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

1,3

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

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}]

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, A373724.

Sequence in context: A240337 A084271 A092086 * A369142 A218478 A223624

Adjacent sequences: A373720 A373721 A373722 * A373724 A373725 A373726

Keyword

nonn

Author

José María Grau Ribas, Jun 15 2024

Extensions

More terms from Robin Visser, Apr 18 2025

Status

approved