2021-06-26
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 GT-polytope. See [AA19a] for more background.
We may compute the corresponding $h^*$-polynomial, and get some interesting properties.
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 [AA19b], 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.
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 poset 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{align*} &\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{align*}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 Werner Schulte, Oct 23 2016.
Schubert charge
The Schubert polynomials generalize the Schur polynomials, so for Grassmann permutations, there should be a bijection to SSYT from pipe dreams. What is the notion of (co)charge on pipe dreams? Does it generalize to arbitrary permutations?
Combinatorial characterization of interval intersections
Roots of staircase Schur polynomials (solved)
I wrote down the following observation here, and Vasu Tewari pointed out the straightforward proof presented below.
Let $\delta_n$ be the staircase partition $(n,n-1,\dotsc,1,0).$ Consider the Schur polynomial indexed by the stretched straircase,
\[ P_{n,k}(t) \coloneqq \schurS_{k \delta_n}(t,1^{n}). \]Then
\[ P_{n,k}(t) = (k+1)^{\binom{n}{2}} ([k+1]_t)^{n}. \]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.
References
- [AA19a] Per Alexandersson and Elie Alhajjar. Ehrhart positivity and Demazure characters. Algebraic and Geometric Combinatorics on Lattice Polytopes. World Scientific, June 2019.
- [AA19b] Per Alexandersson and Nima Amini. The cone of cyclic sieving phenomena. Discrete Mathematics, 342(6):1581–1601, 2019.