%I #11 Oct 06 2021 12:18:46

%S 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,

%T 357,864,2109,1,7,23,43,55,180,269,479,441,1193,2169,2992,5483,13958,

%U 35773,1,8,32,77,106,78,341,734,1354,2153,856,3468,5559,10544,20185,8943,27572,53115,72517,140563,373927

%N 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.

%C A permutation on the list of dimension increments does not modify the number of subspace chains.

%C The length of the enumerated chains is r = len(L), where L is the parameter partition.

%H Álvar Ibeas, <a href="/A348113/b348113.txt">Table of n, a(n) for n = 1..137</a>

%H Álvar Ibeas, <a href="/A348113/a348113.txt">First 16 rows, with gaps</a>

%H Álvar Ibeas, <a href="/A348113/a348113_1.txt">Pseudo-column T(n, L), where L = (n-2, 1, 1), up to n=100</a>

%H Álvar Ibeas, <a href="/A348113/a348113_2.txt">Pseudo-column T(n, L), where L = (n-3, 2, 1), up to n=100</a>

%H Álvar Ibeas, <a href="/A348113/a348113_3.txt">Pseudo-column T(n, L), where L = (n-3, 1, 1, 1), up to n=100</a>

%H Álvar Ibeas, <a href="/A348113/a348113_4.txt">Pseudo-column T(n, L), where L = (n-4, 3, 1), up to n=100</a>

%H Álvar Ibeas, <a href="/A348113/a348113_5.txt">Pseudo-column T(n, L), where L = (n-4, 2, 2), up to n=100</a>

%H Álvar Ibeas, <a href="/A348113/a348113_6.txt">Pseudo-column T(n, L), where L = (n-4, 2, 1, 1), up to n=100</a>

%H Álvar Ibeas, <a href="/A348113/a348113_7.txt">Pseudo-column T(n, L), where L = (n-4, 1, 1, 1, 1), up to n=100</a>

%F 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).

%e For L = (1, 1, 1), there are 21 (= 7 * 3) = A347485(3, 3) subspace chains 0 < V_1 < V_2 < (F_2)^3.

%e The permutations of the three coordinates classify them into 6 = T(3, 3) orbits:

%e <e_1>, <e_1, e_2>; <e_1>, <e_1, e_2 + e_3>;

%e <e_1 + e_2>, <e_1, e_2>; <e_1 + e_2>, <e_1 + e_2, e_3>;

%e <e_1 + e_2>, <e_1 + e_2, e_1 + e_3>; <e_1 + e_2 + e_3>, <e_1 + e_2, e_3>.

%e 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.

%e Triangle begins:

%e k: 1 2 3 4 5 6 7 8 9 10 11

%e ------------------------------------

%e n=1: 1

%e n=2: 1 2

%e n=3: 1 3 6

%e n=4: 1 4 6 13 28

%e n=5: 1 5 10 23 37 85 196

%e n=6: 1 6 16 22 37 87 149 207 357 864 2109

%Y Cf. A076831, A347485.

%K nonn,tabf

%O 1,3

%A _Álvar Ibeas_, Oct 01 2021