A062801 -id:A062801 - cp's OEIS frontend

A183761 Number of 2 X 2 nonsingular 0..n matrices with rows in increasing order.

3, 25, 96, 256, 563, 1073, 1880, 3056, 4715, 6961, 9944, 13752, 18603, 24601, 31936, 40800, 51427, 63937, 78664, 95720, 115435, 138057, 163888, 193064, 226059, 263089, 304480, 350528, 401715, 458145, 520488, 588872, 663803, 745681, 834872, 931736

Offset: 1

R. H. Hardin, Jan 06 2011

nonn

Possibly this sequence gives the number of 2 X 2 matrices with all terms in {0,1,...,n} and positive determinant, as evidenced by a program in the Mathematica section. - Clark Kimberling, Mar 19 2012

This conjecture is true, since half of all 2 X 2 nonsingular 0..n matrices have rows in increasing order, and half of them have positive determinant. - David Radcliffe, Aug 13 2025

			Some solutions for n=2:
..1..0....1..0....1..2....0..2....1..1....1..1....0..1....2..0....0..2....1..2
..2..2....1..2....2..1....1..0....2..0....1..2....2..0....2..1....1..2....2..2
Contribution from _Clark Kimberling_, Mar 19 2012: (Start)
As an example for counting positive determinants (see Comments), the 3 matrices counted by a(1) are
1 0.....1 1.....1 0
0 1.....0 1.....1 1  (End)

R. H. Hardin, Table of n, a(n) for n = 1..200

Cf. A062801.

a = 0; b = n; z1 = 45;
t[n_] := t[n] = Flatten[Table[w*z - x*y, {w, a, b}, {x, a, b}, {y, a, b}, {z, a, b}]]
c[n_, k_] := c[n, k] = Count[t[n], k]
c1[n_, m_] := c1[n, m] = Sum[c[n, k], {k, 0, m}]
Table[c1[n, n^2] - c[n, 0], {n, 0, z1}]
(* Clark Kimberling, Mar 19 2012 *)

a(n) = A062801(n)/2. - David Radcliffe, Aug 13 2025

A064363 Number of 2 X 2 regular integer matrices with elements from {0,...,n} up to row and column permutation.

Original entry on oeis.org

0, 2, 14, 51, 133, 289, 547, 954, 1546, 2380, 3508, 5005, 6915, 9347, 12353, 16028, 20468, 25790, 32054, 39427, 47965, 57833, 69155, 82082, 96682, 113192, 131720, 152429, 175467, 201075, 229305, 260492, 294700, 332182, 373138, 417751, 466201

Offset: 0

Vladeta Jovovic, Sep 25 2001

			There are 2 binary regular matrices up to row and column permutation:
[1 0] [1 1]
[0 1] [1 0].

Cf. A064276, A059306, A062801, A059976, A039623.

A059306[0] = 1; A059306[n_] := Table[{w, x, y, z} /. {ToRules[ Reduce[0 <= x <= n && 0 <= y <= n && 0 <= z <= n && w*z - x*y == 0, {x, y, z}, Integers]]}, {w, 0, n}] // Flatten[#, 1] & // Length; a[n_] := ((n + 1)*(n^3 + 3*n^2 + 4*n + 1) - A059306[n])/4; Table[Print[an = a[n]]; an, {n, 0, 36}] (* Jean-François Alcover, Nov 26 2013 *)

a(n) = ((n+1)*(n^3+3*n^2+4*n+1)-A059306(n))/4.

cp's OEIS Frontend

A183761 Number of 2 X 2 nonsingular 0..n matrices with rows in increasing order.

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