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).

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

Author

Carlos Améndola, Aug 12 2025

Keywords

• hard
• nonn
• more
• tabl

Comments

The range of k is precisely chosen so that T(n,k) is positive. That is, whenever the degree is higher than 2n-1 or lower than n, there are no fundamental models.

Examples

			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
		

Links

• Carlos Améndola, Viet Duc Nguyen, and Janike Oldekop, One-dimensional discrete models of maximum likelihood degree one, arXiv:2507.18686 [math.ST], 2025. See p. 20 (Figure 9).
• Arthur Bik and Orlando Marigliano, Classifying one-dimensional discrete models with maximum likelihood degree one, Adv. Appl. Math., 170 (2025), 102928.

Crossrefs

Columns 1..4 are A143107, A143108, A387029, A386840.