A143109
Let H(2,d) be the space of polynomials p(x,y) of two variables with nonnegative coefficients such that p(x,y)=1 whenever x + y = 1. a(n) is the number of different polynomials in H(2,d) with exactly n distinct monomials and of maximum degree minus two, i.e., of degree 2n-5.
Original entry on oeis.org
0, 0, 0, 11, 38, 88, 198
Offset: 1
Jiri Lebl (jlebl(AT)math.uiuc.edu), Jul 25 2008
- J. P. D'Angelo, Simon Kos and Emily Riehl, A sharp bound for the degree of proper monomial mappings between balls, J. Geom. Anal., 13(4):581-593, 2003.
- J. P. D'Angelo and J. Lebl, Complexity results for CR mappings between spheres, arXiv:0708.3232 [math.CV], 2008.
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- J. Lebl and D. Lichtblau, Uniqueness of certain polynomials constant on a hyperplane, arXiv:0808.0284 [math.CV], 2008-2010.
- J. Lebl and D. Lichtblau, Uniqueness of certain polynomials constant on a hyperplane, Linear Algebra Appl., 433 (2010), no. 4, 824-837
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
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