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

1,3

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.

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

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).

Example

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

Crossrefs

Cf. A076831, A347485.

Sequence in context: A036038 A210237 A078760 * A103280 A046899 A309220

Adjacent sequences: A348110 A348111 A348112 * A348114 A348115 A348116

Keyword

nonn,tabf

Author

Álvar Ibeas, Oct 01 2021

Status

approved