Is there a standard name for this poset

Question

I've run into the following poset and I would expect it has a standard name. Let $n\geq k\geq 0$. Then $P_{n,k}$ consists of all $k$-element subsets of $\{1,\ldots,n\}$ ordered by $X\leq Y$ if $X=\{x_1< \ldots <x_k\}$ and $Y=\{y_1< \ldots <y_k\}$ where $x_i\leq y_i$ for all $i$. Hivert and Thiery call this the product order in one of their papers but I am not sure whether that is the standard name or one created for their paper.

Answer 1

This is a manifestation of the Bruhat order on the Grassmannian $Gr(k,n)$ of $k$-planes in an $n$-dimensional vector space.

In terms of partitions (Young diagrams): Given a set $X$ as above, let $\lambda_X = (\lambda_X^1, \dots, \lambda_X^k)$ be the partition with at most $k$ parts and each part of size at most $n-k$ (so the Young diagram fits inside a $k \times (n-k)$ box) given by: $$\lambda_X^i = x_i - i.$$ Then $X \leq Y$ iff $\lambda_X \leq \lambda_Y$ (i.e. iff $\lambda_X^i \leq \lambda_Y^i$ for all $1 \leq i \leq k$) iff the Young diagram of $\lambda_X$ is contained in the Young diagram of $\lambda_Y$.

There is also an interpretation in terms of the Bruhat order of permutations, restricted to those with at most a descent in the $k$-th position. Given $X$, let $x'_1 < \dots < x'_{n-k}$ be the set of elements of $\{1,\dots,n\} \setminus X$, in increasing order. Then associate to $X$ the permutation $w_X$ whose one-line notation is $x_1, \dots, x_k, x'_1, \dots, x'_{n-k}$.