The variation of a composition.

The number of ones minus the number of zeros of a binary word.

The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue. For example, the cycle on four vertices has distance Laplacian $$ \left(\begin{array}{rrrr} 4 & -1 & -2 & -1 \\ -1 & 4 & -1 & -2 \\ -2 & -1 & 4 & -1 \\ -1 & -2 & -1 & 4 \end{array}\right). $$ Its eigenvalues are $0,4,4,6$, so the statistic is $1$. The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore also statistic $1$. The graphs with statistic $n-1$, $n-2$ and $n-3$ have been characterised, see [1].

The Dyson rank of a partition. This rank is defined as the largest part minus the number of parts. It was introduced by Dyson [1] in connection to Ramanujan's partition congruences $$p(5n+4) \equiv 0 \pmod 5$$ and $$p(7n+6) \equiv 0 \pmod 7.$$

The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue. For example, the cycle on four vertices has distance Laplacian $$ \left(\begin{array}{rrrr} 4 & -1 & -2 & -1 \\ -1 & 4 & -1 & -2 \\ -2 & -1 & 4 & -1 \\ -1 & -2 & -1 & 4 \end{array}\right). $$ Its eigenvalues are $0,4,4,6$, so the statistic is $2$. The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore statistic $1$.

The number of standard desarrangement tableaux of shape equal to the given partition. A '''standard desarrangement tableau''' is a standard tableau whose first ascent is even. Here, an ascent of a standard tableau is an entry $i$ such that $i+1$ appears to the right or above $i$ in the tableau (with respect to English tableau notation). This is also the nullity of the random-to-random operator (and the random-to-top) operator acting on the simple module of the symmetric group indexed by the given partition. See also: * [[St000046]]: The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition * [[St000500]]: Eigenvalues of the random-to-random operator acting on the regular representation.