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open-problems:family-of-constant-term-identities [2019/10/08 19:55] romik |
open-problems:family-of-constant-term-identities [2019/10/08 22:00] (current) romik |
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| A_n = \frac{1!4!7!\cdots (3n-2)!}{n!(n+1)!\cdots(2n-1)!}. | A_n = \frac{1!4!7!\cdots (3n-2)!}{n!(n+1)!\cdots(2n-1)!}. | ||
| $$ | $$ | ||
| - | Define a polynomial $P_n(z_1,\ldots,z_n)$ by | + | Define a multivariate polynomial $P_n(z_1,\ldots,z_n)$ by |
| $$ | $$ | ||
| P_n(z_1,\ldots,z_n) = \prod_{1\le i<j\le n} (z_j-z_i)(1+z_j+z_i z_j) \prod_{j=2}^n (1+z_j). | P_n(z_1,\ldots,z_n) = \prod_{1\le i<j\le n} (z_j-z_i)(1+z_j+z_i z_j) \prod_{j=2}^n (1+z_j). | ||
| $$ | $$ | ||
| Zinn-Justin and Di Francesco (2008) discovered and proved (with a little help from Zeilberger (2007)) the constant term identity | Zinn-Justin and Di Francesco (2008) discovered and proved (with a little help from Zeilberger (2007)) the constant term identity | ||
| - | \begin{equation} | + | $$ |
| - | [z_1^0 z_2^2 \cdots z_n^{2n-2}] P_n(z_1,\ldots,z_n) = A_n, \tag{*} | + | [z_1^0 z_2^2 \cdots z_n^{2n-2}] P_n(z_1,\ldots,z_n) = A_n, \qquad\qquad (*) |
| - | \end{equation} | + | $$ |
| - | where the notation $[z_1^{j_1} \cdots z_n^{j_n}] Q(z_1,\ldots,z_n)$ refers to the coefficient in a multivariate polynomial $Q(z_1,\ldots,z_n)$ of the monomial $z_1^{j_1} \cdots z_n^{j_n}$. | + | where the notation $[z_1^{j_1} \cdots z_n^{j_n}] Q(z_1,\ldots,z_n)$ refers to the coefficientof the monomial $z_1^{j_1} \cdots z_n^{j_n}$ in a multivariate polynomial $Q(z_1,\ldots,z_n)$ . |
| - | Romik (2014) discovered a family of identities generalizing (*). The idea is that, fixing $k\ge 1$ and a "deformation" vector $(a_1,\ldots,a_k)$ of integers, we deform the polynomial $P_n(z_1,\ldots,z_n)$ by replacing it with the slightly modified version | + | Romik (2014) discovered a family of identities generalizing $(*)$. The idea is that, fixing $k\ge 1$ and a "deformation" vector $(a_1,\ldots,a_k)$ of integers, we deform the polynomial $P_n(z_1,\ldots,z_n)$ by replacing it with the slightly modified version |
| $$ | $$ | ||
| P_{n,k}(z_1,\ldots,z_n) = \prod_{1\le i<j\le n} (z_j-z_i)(1+z_j+z_i z_j) \prod_{j=k+2}^n (1+z_j) \qquad (n\ge k+2), | P_{n,k}(z_1,\ldots,z_n) = \prod_{1\le i<j\le n} (z_j-z_i)(1+z_j+z_i z_j) \prod_{j=k+2}^n (1+z_j) \qquad (n\ge k+2), | ||
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| ===== Background and history ===== | ===== Background and history ===== | ||
| - | 1. The numbers $A_n$ are an important sequence of numbers. They enumerate alternating sign matrices, descending plane partitions and other classes of combinatorial objects. See this [[http://oeis.org/A005130|OEIS page]]. | + | 1. The number sequence $(A_n)_{n=1}^\infty = 1,2,7,42,429,\ldots$ is an important sequence of numbers. The $A_n$'s enumerate alternating sign matrices, descending plane partitions and other classes of combinatorial objects. See this [[http://oeis.org/A005130|OEIS page]]. |
| | | ||
| 2. The [[https://en.wikipedia.org/wiki/Dyson_conjecture|Dyson conjecture]] was an early example of a nontrivial constant term identity. Many such identities have been discovered, but no systematic theory for proving them seems to have been developed (that by itself looks like an interesting, though not completely well-defined, open problem). | 2. The [[https://en.wikipedia.org/wiki/Dyson_conjecture|Dyson conjecture]] was an early example of a nontrivial constant term identity. Many such identities have been discovered, but no systematic theory for proving them seems to have been developed (that by itself looks like an interesting, though not completely well-defined, open problem). | ||
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| * D. Romik. [[https://arxiv.org/abs/1303.6341|Connectivity patterns in loop percolation I: the rationality phenomenon and constant term identities]]. //Commun. Math. Phys.// 330 (2014), 499–538. | * D. Romik. [[https://arxiv.org/abs/1303.6341|Connectivity patterns in loop percolation I: the rationality phenomenon and constant term identities]]. //Commun. Math. Phys.// 330 (2014), 499–538. | ||
| * P. Zinn-Justin, P. Di Francesco. [[https://arxiv.org/abs/math-ph/0703015|Quantum Knizhnik-Zamolodchikov equation, totally symmetric self-complementary plane partitions and alternating sign matrices.]] //Theor. Math. Phys.// 154 (2008), 331-348. | * P. Zinn-Justin, P. Di Francesco. [[https://arxiv.org/abs/math-ph/0703015|Quantum Knizhnik-Zamolodchikov equation, totally symmetric self-complementary plane partitions and alternating sign matrices.]] //Theor. Math. Phys.// 154 (2008), 331-348. | ||
| + | * D. Zeilberger. [[http://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/diFrancesco.html|Proof of a Conjecture of Philippe Di Francesco and Paul Zinn-Justin related to the qKZ equation and to Dave Robbins' Two Favorite Combinatorial Objects]]. Online paper (2008). | ||