A387029 Number of fundamental one-dimensional discrete statistical models with rational maximum likelihood estimator supported on n+1 states and of degree 2n-3.
0, 0, 12, 38, 88, 198, 332
Offset: 1
Examples
For n=3 there are a(3)=12 models supported on 3+1=4 states of degree 2*3-3=3. Encoding each model parametrization as a bivariate polynomial shows why the 4th term of A143109 is also 12. Concretely, the following polynomials in x,y with 4 terms and of degree 2*4-5=3 yield the constant 1 when making the substitution y=1-x: 1. x + x^2*y + 2*x*y^2 + y^3, 2. x + x^2*y + y^2 + x*y^2, 3. x + x*y + x*y^2 + y^3, 4. x^2 + 2*x^2*y + 3*x*y^2 + y^3, 5. x^2 + 2*x^2*y + y^2 + 2*x*y^2, 6. x^2 + 2*x*y + x*y^2 + y^3, 7. x^2 + y + x^2*y + x*y^2, 8. x^3 + 2*x*y + x^2*y + y^2, 9. x^3 + 3*x^2*y + 3*x*y^2 + y^3, 10. x^3 + 3*x^2*y + y^2 + 2*x*y^2, 11. x^3 + y + 2*x^2*y + x*y^2, 12. x^3 + y + x*y + x^2*y.
Links
- C. Améndola, V. Nguyen and J. Oldekop, One-dimensional discrete models of maximum likelihood degree one, arXiv:2507.18686 [math.ST] 2025.
- A. Bik and O. Marigliano, Classifying one-dimensional discrete models with maximum likelihood degree one, Adv. Appl. Math., 170 (2025), 102928.
- J. Lebl and D. Lichtblau, Uniqueness of certain polynomials constant on a hyperplane, arXiv:0808.0284 [math.CV], 2008-2010.
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