A373723 Number of strictly totally positive 3 X 3 matrices having all terms in {1,...,n}.

0, 0, 22, 597, 7178, 43090, 207494, 748801, 2321973, 6267631, 15596170, 34784307, 74017706, 147072570, 277965322, 503711791, 884612799, 1491687919, 2458600175, 3925566799, 6133712065, 9388594434, 14121653942, 20783339478, 30178942357, 43156537147, 60868287839, 84699183224, 116688767652

Offset: 1

Author

José María Grau Ribas, Jun 15 2024

Keywords

• nonn

Comments

A matrix is strictly totally positive if all its minors are greater than zero.

Links

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

Crossrefs

Cf. A211059, A373724.

Programs

Mathematica

ispositive1[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],ispositive1[#]&],{n,1,7}]

Sage

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

Extensions

More terms from Robin Visser, Apr 18 2025