This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A348113 #11 Oct 06 2021 12:18:46 %S A348113 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 A348113 357,864,2109,1,7,23,43,55,180,269,479,441,1193,2169,2992,5483,13958, %U A348113 35773,1,8,32,77,106,78,341,734,1354,2153,856,3468,5559,10544,20185,8943,27572,53115,72517,140563,373927 %N 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. %C A348113 A permutation on the list of dimension increments does not modify the number of subspace chains. %C A348113 The length of the enumerated chains is r = len(L), where L is the parameter partition. %H A348113 Álvar Ibeas, <a href="/A348113/b348113.txt">Table of n, a(n) for n = 1..137</a> %H A348113 Álvar Ibeas, <a href="/A348113/a348113.txt">First 16 rows, with gaps</a> %H A348113 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 A348113 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 A348113 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 A348113 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 A348113 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 A348113 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 A348113 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 A348113 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 A348113 For L = (1, 1, 1), there are 21 (= 7 * 3) = A347485(3, 3) subspace chains 0 < V_1 < V_2 < (F_2)^3. %e A348113 The permutations of the three coordinates classify them into 6 = T(3, 3) orbits: %e A348113 <e_1>, <e_1, e_2>; <e_1>, <e_1, e_2 + e_3>; %e A348113 <e_1 + e_2>, <e_1, e_2>; <e_1 + e_2>, <e_1 + e_2, e_3>; %e A348113 <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 A348113 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 A348113 Triangle begins: %e A348113 k: 1 2 3 4 5 6 7 8 9 10 11 %e A348113 ------------------------------------ %e A348113 n=1: 1 %e A348113 n=2: 1 2 %e A348113 n=3: 1 3 6 %e A348113 n=4: 1 4 6 13 28 %e A348113 n=5: 1 5 10 23 37 85 196 %e A348113 n=6: 1 6 16 22 37 87 149 207 357 864 2109 %Y A348113 Cf. A076831, A347485. %K A348113 nonn,tabf %O A348113 1,3 %A A348113 _Álvar Ibeas_, Oct 01 2021