A143108 -id:A143108 - cp's OEIS frontend

A143107 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, i.e., of degree 2n-3.

0, 1, 1, 2, 4, 2, 4, 8, 4, 2, 24, 2

Offset: 1

Jiri Lebl, Jul 25 2008

It is unknown if this sequence is bounded. For all n >= 4, a(n) is at least two. It is unknown if it is 2 for infinitely many n. It is unknown if it is always even for all n >= 2. Note that 2n-3 appears in A143106 if and only if a(n) is 1 or 2.

Also the number of fundamental one-dimensional discrete statistical models with rational maximum likelihood estimator supported on n states and of degree 2n-3. - Carlos Améndola, Aug 05 2025

			a(3) = 1 as x^3 + 3xy + y^3 is the unique polynomial in H(2,d) with 3 terms and of maximum degree (in this case 3).
This corresponds to the discrete model on 3 states parametrized by t-> (t^3, 3t(1-t), (1-t)^3), 0<=t<=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. P. D'Angelo and J. Lebl, Complexity results for CR mappings between spheres, Internat. J. Math. 20 (2009), no. 2, 149-166.
J. P. D'Angelo and J. Lebl, Complexity results for CR mappings between spheres, arXiv:0708.3232 [math.CV], 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. Lebl, Addendum to Uniqueness of certain polynomials constant on a line arxiv 1302.1441 [math.AC], 2013.
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.

Cf. A143106, A143108, A143109.

Mathematica
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See the paper by Lebl and Lichtblau.
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One more term (24), added addendum to and corrected title of paper - Jiri Lebl, Feb 08 2013

Added another term (2) that was computed in the newer version of the addendum. Edited by Jiri Lebl, May 02 2014

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

It is unknown but conjectured that this is a sequence of finite numbers. Note that if we went one degree lower and look at polynomials of degree 2n-6, then there are infinitely many if any exist in H(2,d).

Likely an erroneous version of A387029. - Sean A. Irvine, Aug 13 2025

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.
J. P. D'Angelo and J. Lebl, Complexity results for CR mappings between spheres, Internat. J. Math. 20 (2009), no. 2, 149-166.
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

Cf. A143107, A143108, A387029.

Mathematica
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(* See the paper by Lebl-Lichtblau. *)
```

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

Carlos Améndola, Aug 05 2025

A143109 is likely an erroneous version of this sequence.

Table 2 of Lebl and Lichtblau gives (incorrect) a(3)=11.

			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.

Cf. A143107, A143108, A143109, A386841.

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

Carlos Améndola, Aug 12 2025

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.

			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

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.

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

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

Carlos Améndola, Aug 05 2025

Unlike A143107 and A143108 (and conjecturally A143109), there are infinitely many polynomials in H(2,d) of degree 2n-6. Nevertheless, this sequence consists of finite numbers.

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

Cf. A143107, A143108, A143109, A387029, A386841.

cp's OEIS Frontend

A143107 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, i.e., of degree 2n-3.

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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.

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

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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).

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A386840 Number of fundamental one-dimensional discrete statistical models with rational maximum likelihood estimator supported on n states and of degree 2n-6.

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