cp's OEIS Frontend

A387029 Number of fundamental one-dimensional discrete statistical models with rational maximum likelihood estimator supported on n+1 states and of degree 2n-3.

%I A387029 #14 Aug 20 2025 11:14:42
%S A387029 0,0,12,38,88,198,332
%N A387029 Number of fundamental one-dimensional discrete statistical models with rational maximum likelihood estimator supported on n+1 states and of degree 2n-3.
%C A387029 A143109 is likely an erroneous version of this sequence.
%C A387029 Table 2 of Lebl and Lichtblau gives (incorrect) a(3)=11.
%H A387029 C. Améndola, V. Nguyen and J. 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.
%H A387029 A. Bik and O. 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.
%H A387029 J. Lebl and D. Lichtblau, <a href="http://arxiv.org/abs/0808.0284">Uniqueness of certain polynomials constant on a hyperplane</a>, arXiv:0808.0284 [math.CV], 2008-2010.
%e A387029 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:
%e A387029   1.  x + x^2*y + 2*x*y^2 + y^3,
%e A387029   2.  x + x^2*y + y^2 + x*y^2,
%e A387029   3.  x + x*y + x*y^2 + y^3,
%e A387029   4.  x^2 + 2*x^2*y + 3*x*y^2 + y^3,
%e A387029   5.  x^2 + 2*x^2*y + y^2 + 2*x*y^2,
%e A387029   6.  x^2 + 2*x*y + x*y^2 + y^3,
%e A387029   7.  x^2 + y + x^2*y + x*y^2,
%e A387029   8.  x^3 + 2*x*y + x^2*y + y^2,
%e A387029   9.  x^3 + 3*x^2*y + 3*x*y^2 + y^3,
%e A387029   10. x^3 + 3*x^2*y + y^2 + 2*x*y^2,
%e A387029   11. x^3 + y + 2*x^2*y + x*y^2,
%e A387029   12. x^3 + y + x*y + x^2*y.
%Y A387029 Cf. A143107, A143108, A143109, A386841.
%K A387029 hard,nonn
%O A387029 1,3
%A A387029 _Carlos Améndola_, Aug 05 2025