A348180 Irregular triangle read by rows: T(n, k) is the number of chains of subspaces 0 < V_1 < ... < V_r = (F_5)^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, 4, 1, 9, 40, 1, 19, 56, 279, 1428, 1, 33, 289, 1561, 6345, 35689, 202421, 1, 55, 1358, 4836, 7652, 129505, 615395, 757560, 3620918, 21341449, 125952538, 1, 85, 5771, 80605, 33435, 2362185, 10691648, 53822709, 14039541, 321134138, 1622410155, 1916573757, 9688635876, 57866763847

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

Table of n, a(n) for n=1..43.

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) = A347972(n, a) = A347972(n, n-a).

Example

For L = (1, 1, 1), there are 186 (= 31 * 6) = A347488(3, 3) subspace chains 0 < V_1 < V_2 < (F_5)^3.
The permutations of the three coordinates classify them into 40 = T(3, 3) orbits.
T(3, 2) = 9 refers to partition (2, 1) and counts subspace chains in (F_5)^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
      -------------------------------
n=1:  1
n=2:  1  4
n=3:  1  9  40
n=4:  1 19  56  279 1428
n=5:  1 33 289 1561 6345 35689 202421

Crossrefs

Cf. A347972, A347488, A348113-A348115.

Sequence in context: A193963 A028941 A364109 * A176080 A328302 A346748

Adjacent sequences: A348177 A348178 A348179 * A348181 A348182 A348183

Keyword

nonn,tabf

Author

Álvar Ibeas, Oct 05 2021

Status

approved