1$ on the conjugacy class $(2,1,1)$. Therefore, the statistic on the partition $(2,2)$ is $2$.

The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.