====== A family of constant term identities related to loop percolation ======

===== Statement of the problem =====

Define numbers $A_n$ by
$$ 
A_n = \frac{1!4!7!\cdots (3n-2)!}{n!(n+1)!\cdots(2n-1)!}.
$$
Define a multivariate polynomial $P_n(z_1,\ldots,z_n)$ by
$$
P_n(z_1,\ldots,z_n) = \prod_{1\le i<j\le n} (z_j-z_i)(1+z_j+z_i z_j) \prod_{j=2}^n (1+z_j).
$$
Zinn-Justin and Di Francesco (2008) discovered and proved (with a little help from Zeilberger (2007)) the constant term identity
$$
[z_1^0 z_2^2 \cdots z_n^{2n-2}] P_n(z_1,\ldots,z_n) = A_n,  \qquad\qquad (*)
$$
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
$$
P_{n,k}(z_1,\ldots,z_n) = \prod_{1\le i<j\le n} (z_j-z_i)(1+z_j+z_i z_j) \prod_{j=k+2}^n (1+z_j) \qquad (n\ge k+2),
$$
and also change the monomial whose coefficient the identity is looking at by shifting the exponent vector $(0,2,4,\ldots,2n-2)$ by the fixed vector $(a_1,\ldots,a_k)$ (with no shift beyond the $k$th coordinate). The precise statement (Conjecture 1.14 in Romik's paper) is as follows.

**Conjecture.** For any fixed $k$ and integers $a_1,\ldots,a_k$, there exists a rational function $R(n)=p(n)/q(n)$, where $p(n)$ and $q(n)$ are polynomials with integer coefficients, such that
$$
[z_1^{a_1} z_2^{2+a_2} \cdots z_k^{2k-2+a_k} z_{k+1}^{2k} \cdots z_n^{2n-2}] P_{n,k}(z_1,\ldots,z_n) = A_n \times R(n) \qquad (n\ge k+2).
$$


===== Background and history =====

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).

3. As explained in Romik's paper, the above conjecture has interesting implications to the so-called "rationality phenomenon" in a probabilistic model called "loop percolation". Specifically, it would imply that the probabilities of certain classes of events in this model have probabilities that are rational numbers that can be computed using an explicit algorithm.


===== Further reading =====

  * 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.
  * 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).