A347971 Triangle read by rows: T(n, k) is the number of k-dimensional subspaces in (F_4)^n, counted up to coordinate permutation (n >= 0, 0 <= k <= n).

1, 1, 1, 1, 3, 1, 1, 7, 7, 1, 1, 12, 31, 12, 1, 1, 19, 111, 111, 19, 1, 1, 29, 361, 964, 361, 29, 1, 1, 41, 1068, 8042, 8042, 1068, 41, 1, 1, 56, 2954, 64674, 205065, 64674, 2954, 56, 1, 1, 75, 7681, 492387, 5402621, 5402621, 492387, 7681, 75, 1, 1, 97, 18880, 3507681, 137287827

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    3    1
n=3:  1    7    7    1
n=4:  1   12   31   12    1
n=5:  1   19  111  111   19    1
n=6:  1   29  361  964  361   29    1
There are 5 = A022168(2, 1) one-dimensional subspaces in (F_4)^2, namely, those generated by vectors (0, 1), (1, 0), (1, 1), (1, x), and (1, x + 1), where F_4 = F_2[x] / (x^2 + x + 1). The coordinate swap identifies the first two on the one hand and the last two on the other, while <(1, 1)> is invariant. Hence, T(2, 1) = 3.
		

Links

• Álvar Ibeas, Entries up to T(14, 6)
• H. Fripertinger, Isometry classes of codes
• H. Fripertinger, Number of the isometry classes of all quaternary (n,k)-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
• Álvar Ibeas, Column k=6 up to n=100

Crossrefs

Cf. A022168, A007997, A241926.

Formula

T(n, 1) = T(n - 1, 1) + A007997(n + 5).