A347489 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 = 7.

1, 1, 8, 1, 57, 456, 1, 400, 2850, 22800, 182400, 1, 2801, 140050, 1120400, 7982850, 63862800, 510902400, 1, 19608, 6865251, 48177200, 54922008, 2746100400, 19565965350, 21968803200, 156527722800, 1252221782400, 10017774259200, 1, 137257, 336416907, 16531644851, 2691335256

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_7)^n with dimension increments (e_1,...,e_r).

Examples

			The number of subspace chains 0 < V_1 < V_2 < (F_7)^3 is 456 = T(3, (1, 1, 1)). There are 57 = A022171(3, 1) choices for a one-dimensional subspace V_1 and, for each of them, 8 = A022171(2, 1) extensions to a two-dimensional subspace V_2.
Triangle begins:
  k:  1   2    3     4      5
      -----------------------
n=1:  1
n=2:  1   8
n=3:  1  57  456
n=4:  1 400 2850 22800 182400
		

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), A022171, A015006 (last entry in each row).

Formula

T(n, (n)) = 1. T(n, L) = A022171(n, e) * T(n - e, L \ {e}), if L is a partition of n and e < n is a part of L.