A347487 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 = 4.

1, 1, 5, 1, 21, 105, 1, 85, 357, 1785, 8925, 1, 341, 5797, 28985, 121737, 608685, 3043425, 1, 1365, 93093, 376805, 465465, 7912905, 33234201, 39564525, 166171005, 830855025, 4154275125, 1, 5461, 1490853, 24208613, 7454265, 508380873, 2057732105, 8642474841

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_4)^n with dimension increments (e_1,...,e_r).

Examples

			The number of subspace chains 0 < V_1 < V_2 < (F_4)^3 is 105 = T(3, (1, 1, 1)). There are 21 = A022168(3, 1) choices for a one-dimensional subspace V_1 and, for each of them, 5 = A022168(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   5
n=3:  1  21  105
n=4:  1  85  357  1785   8925
n=5:  1 341 5795 28985 121737 608685 3043425
		

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), A022168, A015002 (last entry in each row).

Formula

T(n, (n)) = 1. T(n, L) = A022168(n, e) * T(n - e, L \ {e}), if L is a partition of n and e < n is a part of L.