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A348113
Irregular triangle read by rows: T(n, k) is the number of chains of subspaces 0 < V_1 < ... < V_r = (F_2)^n, counted up to coordinate permutation, with dimension increments given by (any fixed permutation of) the parts of the k-th partition of n in Abramowitz-Stegun order.
3
1, 1, 2, 1, 3, 6, 1, 4, 6, 13, 28, 1, 5, 10, 23, 37, 85, 196, 1, 6, 16, 22, 37, 87, 149, 207, 357, 864, 2109, 1, 7, 23, 43, 55, 180, 269, 479, 441, 1193, 2169, 2992, 5483, 13958, 35773, 1, 8, 32, 77, 106, 78, 341, 734, 1354, 2153, 856, 3468, 5559, 10544, 20185, 8943, 27572, 53115, 72517, 140563, 373927
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.
FORMULA
If the k-th partition of n in A-St is L = (a, n-a), then T(n, k) = A076831(n, a) = A076831(n, n-a).
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
Sequence in context: A036038 A210237 A078760 * A103280 A046899 A309220
KEYWORD
nonn,tabf
AUTHOR
Álvar Ibeas, Oct 01 2021
STATUS
approved