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1. Polynomials Arising From Combinatorics


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      Problem 1.1.

      Determine the precise locations of the roots of Eulerian Polynomials. In particular, prove an outstanding conjecture of the asymptotic behavior of the largest root (and other roots).
        1. Remark. [Richard Stanley] What is the conjecture? The roots are well-known to be negative real numbers. The sum of the roots of the n-th Eulerian polynomial is -(2^n-n-1). Since there are only n roots, the largest root will be near 2^n. Similarly the second elementary symmetric function of the roots is of the order 3^n. Thus the second largest root is around (3/2)^n. The third largest root is around (4/3)^n, etc.


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              Problem 1.2.

              Let $D(\lambda)$ be the size of the Durfee square of $\lambda$. Define $D_n(x):=\sum_{\lambda\vdash n} x^{D(\lambda)}$. Prove or disprove that $D_n(x)$ has all real roots.
                  It is known that $D_n(x)$ is asymptotically log concave. Asymptotically, the mean and mode differ by at most 1.


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                  Problem 1.3.

                  Study unimodality of families of palindromic polynomials via alternating $\gamma$-vectors. Some polynomials to consider are sums over partitions with distinct parts or ${n \choose k}_q$.


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                      Problem 1.4.

                      Define $\displaystyle\mathcal{A}_n(t) =\sum_{\lambda\vdash n}\prod_{\Box\in\lambda}\frac{h_{\Box}^2+t}{h_{\Box}}$ where $h_\Box$ is the hook length of $\Box$. Consider the problems:
                      • Prove Unimodality
                      • Prove that $\mathcal{A}_n(t)$ has only negative real roots.
                      • Prove that $\mathcal{A}_n(t)$ has only simple negative real roots.
                          If $t\in\Z_{>0}$ these are partitions with $t$ colors. It can be shown that $\displaystyle \mathcal{A}_n(t)=\sum_{\lambda\vdash n}\prod_{\substack{\Box\in \lambda\\ { \Box \tiny \mbox{ has no arms}}}} \frac{h_\Box+t}{h_\Box}$.
                        1. Remark. In the first definition of the polynomials $A_n(t)$, the denominator must also be "squares".


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                              Problem 1.5.

                              Let $C(n,a,b,m)= {\rm ConstantTerm}_{x_1,\cdots,x_n}\left(\prod_{i=1}^{n}(1-x_i)^bx_i^{-a}\prod_{{1\leq i,j\leq n} \\ }(x_i-x_j)^{-m}\right)$. Morris’s identity is $C(n,a,b,m)=C(n,b+1,a-1,m)$.

                              Give a combinatorial proof of Morris’s identity.

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