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3. Lecture Hall Combinatorics: Partitions, Polytopes, and Valuations


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      Problem 3.1.

      Characterize simultaneous core partitions in terms of lecture hall partitions.


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          Problem 3.2.

          Consider deformations of the real braid arrangement by $s$-lecture hall partitions $\left(\frac{x_i}{s_i}=\frac{x_j}{s_j}\right)$. Starting with the case of $s=(1,2,\cdots,n)$, study the theory of generalized permutahedra and other geometric objects arising from this arrangement.


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              Problem 3.3.

              Consider plane lecture hall partitions on the poset $[0,n]\times[0,m]\subset\Z^2$ which are defined to be functions $f$ such that \[ 0\leq \frac{f(x)}{{\rm rank}(x)+1}\leq \frac{f(y)}{{\rm rank}(y)+1} \] when $x\leq y$. Are there nice combinatorics which arise? Start with the case $n=m=2$.
                  If we specify that $ \frac{f(y)}{{\rm rank}(y)+1}\leq 1$, this defines a Gorenstein polytope.


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                  Problem 3.4.

                  Consider valuations of $s$-lecture hall polytopes different than the usual Ehrhart polynomial.


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                      Problem 3.5.

                      Find algebraic interpretation (e.g. Coxeter etc.) of $s$-lecture hall partitions.
                          Promising places to start are the anti lecture hall sequence $(n,n-1,\cdots,2,1)$ or $\ell$-sequences.


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                          Problem 3.6.

                          Consider principle evaluations of symmetric functions in the context of $s$-lecture hall partitions.


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                              Problem 3.7.

                              What is the Hilbert basis for $L_n^{(s)}$ for general $s$?
                                  This is known in the case of $s=(1,2,\cdots,n)$ by M. Beck, B. Braun, M. K$\"{o}$ppe, C. Savage, and Z. Zafeirakopoulos.


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                                  Problem 3.8.

                                  What are the multigraded Betti numbers $\beta_{i,m}$ of $L_n^{(s)}\cap \mathbb{Z}^n$?


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                                      Problem 3.9.

                                      [T. Amdeberhan] The $\ell$-sequences are defined by the recurrence $\{0\}_{\ell,-1}=0, \{1\}_{\ell,-1}=1$ and for $n\geq2$: $$\{n\}_{\ell,-1}=\ell\cdot\{n-1\}_{\ell,-1}-\{n-2\}_{\ell,-1}.$$ Write $\{n\}$ short for $\{n\}_{\ell,-1}$. The Generalized Lecture Hall Theorem (due to Mireille Bousquet-Melou and Kimmo Eriksson) depends on a polynomial analogue of the fact that for $\ell$-sequences, the following ratio is integral for all $n\geq1$: $$\frac{\{n\}(\{n\}+\{n-1\})\cdots(\{n\}+\cdots+\{1\})}{\{n\}\{n-1\}\cdots\{1\}}.$$ Conjecture: For integers $n, k\geq1$ and $\ell\geq2$, the following ratios $$\prod_{j=1}^n\frac{\{n\}^{2k-1}+\cdots+\{j\}^{2k-1}}{\{j\}}, \qquad \frac{\{n\}^{2k-1}+\cdots+\{1\}^{2k-1}}{\{n\}+\cdots+\{1\}}, \qquad \frac{\{n\}^{3(4k-3)}+\cdots+\{1\}^{3(4k-3)}}{\{n\}^3+\cdots+\{1\}^3}$$ are all integrals. m ̊The last assertion holds only for $\ell=2$.

                                      A stronger conjecture: The following are also integers. $$\prod_{j=1}^n\frac{\{2n\}^{2k-1}+\{2n-2\}^{2k-1}+\cdots+\{2j\}^{2k-1}}{\{2j\}}, \\ \prod_{j=1}^n\frac{\{2n-1\}^{2k-1}+\{2n-3\}^{2k-1}+\cdots+\{2j-1\}^{2k-1}}{\{2j-1\}},\\ \prod_{j=1}^n\frac{\{2n\}^{2k-1}+\{2n-1\}^{2k-1}+\cdots+\{2j\}^{2k-1}}{\{2j-1\}}, \\ \prod_{j=1}^n\frac{\{rn\}^{2k-1}+\{r(n-1)\}^{2k-1}+\cdots+\{rj\}^{2k-1}}{\{rj\}}.$$ \bf Some special cases. m ̊We have integrality of $$\frac{\sum_{j=1}^n\{2j\}^3}{\{n\}^2\{n+1\}^2}, \qquad \frac{\sum_{j=1}^n\{2j-1\}^3}{ \{n\}^2}.$$ $$\frac{\sum_{j=1}^n\{2j\}^{2k-1}}{\{n\}\{n+1\}}, \qquad \frac{\sum_{j=1}^n\{2j-1\}^{2k-1}}{\{n\}}.$$

                                      Question: Is there a combinatorial reason why?

                                          Cite this as: AimPL: Polyhedral geometry and partition theory, available at http://aimpl.org/polypartition.