The symmetric functions catalog

An overview of symmetric functions and related topics


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.

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 [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.

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 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.

See this MO-post also

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,

\[ P_{n,k}(t) \coloneqq \schurS_{k \delta_n}(t,1^{n}). \]


\[ P_{n,k}(t) = (k+1)^{\binom{n}{2}} ([k+1]_t)^{n}. \]

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.