A348115 Irregular triangle read by rows: T(n, k) is the number of chains of subspaces 0 < V_1 < ... < V_r = (F_4)^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, 3, 1, 7, 24, 1, 12, 31, 117, 469, 1, 19, 111, 458, 1435, 6356, 28753, 1, 29, 361, 964, 1579, 15266, 55470, 71660, 264300, 1267174, 6105030, 1, 41, 1068, 8042, 4886, 145628, 494779, 1952843, 705790, 9589197, 38323695, 47157299, 188963325, 932529235

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) = A347971(n, a) = A347971(n, n-a).

Example

For L = (1, 1, 1), there are 105 (= 21 * 5) = A347487(3, 3) subspace chains 0 < V_1 < V_2 < (F_4)^3.
The permutations of the three coordinates classify them into 24 = T(3, 3) orbits.
T(3, 2) = 7 refers to partition (2, 1) and counts subspace chains in (F_4)^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  3
n=3:  1  7  24
n=4:  1 12  31 117  469
n=5:  1 19 111 458 1435 6356 28753

Crossrefs

Cf. A347971, A347487.

Sequence in context: A347485 A073676 A232149 * A282422 A282685 A194583

Adjacent sequences: A348112 A348113 A348114 * A348116 A348117 A348118

Keyword

nonn,tabf

Author

Álvar Ibeas, Oct 01 2021

Status

approved