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

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 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
		

Links

• 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. A347971, A347487.

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).