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

1,3

Comments

A143109 is likely an erroneous version of this sequence.

Table 2 of Lebl and Lichtblau gives (incorrect) a(3)=11.

Links

Table of n, a(n) for n=1..7.

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.

Example

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.

Crossrefs

Cf. A143107, A143108, A143109, A386841.

Sequence in context: A079539 A242720 A212510 * A213490 A259517 A303619

Adjacent sequences: A387026 A387027 A387028 * A387030 A387031 A387032

Keyword

hard,nonn

Author

Carlos Améndola, Aug 05 2025

Status

approved