A348114 - OEIS

%I #12 Oct 06 2021 12:18:54

%S 1,1,3,1,5,15,1,8,16,49,154,1,11,39,126,288,964,3275,1,15,87,168,291,

%T 1412,3600,4957,12865,46400,168862,1,19,176,644,608,6101,14001,38996,

%U 22294,146064,418072,549894,1586761,6045724,23115063,1,24,338,2348,4849,1195,24329

%N Irregular triangle read by rows: T(n, k) is the number of chains of subspaces 0 < V_1 < ... < V_r = (F_3)^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="/A348114/b348114.txt">Table of n, a(n) for n = 1..65</a>

%H Álvar Ibeas, <a href="/A348114/a348114.txt">First 16 rows, with gaps</a>

%H Álvar Ibeas, <a href="/A348114/a348114_1.txt">Pseudo-column T(n, L), where L = (n-2, 1, 1), up to n=100</a>

%H Álvar Ibeas, <a href="/A348114/a348114_2.txt">Pseudo-column T(n, L), where L = (n-3, 2, 1), up to n=100</a>

%H Álvar Ibeas, <a href="/A348114/a348114_3.txt">Pseudo-column T(n, L), where L = (n-3, 1, 1, 1), up to n=100</a>

%H Álvar Ibeas, <a href="/A348114/a348114_4.txt">Pseudo-column T(n, L), where L = (n-4, 3, 1), up to n=100</a>

%H Álvar Ibeas, <a href="/A348114/a348114_5.txt">Pseudo-column T(n, L), where L = (n-4, 2, 2), up to n=100</a>

%H Álvar Ibeas, <a href="/A348114/a348114_6.txt">Pseudo-column T(n, L), where L = (n-4, 2, 1, 1), up to n=100</a>

%H Álvar Ibeas, <a href="/A348114/a348114_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) = A347970(n, a) = A347970(n, n-a).

%e For L = (1, 1, 1), there are 52 (= 13 * 4) = A347486(3, 3) subspace chains 0 < V_1 < V_2 < (F_3)^3.

%e The permutations of the three coordinates classify them into 15 = T(3, 3) orbits.

%e T(3, 2) = 5 refers to partition (2, 1) and counts subspace chains in (F_3)^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  5 15

%e n=4:  1  8 16  49 154

%e n=5:  1 11 39 126 288 964 3275

%Y Cf. A347970, A347486.

%K nonn,tabf

%O 1,3

%A _Álvar Ibeas_, Oct 01 2021