The sorting index of a permutation. The sorting index counts the total distance that symbols move during a selection sort of a permutation. This sorting algorithm swaps symbol n into index n and then recursively sorts the first n-1 symbols. Compare this to [[St000018]], the number of inversions of a permutation, which is also the total distance that elements move during a bubble sort.

The sorting index of a signed permutation. A signed permutation $\sigma = [\sigma(1),\ldots,\sigma(n)]$ can be sorted $[1,\ldots,n]$ by signed transpositions in the following way: First move $\pm n$ to its position and swap the sign if needed, then $\pm (n-1), \pm (n-2)$ and so on. For example for $[2,-4,5,-1,-3]$ we have the swaps $$ [2,-4,5,-1,-3] \rightarrow [2,-4,-3,-1,5] \rightarrow [2,1,-3,4,5] \rightarrow [2,1,3,4,5] \rightarrow [1,2,3,4,5] $$ given by the signed transpositions $(3,5), (-2,4), (-3,3), (1,2)$. If $(i_1,j_1),\ldots,(i_n,j_n)$ is the decomposition of $\sigma$ obtained this way (including trivial transpositions) then the sorting index of $\sigma$ is defined as $$ \operatorname{sor}_B(\sigma) = \sum_{k=1}^{n-1} j_k - i_k - \chi(i_k < 0), $$ where $\chi(i_k < 0)$ is 1 if $i_k$ is negative and 0 otherwise. For $\sigma = [2,-4,5,-1,-3]$ we have $$ \operatorname{sor}_B(\sigma) = (5-3) + (4-(-2)-1) + (3-(-3)-1) + (2-1) = 13. $$

The number of atoms of a lattice. An element of a lattice is an '''atom''' if it covers the least element.

Number of indecomposable injective modules with projective dimension 2.

The number of simple modules with projective dimension at most 1.

The segment statistic of a semistandard tableau. Let ''T'' be a tableau. A ''k''-segment of ''T'' (in the ''i''th row) is defined to be a maximal consecutive sequence of ''k''-boxes in the ith row. Note that the possible ''i''-boxes in the ''i''th row are not considered to be ''i''-segments. Then seg(''T'') is the total number of ''k''-segments in ''T'' as ''k'' varies over all possible values.

The flush statistic of a semistandard tableau. Let $T$ be a tableaux with $r$ rows such that each row is longer than the row beneath it by at least one box. Let $1 \leq i < k \leq r+1$ and suppose $l$ is the smallest integer greater than $k$ such that there exists an $l$-segment in the $(i+1)$-st row of $T$. A $k$-segment in the $i$-th row of $T$ is called '''flush''' if the leftmost box in the $k$-segment and the leftmost box of the $l$-segment are in the same column of $T$. If, however, no such $l$ exists, then this $k$-segment is said to be flush if the number of boxes in the $k$-segment is equal to difference of the number of boxes between the $i$-th row and $(i+1)$-st row. The flush statistic is given by the number of $k$-segments in $T$.

The number of distinct eigenvalues of the distance Laplacian of a connected graph.

The pebbling number of a connected graph.

The diameter of a connected graph. This is the greatest distance between any pair of vertices.