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A347973
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).
2
1, 1, 1, 1, 5, 1, 1, 15, 15, 1, 1, 37, 162, 37, 1, 1, 79, 1538, 1538, 79, 1, 1, 159, 13237, 74830, 13237, 159, 1, 1, 291, 102019, 3546909, 3546909, 102019, 291, 1, 1, 508, 708712, 153181682, 1010416196, 153181682, 708712, 508, 1, 1, 843, 4473998, 5954653026, 267444866627
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
H. Fripertinger, Isometry classes of codes
FORMULA
T(n, 1) = T(n - 1, 1) + A032191(n + 6).
EXAMPLE
Triangle begins:
k: 0 1 2 3 4 5
--------------------------
n=0: 1
n=1: 1 1
n=2: 1 5 1
n=3: 1 15 15 1
n=4: 1 37 162 37 1
n=5: 1 79 1538 1538 79 1
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.
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Álvar Ibeas, Sep 21 2021
STATUS
approved