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.

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

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_9)^n with dimension increments (e_1,...,e_r).

Examples

			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
		

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), A022173, A015008 (last entry in each row).

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.