A348114 Irregular triangle read by rows: T(n, k) is the number of chains of subspaces 0 < V_1 < ... < V_r = (F_3)^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, 5, 15, 1, 8, 16, 49, 154, 1, 11, 39, 126, 288, 964, 3275, 1, 15, 87, 168, 291, 1412, 3600, 4957, 12865, 46400, 168862, 1, 19, 176, 644, 608, 6101, 14001, 38996, 22294, 146064, 418072, 549894, 1586761, 6045724, 23115063, 1, 24, 338, 2348, 4849, 1195, 24329

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..65

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) = A347970(n, a) = A347970(n, n-a).

Example

For L = (1, 1, 1), there are 52 (= 13 * 4) = A347486(3, 3) subspace chains 0 < V_1 < V_2 < (F_3)^3.
The permutations of the three coordinates classify them into 15 = T(3, 3) orbits.
T(3, 2) = 5 refers to partition (2, 1) and counts subspace chains in (F_3)^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  5 15
n=4:  1  8 16  49 154
n=5:  1 11 39 126 288 964 3275

Crossrefs

Cf. A347970, A347486.

Sequence in context: A370380 A091623 A215474 * A309498 A059616 A125053

Adjacent sequences: A348111 A348112 A348113 * A348115 A348116 A348117

Keyword

nonn,tabf

Author

Álvar Ibeas, Oct 01 2021

Status

approved