A373724 Number of totally positive 3 X 3 matrices having all terms in {1,...,n}.

1, 61, 797, 6490, 32744, 146441, 492277, 1521123, 4105795, 10194558, 22922408, 49594408, 98935110, 190221734, 350417949, 621178227, 1058404994, 1764873413, 2845696865, 4506618651, 6966717779, 10552756376, 15670141644, 22984055065, 33094853060, 47016605050, 65934960254, 91414399149

Offset

1,2

Comments

A matrix is totally positive if all its minors are nonnegative.

Links

Robin Visser, Table of n, a(n) for n = 1..40

Mathematica

ispositive2[M_]:=ispositive1[M]=Union@Table[Select[Union@Flatten@Minors[M, r], (#<= 0)&]=={}, {r, 1, Length[M]}]=={True};
W[n_]:=W[n]=Flatten[Table[{{a11, a12, a13}, {a21, a22, a23}, {a31, a32, a33}}, {a11, 1, n}, {a12, 1, n}, {a13, 1, n}, {a21, 1, n}, {a22, 1, n}, {a23, 1, n}, {a31, 1, n}, {a32, 1, n}, {a33, 1, n}], 8];
Table[Length@Select[W[n], ispositive2[#]&], {n, 1, 6}]

Prog

(SageMath)
import itertools
def a(n):
    ans, W = 0, itertools.product(range(1, n+1), repeat=9)
    for w in W:
        M = Matrix(ZZ, 3, 3, w)
        if (min(M.minors(2)) >= 0) and (M.det() >= 0): ans += 1
    return ans  # Robin Visser, Apr 18 2025

Crossrefs

Cf. A211059, A373723.

Sequence in context: A372963 A251426 A232858 * A093473 A000506 A008358

Adjacent sequences: A373721 A373722 A373723 * A373725 A373726 A373727

Keyword

nonn

Author

José María Grau Ribas, Jun 15 2024

Extensions

More terms from Robin Visser, Apr 18 2025

Status

approved