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%I A380660 #23 Apr 14 2025 05:44:23
%S A380660 5,16,27,48,65,84,119,144,171,200,253,288,325,364,405,480,527,576,627,
%T A380660 680,735,836,897,960,1025,1092,1161,1232,1363,1440,1519,1600,1683,
%U A380660 1768,1855,1944,2109,2204,2301,2400,2501,2604,2709,2816,2925,3128,3243,3360
%N 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.
%C A380660 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.
%C A380660 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).
%C A380660 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.
%C A380660 We have:
%C A380660 pos(i,j,s,n) is the sum of the n!/2 products for which p(u) = 0, and
%C A380660 neg(i,j,s,n) is the sum of the n!/2 products for which p(u) = 1.
%C A380660 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).
%D A380660 S. Lang, Algebra, 2nd ed., Addison-Wesley, 1984, 452-453.
%H A380660 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%e A380660 Corner of pos(i,j,1,2):
%e A380660 5 16 48 119 253 480 836 1363 2109
%e A380660 27 65 144 288 527 897 1440 2204 3243
%e A380660 84 171 325 576 960 1519 2301 3360 4756
%e A380660 200 364 627 1025 1600 2400 3479 4897 6720
%e A380660 405 680 1092 1683 2501 3600 5040 6887 9213
%e A380660 735 1161 1768 2604 3723 5185 7056 9408 12319
%e A380660 1232 1855 2709 3848 5332 7227 9605 12544 16128
%e A380660 1944 2816 3975 5481 7400 9804 12771 16385 20736
%e A380660 2925 4104 5632 7575 10005 13000 16644 21027 26245
%e A380660 4235 5785 7752 10208 13231 16905 21320 26572 32763
%e A380660 5940 7931 10413 13464 17168 21615 26901 33128 40404
%e A380660 8112 10620 13699 17433 21912 27232 33495 40809 49288
%e A380660 M(1,1,1,2) is the matrix with (row 1) = (1,2), (row 2) =(3,5), so that
%e A380660 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.
%t A380660 s = 1; n = 2; z = 12;
%t A380660 r[n_, k_] := n + (n + k - 2)*(n + k - 1)/2 (* Array A000027 *)
%t A380660 Grid[Table[r[n, k], {n, 1, z}, {k, 1, z}]]
%t A380660 t[i_, j_] := Table[r[i, j + k*s], {k, 0, n - 1}];
%t A380660 d[i_, j_] := Det[Table[t[i + k*s, j], {k, 0, n - 1}]]; (* D(i,j,s,n) *)
%t A380660 p[i_, j_] := Permanent[Table[t[i + k*s, j], {k, 0, n - 1}]]; (* P(i,j,s,n) *)
%t A380660 pos[i_, j_] := (p[i, j] + d[i, j])/2;
%t A380660 neg[i_, j_] := (p[i, j] - d[i, j])/2;
%t A380660 Grid[Table[pos[i, j], {i, 1, z}, {j, 1, z}]] (* A380660 array *)
%t A380660 Grid[Table[neg[i, j], {i, 1, z}, {j, 1, z}]] (* A380661 array *)
%t A380660 FindLinearRecurrence[Table[pos[1, k], {k, 1, 20}]] (* row recurrence, all rows *)
%t A380660 FindLinearRecurrence[Table[neg[7, k], {k, 1, 20}]] (* row recurrence, all rows *)
%t A380660 Table[pos[k, m - k], {m, 2, z}, {k, 1, m - 1}] // Flatten (* A380660 sequence *)
%t A380660 Table[neg[k, m - k], {m, 2, z}, {k, 1, m - 1}] // Flatten (* A380661 sequence *)
%Y A380660 Cf. A000027, A380649, A380661.
%K A380660 nonn,tabl
%O A380660 1,1
%A A380660 _Clark Kimberling_, Feb 04 2025