Carlos Améndola has authored 3 sequences.
A386841
Triangle read by rows: T(n,k) is the number of fundamental one-dimensional discrete statistical models with rational maximum likelihood estimator supported on the n-dimensional probability simplex and of degree 2n-k (n>=1, 1<=k<=n).
Original entry on oeis.org
1, 1, 3, 2, 4, 12, 4, 10, 38, 82, 2, 24, 88, 254, 602, 4, 32, 198, 643, 2421, 6710, 8, 56, 332, 1442, 6445, 23285, 83906, 4
Offset: 1
When n=1 then k=1 and the unique model T(1,1)=1 corresponds to the model described by a Bernoulli random variable that assigns probabilities 1-t and t to two possible states, 0<=t<=1. This line segment parametrizes the 1-dimensional probability simplex.
When n=2 we have 1<=k<=2. The T(2,1)=1 unique fundamental model with degree 3 corresponds to the parametrization t -> ((1-t)^3, 3t(1-t), t^3) and the T(2,2)=3 fundamental models of degree 2 correspond to the parametrizations ((1-t)^2, 2t(1-t), t^2) , (1-t, t(1-t), t^2) and ((1-t)^2, t(1-t), t).
Continuing in this way, the first five rows (1<=n<=5) of the fundamental models triangle are:
1
1 3
2 4 12
4 10 38 82
2 24 88 254 602
A386840
Number of fundamental one-dimensional discrete statistical models with rational maximum likelihood estimator supported on n states and of degree 2n-6.
Original entry on oeis.org
0, 0, 0, 0, 82, 254, 643, 1442
Offset: 1
- 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, Linear Algebra Appl., 433 (2010), no. 4, 824-837
A387029
Number of fundamental one-dimensional discrete statistical models with rational maximum likelihood estimator supported on n+1 states and of degree 2n-3.
Original entry on oeis.org
0, 0, 12, 38, 88, 198, 332
Offset: 1
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.
- 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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