---
tag: NDYA85381W
kind: statement
role: theorem
name: Dyson conjecture
msc: [05A19, 33C52]
uses: [1R5P656GS6, 0GBZM0MT0F]
about: [0GBZM0MT0F]
sources:
  - url: https://en.wikipedia.org/wiki/Dyson_conjecture
same_as:
  wikidata: Q5319505
  wikipedia: Dyson conjecture
---

## Statement

Let $n \ge 1$ and let $a_1, \dots, a_n$ be nonnegative integers.
In the [[ring of Laurent polynomials|1R5P656GS6]] $\Q[x_1^{\pm 1}, \dots, x_n^{\pm 1}]$, the constant term of
$$\prod_{1 \le i \ne j \le n} \left( 1 - \frac{x_i}{x_j} \right)^{a_i}$$
is the multinomial coefficient
$$\frac{(a_1 + a_2 + \cdots + a_n)!}{a_1!\, a_2! \cdots a_n!}.$$

## Remarks

Freeman Dyson stated this as a conjecture in 1962, in work on the statistical theory of energy levels.
It was proved the same year by Gunson and by Wilson, and Good gave a short proof in 1970.
Good's proof shows that both sides satisfy the recurrence $F(a) = \sum_{i} F(a - e_i)$ when all $a_i \ge 1$, and that both reduce to the case of $n - 1$ variables when some $a_i = 0$.
The recurrence for the left side comes from the identity $\sum_{i=1}^n \prod_{j \ne i} \left(1 - \frac{x_j}{x_i}\right)^{-1} = 1$, a case of Lagrange interpolation.
Example: $n = 2$.
The constant term of $(1 - x_1/x_2)^{a_1} (1 - x_2/x_1)^{a_2}$ collects the products of the terms $(x_1/x_2)^k$ and $(x_2/x_1)^k$, and equals $\sum_{k} \binom{a_1}{k}\binom{a_2}{k} = \binom{a_1 + a_2}{a_1}$ by [[Vandermonde's identity|01JADB9VW5]].

## Explanations

### L1 human

Multiply out a product of many factors of the form $1 - x_i/x_j$ and keep only the terms with no variable left.
The Dyson conjecture, now a theorem, says that the sum of these terms is a multinomial coefficient, the number of ways to arrange $a_1 + \cdots + a_n$ objects of $n$ colours.

### L2 human

The case where all $a_i$ equal $k$ gives $\frac{(nk)!}{(k!)^n}$, which computes the normalizing constant of the circular ensembles of [[random matrices|1HS4RT7HK5]] through an integral over the torus.
Andrews proposed a $q$-analogue, proved by Zeilberger and Bressoud.
The equal-parameter case is the type $A$ case of the [[Macdonald constant term conjecture|5J25W265JA]].