Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. Given a partition $\lambda$ with $r$ parts, the number of semi-standard Young-tableaux of shape $k\lambda$ and boxes with values in $[r]$ grows as a polynomial in $k$. This follows by setting $q=1$ in (7.105) on page 375 of [1], which yields the polynomial $$p(k) = \prod_{i < j}\frac{k(\lambda_j-\lambda_i)+j-i}{j-i}.$$ The statistic of the degree of this polynomial. For example, the partition $(3, 2, 1, 1, 1)$ gives $$p(k) = \frac{-1}{36} (k - 3) (2k - 3) (k - 2)^2 (k - 1)^3$$ which has degree 7 in $k$. Thus, $[3, 2, 1, 1, 1] \mapsto 7$. This is the same as the number of unordered pairs of different parts, which follows from: $$\deg p(k)=\sum_{i < j}\begin{cases}1& \lambda_j \neq \lambda_i\\0&\lambda_i=\lambda_j\end{cases}=\sum_{\stackrel{i < j}{\lambda_j \neq \lambda_i}} 1$$

The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block.

The number of inversions of an integer composition. This is the number of pairs $(i,j)$ such that $i < j$ and $c_i > c_j$.

The major index of a composition regarded as a word. This is the sum of the positions of the descents of the composition. For the statistic which interprets the composition as a descent set, see [[St000008]].

The interlacing number of a set partition. Let $\pi$ be a set partition of $\{1,\dots,n\}$ with $k$ blocks. To each block of $\pi$ we add the element $\infty$, which is larger than $n$. Then, an interlacing of $\pi$ is a pair of blocks $B=(B_1 < \dots < B_b < B_{b+1} = \infty)$ and $C=(C_1 < \dots < C_c < C_{c+1} = \infty)$ together with an index $1\leq i\leq \min(b, c)$, such that $B_i < C_i < B_{i+1} < C_{i+1}$.

The second largest eigenvalue of a graph if it is integral. This statistic is undefined if the second largest eigenvalue of the graph is not integral. Chapter 4 of [1] provides lots of context.

The number of minimal elements in a poset.

The number of connected components of short braid edges in the graph of braid moves of a permutation. Given a permutation $\pi$, let $\operatorname{Red}(\pi)$ denote the set of reduced words for $\pi$ in terms of simple transpositions $s_i = (i,i+1)$. We now say that two reduced words are connected by a short braid move if they are obtained from each other by a modification of the form $s_i s_j \leftrightarrow s_j s_i$ for $|i-j| > 1$ as a consecutive subword of a reduced word. For example, the two reduced words $s_1s_3s_2$ and $s_3s_1s_2$ for $$(1243) = (12)(34)(23) = (34)(12)(23)$$ share an edge because they are obtained from each other by interchanging $s_1s_3 \leftrightarrow s_3s_1$. This statistic counts the number connected components of such short braid moves among all reduced words.

The variation of a composition.

The descent variation of a composition. Defined in [1].