Name for the cell poset of the staircase partition Is there a standard name for the cell poset of the staircase partition $(n,n-1,\dots,1)$, where, in English notation, a cell covers the adjacent cell in the row above and in the column to the right?  That is, elements are $\{(i,j) | 1\leq i\leq n, 1\leq j\leq n+1-i\}$ and the cover relations are $(i,j)<\!\!\cdot(i-1,j)$ and $(i,j)<\!\!\cdot(i,j+1)$.

I'd also be interested in contexts where it arises.
Update: I now found the appropriate context for my purposes: the antichains in this poset are the $2^n$ lattice paths which begin at the origin $(1,1)$ and take $n$ up and down steps.
 A: Regarding the name of this poset: as Richard Stanley mentioned in a comment, one way to denote this poset — which emphasizes that it is a distributive lattice - is as $J(2 \times n)$. But I think the most common way to refer to it is as the shifted staircase. This is because for a Young diagram shape (or skew shape, or shifted shape, etc.) the natural partial order on cells is the one where cell $u$ is less than or equal to cell $v$ if $u$ is weakly north west (in English notation) of $v$. With this partial order, standard Young tableaux correspond to linear extensions of the poset.
Regarding contexts in which this poset arises: this is a minuscule poset in the sense of Proctor, “Bruhat Lattices, Plane Partition Generating Functions, and Minuscule Representations” (https://doi.org/10.1016/S0195-6698(84)80037-2). The minuscule posets have many remarkable combinatorial properties: for instance, there is a product formula for the number of order ideals (in the case of the shifted staircase, $2^n$, as noted above), a product formula for the number of linear extensions, and more generally the entire order polynomial has a product formula. In fact, the minuscule posets are one of the very few families of posets whose order polynomials have product formulas: see Proctor’s conjecture about Gaussian posets from the above paper, or my survey article https://arxiv.org/abs/2006.01568.
The remarkable combinatorial properties of the minuscule posets come from representation theory and geometry. For example, the most prominent minuscule poset is the rectangle shape $a \times b$, and it is well known how the combinatorics of the rectangle governs the geometry of the (usual, Type A) Grassmannian. In exactly the same way, the combinatorics of the shifted staircase governs the geometry of the Type B maximal orthogonal Grassmannian. This can be seen, for instance, from Standard Monomial Theory. One way to realize the shifted staircase $J(2\times n)$ is to quotient the square rectangle $n\times n$ by its obvious symmetry, and this is a shadow of the way of obtaining the Type B Grassmannian from the Type A one by “folding.”
Finally, let me mention that the unshifted staircase - a different poset than the one you asked about - also has many of the same remarkable combinatorial properties even though it is not quite a minuscule poset. It is related to the geometry of the Lagrangian Grassmannian, which is not quite a “minuscule variety.” That poset is also the same (at least, up to duality) as the Type A root poset. There is a strong connection between root posets and minuscule posets: all minuscule posets are principal order filters in their corresponding root posets.
