A380661 Rectangular array neg(i,j,1,2) read by descending antidiagonals: pos() and neg() denote the positive part and negative part of a determinant; see Comments.

6, 20, 30, 56, 72, 90, 132, 156, 182, 210, 272, 306, 342, 380, 420, 506, 552, 600, 650, 702, 756, 870, 930, 992, 1056, 1122, 1190, 1260, 1406, 1482, 1560, 1640, 1722, 1806, 1892, 1980, 2162, 2256, 2352, 2450, 2550, 2652, 2756, 2862, 2970, 3192, 3306, 3422

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 neg(i,j,1,2):
     6     20    56     132    272    506    870   1406   2162   3192
    30     72    156    306    552    930   1482   2256   3306   4692
    90    182    342    600    992   1560   2352   3422   4830   6642
   210    380    650   1056   1640   2450   3540   4970   6806   9120
   420    702   1122   1722   2550   3660   5112   6972   9312  12210
   756   1190   1806   2652   3782   5256   7140   9506  12432  16002
  1260   1892   2756   3906   5402   7310   9702  12656  16256  20592
  1980   2862   4032   5550   7482   9900  12882  16512  20880  26082
  2970   4160   5700   7656  10100  13110  16770  21170  26406  32580
  4290   5852   7832  10302  13340  17030  21462  26732  32942  40200
  6006   8010  10506  13572  17292  21756  27060  33306  40602  49062
  8190  10712  13806  17556  22052  27390  33672  41006  49506  59292
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, A380660.

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 *)