The weighted size of a partition. Let $\lambda = (\lambda_0\geq\lambda_1 \geq \dots\geq\lambda_m)$ be an integer partition. Then the weighted size of $\lambda$ is $$\sum_{i=0}^m i \cdot \lambda_i.$$ This is also the sum of the leg lengths of the cells in $\lambda$, or $$ \sum_i \binom{\lambda^{\prime}_i}{2} $$ where $\lambda^{\prime}$ is the conjugate partition of $\lambda$. This is the minimal number of inversions a permutation with the given shape can have, see [1, cor.2.2]. This is also the smallest possible sum of the entries of a semistandard tableau (allowing 0 as a part) of shape $\lambda=(\lambda_0,\lambda_1,\ldots,\lambda_m)$, obtained uniquely by placing $i-1$ in all the cells of the $i$th row of $\lambda$, see [2, eq.7.103].

The size of a partition minus its first part. This is the number of boxes in its diagram that are not in the first row.

The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. For example, $e_{22} = s_{1111} + s_{211} + s_{22}$, so the statistic on the partition $22$ is $3$.

The number of even parts of a partition.

The largest part of an integer partition.

The cocharge of a standard tableau. The '''cocharge''' of a standard tableau $T$, denoted $\mathrm{cc}(T)$, is defined to be the cocharge of the reading word of the tableau. The cocharge of a permutation $w_1 w_2\cdots w_n$ can be computed by the following algorithm: 1) Starting from $w_n$, scan the entries right-to-left until finding the entry $1$ with a superscript $0$. 2) Continue scanning until the $2$ is found, and label this with a superscript $1$. Then scan until the $3$ is found, labeling with a $2$, and so on, incrementing the label each time, until the beginning of the word is reached. Then go back to the end and scan again from right to left, and *do not* increment the superscript label for the first number found in the next scan. Then continue scanning and labeling, each time incrementing the superscript only if we have not cycled around the word since the last labeling. 3) The cocharge is defined as the sum of the superscript labels on the letters.

The size of a partition. This statistic is the constant statistic of the level sets.

The (standard) major index of a standard tableau. A descent of a standard tableau $T$ is an index $i$ such that $i+1$ appears in a row strictly below the row of $i$. The (standard) major index is the the sum of the descents.

The leg major index of a standard tableau. The leg length of a cell is the number of cells strictly below in the same column. This statistic is the sum of all leg lengths. Therefore, this is actually a statistic on the underlying integer partition. It happens to coincide with the (leg) major index of a tabloid restricted to standard Young tableaux, defined as follows: the descent set of a tabloid is the set of cells, not in the top row, whose entry is strictly larger than the entry directly above it. The leg major index is the sum of the leg lengths of the descents plus the number of descents.

The maximal part of the shifted composition of an integer partition. A partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ is shifted into a composition by adding $i-1$ to the $i$-th part. The statistic is then $\operatorname{max}_i\{ \lambda_i + i - 1 \}$. See also [[St000380]].