%I #9 Sep 30 2021 11:42:05
%S 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,
%T 29,1,1,41,1068,8042,8042,1068,41,1,1,56,2954,64674,205065,64674,2954,
%U 56,1,1,75,7681,492387,5402621,5402621,492387,7681,75,1,1,97,18880,3507681,137287827
%N 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).
%C Columns can be computed by a method analogous to that of Fripertinger for isometry classes of linear codes, disallowing scalar transformation of individual coordinates.
%C 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.
%H Álvar Ibeas, <a href="/A347971/b347971.txt">Entries up to T(14, 6)</a>
%H H. Fripertinger, <a href="http://www.mathe2.uni-bayreuth.de/frib/codes/tables.html">Isometry classes of codes</a>
%H H. Fripertinger, <a href="http://www.mathe2.uni-bayreuth.de/frib/codes/tables_23.html">Number of the isometry classes of all quaternary (n,k)-codes</a>
%H Álvar Ibeas, <a href="/A347971/a347971.txt">Column k=1 up to n=100</a>
%H Álvar Ibeas, <a href="/A347971/a347971_1.txt">Column k=2 up to n=100</a>
%H Álvar Ibeas, <a href="/A347971/a347971_2.txt">Column k=3 up to n=100</a>
%H Álvar Ibeas, <a href="/A347971/a347971_3.txt">Column k=4 up to n=100</a>
%H Álvar Ibeas, <a href="/A347971/a347971_4.txt">Column k=5 up to n=100</a>
%H Álvar Ibeas, <a href="/A347971/a347971_5.txt">Column k=6 up to n=100</a>
%F T(n, 1) = T(n - 1, 1) + A007997(n + 5).
%e Triangle begins:
%e k: 0 1 2 3 4 5 6
%e -------------------------------
%e n=0: 1
%e n=1: 1 1
%e n=2: 1 3 1
%e n=3: 1 7 7 1
%e n=4: 1 12 31 12 1
%e n=5: 1 19 111 111 19 1
%e n=6: 1 29 361 964 361 29 1
%e 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.
%Y Cf. A022168, A007997, A241926.
%K nonn,tabl
%O 0,5
%A _Álvar Ibeas_, Sep 21 2021