A380660 Rectangular array pos(i,j,1,2) read by descending antidiagonals: pos( ) and neg() denote the positive part and negative part of a determinant; see Comments.

5, 16, 27, 48, 65, 84, 119, 144, 171, 200, 253, 288, 325, 364, 405, 480, 527, 576, 627, 680, 735, 836, 897, 960, 1025, 1092, 1161, 1232, 1363, 1440, 1519, 1600, 1683, 1768, 1855, 1944, 2109, 2204, 2301, 2400, 2501, 2604, 2709, 2816, 2925, 3128, 3243, 3360

Offset: 1

Author

Clark Kimberling, Feb 04 2025

Keywords

• nonn
• tabl

Comments

Suppose that (m(i,j)) is a rectangular array of infinitely many rows and infinitely many columns. For integers s>=1 and n>=1, let M(i,j,s,n) be the nXn matrix (m(i+h*s,j+k*s)), where h=0..n-1, k=0..n-1.

Let D(i,j,s,n) and P(i,j,s,n) denote the determinant and permanent of M(i,j,s,n), respectively. Define arrays pos(i,j,s,n) and neg(i,j,s,n) by pos(i,j,s,n) = (P(i,j,s,n)+D(i,j,s,n))/2 and neg(i,j,s,n) = (P(i,j,s,n)-D(i,j,s,n))/2, so that P(i,j,s,n) = pos(i,j,s,n)+neg(i,j,s,n) and D(i,j,s,n) = pos(i,j,s,n)-neg(i,j,s,n).

A definition of determinant of an nXn matrix (a(i,j)) is the sum of the products (-1)^p(u) a(1,j(1))*a(2,j(2))*...*a(n,j(n)) over the n! permutations u = (j(1),j(2),...,j(n)) of (1,2,...,n), where p(u) is the parity of u; i.e., p(u) = 0 or 1 according as u is an even or odd permutation; see Lang, pp. 452-3, especially Proposition 4.8.

We have:

pos(i,j,s,n) is the sum of the n!/2 products for which p(u) = 0, and

neg(i,j,s,n) is the sum of the n!/2 products for which p(u) = 1.

Here, the foundational array (m(i,j)) is the natural number array (see A000027, A185787, A144112). The row sequences of pos(i,j,s,n) and neg(i,j,s,n) are linearly recurrent with signature (5, -10, 10, -5, 1).

Examples

			Corner of pos(i,j,1,2):
     5     16     48    119    253    480    836   1363   2109
    27     65    144    288    527    897   1440   2204   3243
    84    171    325    576    960   1519   2301   3360   4756
   200    364    627   1025   1600   2400   3479   4897   6720
   405    680   1092   1683   2501   3600   5040   6887   9213
   735   1161   1768   2604   3723   5185   7056   9408  12319
  1232   1855   2709   3848   5332   7227   9605  12544  16128
  1944   2816   3975   5481   7400   9804  12771  16385  20736
  2925   4104   5632   7575  10005  13000  16644  21027  26245
  4235   5785   7752  10208  13231  16905  21320  26572  32763
  5940   7931  10413  13464  17168  21615  26901  33128  40404
  8112  10620  13699  17433  21912  27232  33495  40809  49288
M(1,1,1,2) is the matrix with (row 1) = (1,2), (row 2) =(3,5), so that
pos(1,1,1,2) = 1*5 = 5; neg(1,1,1,2) = 2*3 = 6; D(1,1,1,2) = -1; P(1,1,1,2) = 11.
		

References

• S. Lang, Algebra, 2nd ed., Addison-Wesley, 1984, 452-453.

Links

• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A000027, A380649, A380661.

Programs

Mathematica

s = 1; n = 2; z = 12;
r[n_, k_] := n + (n + k - 2)*(n + k - 1)/2 (* Array A000027 *)
Grid[Table[r[n, k], {n, 1, z}, {k, 1, z}]]
t[i_, j_] := Table[r[i, j + k*s], {k, 0, n - 1}];
d[i_, j_] := Det[Table[t[i + k*s, j], {k, 0, n - 1}]];  (* D(i,j,s,n) *)
p[i_, j_] := Permanent[Table[t[i + k*s, j], {k, 0, n - 1}]];  (* P(i,j,s,n) *)
pos[i_, j_] := (p[i, j] + d[i, j])/2;
neg[i_, j_] := (p[i, j] - d[i, j])/2;
Grid[Table[pos[i, j], {i, 1, z}, {j, 1, z}]]  (* A380660 array *)
Grid[Table[neg[i, j], {i, 1, z}, {j, 1, z}]]  (* A380661 array *)
FindLinearRecurrence[Table[pos[1, k], {k, 1, 20}]] (* row recurrence, all rows *)
FindLinearRecurrence[Table[neg[7, k], {k, 1, 20}]] (* row recurrence, all rows *)
Table[pos[k, m - k], {m, 2, z}, {k, 1, m - 1}] // Flatten (* A380660 sequence *)
Table[neg[k, m - k], {m, 2, z}, {k, 1, m - 1}] // Flatten (* A380661 sequence *)