The semiperimeter of the associated bargraph. Interpret the composition as the sequence of heights of the bars of a bargraph. This statistic is the semiperimeter of the polygon determined by the axis and the bargraph. Put differently, it is the sum of the number of up steps and the number of horizontal steps when regarding the bargraph as a path with up, horizontal and down steps.

The number of vertices of the unicellular map given by a perfect matching. If the perfect matching of $2n$ elements is viewed as a fixed point-free involution $\epsilon$ This statistic is counting the number of cycles of the permutation $\gamma \circ \epsilon$ where $\gamma$ is the long cycle $(1,2,3,\ldots,2n)$. '''Example''' The perfect matching $[(1,3),(2,4)]$ corresponds to the permutation in $S_4$ with disjoint cycle decomposition $(1,3)(2,4)$. Then the permutation $(1,2,3,4)\circ (1,3)(2,4) = (1,4,3,2)$ has only one cycle. Let $\epsilon_v(n)$ is the number of matchings of $2n$ such that yield $v$ cycles in the process described above. Harer and Zagier [1] gave the following expression for the generating series of the numbers $\epsilon_v(n)$. $$\sum_{v=1}^{n+1} \epsilon_{v}(n) N^v = (2n-1)!! \sum_{k\geq 0}^n \binom{N}{k+1}\binom{n}{k}2^k.$$

The length of the path to the largest entry in a standard Young tableau.

The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.

The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order.