The number of addable cells of the Ferrers diagram of an integer partition.

The number of distinct parts of the integer partition. This statistic is also the number of removeable cells of the partition, and the number of valleys of the Dyck path tracing the shape of the partition.

The number of non-final maximal constant sub-paths of length greater than one. This is the total number of occurrences of the patterns $110$ and $001$.

The multiplicity 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 $g_{\lambda,\lambda}^{(n-1)1}$, for $\lambda\vdash n > 1$. For $n\leq1$ the statistic is undefined. It follows from [3, Prop.4.1] (or, slightly easier from [3, Thm.4.2]) that this is one less than [[St000159]], the number of distinct parts of the partition.

The number of descents of type 2 in a permutation. A position $i\in[1,n-1]$ is a descent of type 2 of a permutation $\pi$ of $n$ letters, if it is a descent and if $\pi(j) < \pi(i)$ for all $j < i$.

The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module