|
%I #8 Sep 30 2021 11:42:19
%S 1,1,1,1,5,1,1,15,15,1,1,37,162,37,1,1,79,1538,1538,79,1,1,159,13237,
%T 74830,13237,159,1,1,291,102019,3546909,3546909,102019,291,1,1,508,
%U 708712,153181682,1010416196,153181682,708712,508,1,1,843,4473998,5954653026,267444866627
%N Triangle read by rows: T(n, k) is the number of k-dimensional subspaces in (F_7)^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="/A347973/b347973.txt">Entries up to T(10, 4)</a>
%H H. Fripertinger, <a href="http://www.mathe2.uni-bayreuth.de/frib/codes/tables.html">Isometry classes of codes</a>
%H Álvar Ibeas, <a href="/A347973/a347973.txt">Column k=1 up to n=100</a>
%H Álvar Ibeas, <a href="/A347973/a347973_1.txt">Column k=2 up to n=100</a>
%H Álvar Ibeas, <a href="/A347973/a347973_2.txt">Column k=3 up to n=100</a>
%H Álvar Ibeas, <a href="/A347973/a347973_3.txt">Column k=4 up to n=100</a>
%F T(n, 1) = T(n - 1, 1) + A032191(n + 6).
%e Triangle begins:
%e k: 0 1 2 3 4 5
%e --------------------------
%e n=0: 1
%e n=1: 1 1
%e n=2: 1 5 1
%e n=3: 1 15 15 1
%e n=4: 1 37 162 37 1
%e n=5: 1 79 1538 1538 79 1
%e There are 8 = A022171(2, 1) one-dimensional subspaces in (F_7)^2. Two of them (<(1, 1)> and <(1, 6)>) are invariant by coordinate swap, while the rest are grouped in orbits of size two. Hence, T(2, 1) = 5.
%Y Cf. A022171, A032191, A241926.
%K nonn,tabl
%O 0,5
%A _Álvar Ibeas_, Sep 21 2021
|
