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A347486 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 = 3. 3
1, 1, 4, 1, 13, 52, 1, 40, 130, 520, 2080, 1, 121, 1210, 4840, 15730, 62920, 251680, 1, 364, 11011, 33880, 44044, 440440, 1431430, 1761760, 5725720, 22902880, 91611520, 1, 1093, 99463, 925771, 397852, 12035023, 37030840, 120350230, 48140092, 481400920, 1564552990 (list; graph; refs; listen; history; text; internal format)
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
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
Cf. A036038 (q = 1), A022167, A015001 (last entry in each row).
Sequence in context: A115154 A292270 A051928 * A335337 A226906 A327352
KEYWORD
nonn,tabf
AUTHOR
Álvar Ibeas, Sep 03 2021
STATUS
approved

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Last modified April 17 10:33 EDT 2024. Contains 371763 sequences. (Running on oeis4.)