%I #14 Oct 06 2021 12:19:09

%S 1,1,3,1,7,24,1,12,31,117,469,1,19,111,458,1435,6356,28753,1,29,361,

%T 964,1579,15266,55470,71660,264300,1267174,6105030,1,41,1068,8042,

%U 4886,145628,494779,1952843,705790,9589197,38323695,47157299,188963325,932529235

%N Irregular triangle read by rows: T(n, k) is the number of chains of subspaces 0 < V_1 < ... < V_r = (F_4)^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="/A348115/a348115.txt">First 16 rows, with gaps</a>

%H Álvar Ibeas, <a href="/A348115/a348115_1.txt">Pseudo-column T(n, L), where L = (n-2, 1, 1), up to n=100</a>

%H Álvar Ibeas, <a href="/A348115/a348115_2.txt">Pseudo-column T(n, L), where L = (n-3, 2, 1), up to n=100</a>

%H Álvar Ibeas, <a href="/A348115/a348115_3.txt">Pseudo-column T(n, L), where L = (n-3, 1, 1, 1), up to n=100</a>

%H Álvar Ibeas, <a href="/A348115/a348115_4.txt">Pseudo-column T(n, L), where L = (n-4, 3, 1), up to n=100</a>

%H Álvar Ibeas, <a href="/A348115/a348115_5.txt">Pseudo-column T(n, L), where L = (n-4, 2, 2), up to n=100</a>

%H Álvar Ibeas, <a href="/A348115/a348115_6.txt">Pseudo-column T(n, L), where L = (n-4, 2, 1, 1), up to n=100</a>

%H Álvar Ibeas, <a href="/A348115/a348115_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) = A347971(n, a) = A347971(n, n-a).

%e For L = (1, 1, 1), there are 105 (= 21 * 5) = A347487(3, 3) subspace chains 0 < V_1 < V_2 < (F_4)^3.

%e The permutations of the three coordinates classify them into 24 = T(3, 3) orbits.

%e T(3, 2) = 7 refers to partition (2, 1) and counts subspace chains in (F_4)^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

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

%e n=1: 1

%e n=2: 1 3

%e n=3: 1 7 24

%e n=4: 1 12 31 117 469

%e n=5: 1 19 111 458 1435 6356 28753

%Y Cf. A347971, A347487.

%K nonn,tabf

%O 1,3

%A _Álvar Ibeas_, Oct 01 2021