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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).

%I A386841 #31 Aug 20 2025 21:53:44
%S A386841 1,1,3,2,4,12,4,10,38,82,2,24,88,254,602,4,32,198,643,2421,6710,8,56,
%T A386841 332,1442,6445,23285,83906,4
%N 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).
%C A386841 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.
%H A386841 Carlos Améndola, Viet Duc Nguyen, and Janike Oldekop, <a href="https://arxiv.org/abs/2507.18686">One-dimensional discrete models of maximum likelihood degree one</a>, arXiv:2507.18686 [math.ST], 2025. See p. 20 (Figure 9).
%H A386841 Arthur Bik and Orlando Marigliano, <a href="https://doi.org/10.1016/j.aam.2025.102928">Classifying one-dimensional discrete models with maximum likelihood degree one</a>, Adv. Appl. Math., 170 (2025), 102928.
%e A386841 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.
%e A386841 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).
%e A386841 Continuing in this way, the first five rows (1<=n<=5) of the fundamental models triangle are:
%e A386841   1
%e A386841   1 3
%e A386841   2 4 12
%e A386841   4 10 38 82
%e A386841   2 24 88 254 602
%Y A386841 Columns 1..4 are  A143107, A143108, A387029, A386840.
%K A386841 hard,nonn,more,tabl
%O A386841 1,3
%A A386841 _Carlos Améndola_, Aug 12 2025