# Order polynomial of shifted double staircase

This question is related to my earlier question looking for posets with product formulas for their order polynomials.

Recall that the order polynomial $$\Omega_P(m)$$ of a finite poset $$P$$ is defined by $$\Omega_P(m) := \# \textrm{ weakly order preserving maps P\to \{1,2,\ldots,m\}}.$$

Now let $$\lambda = (n,n-1,n-2,...,1) + (k,k-1,k-2,...,1)$$ for $$0 \leq k < n$$ be a shifted double staircase'' shape (see e.g. Figure 6(c) in Stanley's paper "Promotion and Evacuation", which is linked below). And let $$P$$ be the poset corresponding to $$\lambda$$ (i.e., the poset on the boxes of $$\lambda$$ viewed as a shifted shape, with $$u \lessdot v$$ if the box $$v$$ is directly right of or directly below the box $$u$$).

Question: Is it true that for this $$P$$ we have $$\Omega_P(m) = \prod_{1 \leq i \leq j \leq n} \frac{(m+i+j-2)}{(i+j-1)}\cdot \prod_{1 \leq i \leq j \leq k} \frac{(m+i+j-1)}{(i+j)}?$$

Testing some small cases it looks like this formula works, and this is not an example I have seen in the literature (but a pointer to a place where it is addressed would also definitely be appreciated!).

For context, let me explain some similar formulas which are known.

If $$P$$ is the poset associated to the (unshifted) staircase $$\lambda = (n,n-1,n-2,...,1)$$ (this poset is also the Type A root poset), then $$\Omega_{P}(m) = \prod_{1\leq i \leq j \leq n} \frac{i+j+2m-2}{i+j}.$$ While if $$P$$ is the poset associated to the shifted staircase $$\lambda = (n,n-1,n-2,...,1)$$ (this poset is the Type B/D minuscule poset), then $$\Omega_P(m) = \prod_{1 \leq i \leq j \leq n} \frac{(m+i+j-2)}{(i+j-1)}.$$ Both of these formulas can be seen for instance in the paper "New Symmetric Plane Partition Identities from Invariant Theory Work of De Concini and Procesi " by Proctor (linked below). Note that the shifted staircase is just the case $$k=0$$ of the shifted double staircase and the conjectured formula agrees with the known formula in this case. The formula for the case $$k=n-1$$ is also known: in this case the poset is the Type B root poset; see for instance the abstract by Hamaker and Williams linked below.

Hamaker, Zachary; Williams, Nathan, Subwords and plane partitions, Proceedings of the 27th international conference on formal power series and algebraic combinatorics, FPSAC 2015, Daejeon, South Korea, July 6–10, 2015. Nancy: The Association. Discrete Mathematics & Theoretical Computer Science (DMTCS). Discrete Mathematics and Theoretical Computer Science. Proceedings, 241-252 (2015). ZBL1335.05177.

Proctor, Robert A., New symmetric plane partition identities from invariant theory work of De Concini and Procesi, Eur. J. Comb. 11, No. 3, 289-300 (1990). ZBL0726.05008.

Stanley, Richard P., Promotion and evacuation, Electron. J. Comb. 16, No. 2, Research Paper R9, 24 p. (2009). ZBL1169.06002.

• What is the order polynomial of a staircase (and of a reflected staircase)? Oct 3, 2019 at 9:04
• @IlyaBogdanov: I included this information. I also updated to include a conjectured formula for the shifted double staircase order polynomial, since based on my data it is very simple. Oct 3, 2019 at 14:30
• This conjecture is now mentioned in arxiv.org/abs/2006.01568. Jun 5, 2020 at 11:20

Tri Lai and I proved this conjecture, using the techniques from the theory of lozenge tilings. Indeed, this result is almost already proved by Ciucu in https://arxiv.org/abs/1906.02021. We just need to allow slightly more general parameters for the "flashlight" region of the triangular lattice which he considers- and the techniques he developed there suffice to do that. What we are able to show specifically is that for the region: the number of lozenge tilings of $$F(x,y,z,t)$$ is $$\prod_{1\leq i \leq j\leq y+z}\frac{x+i+j-1}{i+j-1}\prod_{1\leq i \leq j \leq z} \frac{x+i+j}{i+j} \prod_{i=1}^{t}\prod_{j=1}^{z}\frac{(x+z+2i+j)}{(x+2i+j-1)}.$$ The case $$t=0$$ corresponds to the order polynomial for the shifted double staircase (where $$y+z=n$$, $$z=k$$, and $$x=m-1$$ in the notation of the original question).