A347972 Triangle read by rows: T(n, k) is the number of k-dimensional subspaces in (F_5)^n, counted up to coordinate permutation (n >= 0, 0 <= k <= n).

1, 1, 1, 1, 4, 1, 1, 9, 9, 1, 1, 19, 56, 19, 1, 1, 33, 289, 289, 33, 1, 1, 55, 1358, 4836, 1358, 55, 1, 1, 85, 5771, 80605, 80605, 5771, 85, 1, 1, 128, 22594, 1271870, 5525686, 1271870, 22594, 128, 1, 1, 183, 81802, 18478460, 372302962, 372302962, 18478460, 81802, 183, 1

Offset: 0

Author

Álvar Ibeas, Sep 21 2021

Keywords

• nonn
• tabl

Comments

Columns can be computed by a method analogous to that of Fripertinger for isometry classes of linear codes, disallowing scalar transformation of individual coordinates.

Regarding the formula for column k = 1, note that A241926(q - 1, n) counts, up to coordinate permutation, one-dimensional subspaces of (F_q)^n generated by a vector with no zero component.

Examples

			Triangle begins:
  k:  0    1    2    3    4    5    6
      -------------------------------
n=0:  1
n=1:  1    1
n=2:  1    4    1
n=3:  1    9    9    1
n=4:  1   19   56   19    1
n=5:  1   33  289  289   33    1
n=6:  1   55 1358 4836 1358   55    1
There are 6 = A022169(2, 1) one-dimensional subspaces in (F_5)^2. By coordinate swap, <(0, 1)> is identified with <(1, 0)> and <(1, 2)> with <(1, 3)>, while <(1, 1)> and <(1, 4)> rest invariant. Hence, T(2, 1) = 4.
		

Links

• Álvar Ibeas, Entries up to T(12, 5)
• H. Fripertinger, Isometry classes of codes
• Álvar Ibeas, Column k=1 up to n=100
• Álvar Ibeas, Column k=2 up to n=100
• Álvar Ibeas, Column k=3 up to n=100
• Álvar Ibeas, Column k=4 up to n=100
• Álvar Ibeas, Column k=5 up to n=100

Crossrefs

Cf. A022169, A008610, A241926.

Formula

T(n, 1) = T(n - 1, 1) + A008610(n).