The number of permutations less than or equal to a permutation in left weak order. This is the same as the number of permutations less than or equal to the given permutation in right weak order.

The product of the factorials of the parts.

The size of the automorphism group of the rooted tree underlying the ordered tree.

The number of regions of the inversion arrangement of a permutation. The inversion arrangement $\mathcal{A}_w$ consists of the hyperplanes $x_i-x_j=0$ such that $(i,j)$ is an inversion of $w$. Postnikov [4] conjectured that the number of regions in $\mathcal{A}_w$ equals the number of permutations in the interval $[id,w]$ in the strong Bruhat order if and only if $w$ avoids $4231$, $35142$, $42513$, $351624$. This conjecture was proved by Hultman-Linusson-Shareshian-Sjöstrand [1]. Oh-Postnikov-Yoo [3] showed that the number of regions of $\mathcal{A}_w$ is $|\chi_{G_w}(-1)|$ where $\chi_{G_w}$ is the chromatic polynomial of the inversion graph $G_w$. This is the graph with vertices ${1,2,\ldots,n}$ and edges $(i,j)$ for $i\lneq j$ $w_i\gneq w_j$. For a permutation $w=w_1\cdots w_n$, Lewis-Morales [2] and Hultman (see appendix in [2]) showed that this number equals the number of placements of $n$ non-attacking rooks on the south-west Rothe diagram of $w$.

The number of elements less than or equal to the given element in Bruhat order.

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