A347488 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 = 5.

1, 1, 6, 1, 31, 186, 1, 156, 806, 4836, 29016, 1, 781, 20306, 121836, 629486, 3776916, 22661496, 1, 3906, 508431, 2558556, 3050586, 79315236, 409795386, 475891416, 2458772316, 14752633896, 88515803376, 1, 19531, 12714681, 320327931, 76288086

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_5)^n with dimension increments (e_1,...,e_r).

Examples

			The number of subspace chains 0 < V_1 < V_2 < (F_5)^3 is 186 = T(3, (1, 1, 1)). There are 31 = A022169(3, 1) choices for a one-dimensional subspace V_1 and, for each of them, 6 = A022169(2, 1) extensions to a two-dimensional subspace V_2.
Triangle begins:
  k:  1   2    3     4      5
      -----------------------
n=1:  1
n=2:  1   6
n=3:  1  31  186
n=4:  1 156  806  4836  29016
		

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), A022169, A015004 (last entry in each row).

Formula

T(n, (n)) = 1. T(n, L) = A022169(n, e) * T(n - e, L \ {e}), if L is a partition of n and e < n is a part of L.