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.

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

Author

Álvar Ibeas, Oct 01 2021

Keywords

• nonn
• tabf

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.

Examples

			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
		

Links

• Álvar Ibeas, Table of n, a(n) for n = 1..137
• Álvar Ibeas, First 16 rows, with gaps
• Álvar Ibeas, Pseudo-column T(n, L), where L = (n-2, 1, 1), up to n=100
• Álvar Ibeas, Pseudo-column T(n, L), where L = (n-3, 2, 1), up to n=100
• Álvar Ibeas, Pseudo-column T(n, L), where L = (n-3, 1, 1, 1), up to n=100
• Álvar Ibeas, Pseudo-column T(n, L), where L = (n-4, 3, 1), up to n=100
• Álvar Ibeas, Pseudo-column T(n, L), where L = (n-4, 2, 2), up to n=100
• Álvar Ibeas, Pseudo-column T(n, L), where L = (n-4, 2, 1, 1), up to n=100
• Álvar Ibeas, Pseudo-column T(n, L), where L = (n-4, 1, 1, 1, 1), up to n=100

Crossrefs

Cf. A076831, A347485.

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).