A347485 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 = 2.

1, 1, 3, 1, 7, 21, 1, 15, 35, 105, 315, 1, 31, 155, 465, 1085, 3255, 9765, 1, 63, 651, 1395, 1953, 9765, 22785, 29295, 68355, 205065, 615195, 1, 127, 2667, 11811, 8001, 82677, 177165, 413385, 248031, 1240155, 2893695, 3720465, 8681085, 26043255, 78129765

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_2)^n with dimension increments (e_1,...,e_r).

Examples

			The number of subspace chains 0 < V_1 < V_2 < (F_2)^3 is 21 = T(3, (1, 1, 1)). There are 7 = A022166(3, 1) choices for a one-dimensional subspace V_1 and, for each of them, 3 = A022166(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  3
n=3:  1  7  21
n=4:  1 15  35 105  315
n=5:  1 31 155 465 1085 3255 9765
		

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), A022166, A005329 (last entry in each row).

Formula

T(n, (n)) = 1. T(n, L) = A022166(n, e) * T(n - e, L \ {e}), if L is a partition of n and e < n is a part of L.