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A347491
Irregular triangle read by rows: T(n, k) is the q-multinomial coefficient defined by the k-th partition of n in Abramowitz-Stegun order, evaluated at q = 9.
2
1, 1, 10, 1, 91, 910, 1, 820, 7462, 74620, 746200, 1, 7381, 605242, 6052420, 55077022, 550770220, 5507702200, 1, 66430, 49031983, 441826660, 490319830, 40206226060, 365876657146, 402062260600, 3658766571460, 36587665714600, 365876657146000, 1, 597871
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_9)^n with dimension increments (e_1,...,e_r).
REFERENCES
R. P. Stanley, Enumerative Combinatorics (vol. 1), Cambridge University Press (1997), Section 1.3.
LINKS
FORMULA
T(n, (n)) = 1. T(n, L) = A022173(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_9)^3 is 910 = T(3, (1, 1, 1)). There are 91 = A022173(3, 1) choices for a one-dimensional subspace V_1 and, for each of them, 10 = A022173(2, 1) extensions to a two-dimensional subspace V_2.
Triangle begins:
k: 1 2 3 4 5
-----------------------
n=1: 1
n=2: 1 10
n=3: 1 91 910
n=4: 1 820 7462 74620 746200
CROSSREFS
Cf. A036038 (q = 1), A022173, A015008 (last entry in each row).
Sequence in context: A327003 A206819 A178865 * A164881 A276379 A165293
KEYWORD
nonn,tabf
AUTHOR
Álvar Ibeas, Sep 03 2021
STATUS
approved