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.

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

Offset: 1

Author

Álvar Ibeas, Sep 03 2021

Keywords

• nonn
• tabf

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).

Examples

			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
		

References

• R. P. Stanley, Enumerative Combinatorics (vol. 1), Cambridge University Press (1997), Section 1.3.

Links

• Álvar Ibeas, First 20 rows, flattened

Crossrefs

Cf. A036038 (q = 1), A022167, A015001 (last entry in each row).

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.