The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1).

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 rises of length 2 of a Dyck path. This is also the number of $(1,1)$ steps of the associated Łukasiewicz path, see [1]. A related statistic is the number of double rises in a Dyck path, [[St000024]].

The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. The statistic counting all pairs of distinct tunnels is the area of a Dyck path [[St000012]].

The number of occurrences of hills of size 2 in a Dyck path. A hill of size two is a subpath beginning at height zero, consisting of two up steps followed by two down steps.