|
| |
|
|
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).
|
|
3
|
|
|
|
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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,5
|
|
|
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.
|
|
|
LINKS
|
|
|
|
FORMULA
|
T(n, 1) = T(n - 1, 1) + A007997(n + 5).
|
|
|
EXAMPLE
|
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.
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
