#Minor problems
#Interlacing roots and key polynomials
Let \(\key_{\lambda,\sigma}\) denote a key polynomial, and define \[P_{\sigma}(\lambda_1,\dotsc,\lambda_n;k) \coloneqq \key_{k\lambda,\sigma}(1^n).\] This is a polynomial in \(\setQ[\lambda_1,\dotsc,\lambda_n,k],\) and for fixed \(\lambda,\) this is an Ehrhart polynomial of a certain union of faces in a Gelfand–Tsetlin polytope. See [AA19] for more background.
We may compute the corresponding \(h^*\)-polynomial, and get some interesting properties.
Conjecture (Alexandersson (May 2020)).
Let \(\lambda\) and \(\sigma\) be fixed and define \(H_{\lambda,\sigma}(t) \in \setN[t]\) via \[\sum_{k\geq 0} P_{\lambda,\sigma}(k) t^k = \frac{H_{\lambda,\sigma}(t)}{(1-t)^{d+1}}\] where \(d\) is the degree (in \(k\)) of \(P_{\lambda,\sigma}(k).\) Then \(H_{\lambda,\sigma}(t)\) is a real-rooted polynomial. Moreover, if \(\sigma_1,\sigma_2,\dotsc,\sigma_\ell\) is a saturated chain in the Bruhat order, then \[P_{\lambda,\sigma_1}(k), P_{\lambda,\sigma_2}(k),\dotsc,P_{\lambda,\sigma_\ell}(k)\] is a sequence of interlacing polynomials.
I have checked this for some small cases.
#A \(q\)-generalization of an inequality
In [AA19], we used the following inequality, where \(a,b\geq 0\) and \(k \geq j \ge 0.\) \[\binom{ka+kb}{ka}^j \geq \binom{ja+jb}{ja}^k.\] This is not very hard to prove. I realized that there might be a \(q\)-analogue of this inequality.
Conjecture (Alexandersson, 2019).
Suppose \(a,b\geq 0\) and \(k \geq j \ge 0.\) Then \[q^{kab \binom{j}{2}} \qbinom{ka+kb}{ka}_q^j - q^{jab \binom{k}{2}} \qbinom{ja+jb}{ja}_q^k\] is a polynomial in \(\setN[q].\)
This was posted on MathOverflow, and there I sketched a proof that shows that this is true whenever \(j\) divides \(k.\)
#A Schur-positive expansion?
Let \(\BST(\lambda,\mu)\) be the set of border-strip tableaux of shape \(\lambda\) and strip-sizes \(\mu.\) Define \[T_\lambda(\xvec) \coloneqq \sum_{\mu} |\BST(\lambda,\mu)| \powerSum_\mu(\xvec).\] Show that \(T_\lambda(\xvec)\) is Schur-positive. Note the close resemblance with the usual power-sum expansion of Schur polynomials.
#On A189912
The sequence A189912 is defined as \[a_n \coloneqq \sum_{k=0}^n \frac{n!}{(n-k)! (\lfloor k/2 \rfloor!)^2 (\lfloor k/2 \rfloor +1)}.\] Let us split this sum into even and odd \(k.\) We get \[\begin{aligned} &\sum_{k=0}^n \frac{n!}{(n-2k)! (\lfloor 2k/2 \rfloor!)^2 (\lfloor 2k/2 \rfloor +1)} + \\ &\sum_{k=0}^n \frac{n!}{(n-(2k+1))! (\lfloor (2k+1)/2 \rfloor!)^2 (\lfloor (2k+1)/2 \rfloor +1)}. \end{aligned}\] Simplification and reindexing leads to \[\sum_{k=0}^n \left( \frac{n!}{(n-2k)! (k!)^2 (k+1)} + \frac{n!}{(n-2k-1)! (k!)^2 (k+1)} \right).\] Rewriting gives \[\sum_{k=0}^n \frac{n!}{ (k!) (k+1)!}\left( \frac{1}{(n-2k)!} + \frac{1}{(n-2k-1)!} \right) = \sum_{k=0}^n \frac{n!}{(k!) (k+1)!}\left( \frac{1}{(n-2k)!} + \frac{n-2k}{(n-2k)!} \right)\] so we end up with \[\sum_{k=0}^n (n+1-2k) \frac{n!}{ (k!) (k+1)! (n-2k)!}.\] The expression \(\frac{n!}{(k!) (k+1)! (n-2k)!}\) is exactly A055151, so this verifies the conjecture by W. Schulte, Oct 23 2016.
#Schubert charge
The Schubert polynomials generalize the Schur polynomials, so for Grassmannian permutations there should be a bijection from pipe dreams to SSYT. What is the notion of (co)charge on pipe dreams? Does it generalize to arbitrary permutations?
#Combinatorial characterization of interval intersections
See this MathOverflow discussion.
#Roots of staircase Schur polynomials (solved)
I wrote down the following observation here, and V. Tewari pointed out the straightforward proof presented below.
Proposition
Let \(\delta_n\) be the staircase partition \((n,n-1,\dotsc,1,0).\) Consider the Schur polynomial indexed by the stretched staircase, \[P_{n,k}(t) \coloneqq \schurS_{k \delta_n}(t,1^{n}).\] Then \[P_{n,k}(t) = (k+1)^{\binom{n}{2}} ([k+1]_t)^{n}.\]
Proof
By the Vandermonde determinant formula, \[\schurS_{k \delta_n}(x_1,\dotsc,x_n) = \prod_{1 \leq i \lt j \leq n+1} \frac{x_i^{k+1}-x_j^{k+1}}{x_i-x_j}.\] This then implies the claim.
Bibliography
- [AA19]Per Alexandersson and Elie Alhajjar. Ehrhart positivity and Demazure characters. Algebraic and geometric combinatorics on lattice polytopes, June 2019.
.bib
@inproceedings{AlexanderssonAlhajjar2018, doi = {10.1142/9789811200489_0003}, url2 = {https://doi.org/10.1142/9789811200489_0003}, year = {2019}, month = jun, publisher = {World Scientific}, author = {Per Alexandersson and Elie Alhajjar}, title = {Ehrhart positivity and {D}emazure characters}, booktitle = {Algebraic and Geometric Combinatorics on Lattice Polytopes} } - [AA19]Per Alexandersson and Nima Amini. The cone of cyclic sieving phenomena. Discrete Mathematics, 342(6):1581–1601, 2019.
.bib
@article{AlexanderssonAmini2018, Author = {Per Alexandersson and Nima Amini}, Title = {The Cone of Cyclic Sieving Phenomena}, Year = {2019}, journal = {Discrete Mathematics}, volume = {342}, number = {6}, pages = {1581--1601}, issn = {0012-365X}, doi = {10.1016/j.disc.2019.01.037}, url2 = {http://www.sciencedirect.com/science/article/pii/S0012365X19300482} }