OFFSET
1,3
COMMENTS
Abuse of notation: we write T(n, L) for T(n, k), where L is the k-th partition of n in A-St order.
For any permutation (e_1,...,e_r) of the parts of L, T(n, L) is the number of chains of subspaces 0 < V_1 < ··· < V_r = (F_3)^n with dimension increments (e_1,...,e_r).
REFERENCES
R. P. Stanley, Enumerative Combinatorics (vol. 1), Cambridge University Press (1997), Section 1.3.
LINKS
Álvar Ibeas, First 20 rows, flattened
FORMULA
T(n, (n)) = 1. T(n, L) = A022167(n, e) * T(n - e, L \ {e}), if L is a partition of n and e < n is a part of L.
EXAMPLE
The number of subspace chains 0 < V_1 < V_2 < (F_3)^3 is 52 = T(3, (1, 1, 1)). There are 13 = A022167(3, 1) choices for a one-dimensional subspace V_1 and, for each of them, 4 = A022167(2, 1) extensions to a two-dimensional subspace V_2.
Triangle begins:
k: 1 2 3 4 5 6 7
----------------------------------
n=1: 1
n=2: 1 4
n=3: 1 13 52
n=4: 1 40 130 520 2080
n=5: 1 121 1210 4840 15730 62920 251680
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Álvar Ibeas, Sep 03 2021
STATUS
approved