OFFSET
1,3
COMMENTS
A permutation on the list of dimension increments does not modify the number of subspace chains.
The length of the enumerated chains is r = len(L), where L is the parameter partition.
LINKS
Álvar Ibeas, Table of n, a(n) for n = 1..137
Álvar Ibeas, First 16 rows, with gaps
FORMULA
EXAMPLE
For L = (1, 1, 1), there are 21 (= 7 * 3) = A347485(3, 3) subspace chains 0 < V_1 < V_2 < (F_2)^3.
The permutations of the three coordinates classify them into 6 = T(3, 3) orbits:
<e_1>, <e_1, e_2>; <e_1>, <e_1, e_2 + e_3>;
<e_1 + e_2>, <e_1, e_2>; <e_1 + e_2>, <e_1 + e_2, e_3>;
<e_1 + e_2>, <e_1 + e_2, e_1 + e_3>; <e_1 + e_2 + e_3>, <e_1 + e_2, e_3>.
T(3, 2) = 3 refers to partition (2, 1) and counts subspace chains in (F_2)^2 with dimensions (0, 2, 3), i.e., 2-dimensional subspaces. It also counts chains with dimensions (0, 1, 3), i.e., 1-dimensional subspaces.
Triangle begins:
k: 1 2 3 4 5 6 7 8 9 10 11
------------------------------------
n=1: 1
n=2: 1 2
n=3: 1 3 6
n=4: 1 4 6 13 28
n=5: 1 5 10 23 37 85 196
n=6: 1 6 16 22 37 87 149 207 357 864 2109
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Álvar Ibeas, Oct 01 2021
STATUS
approved