A347490 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 = 8.

1, 1, 9, 1, 73, 657, 1, 585, 4745, 42705, 384345, 1, 4681, 304265, 2738385, 22211345, 199902105, 1799118945, 1, 37449, 19477641, 156087945, 175298769, 11394419985, 92421406545, 102549779865, 831792658905, 7486133930145, 67375205371305, 1, 299593

Offset

1,3

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_8)^n with dimension increments (e_1,...,e_r).

References

R. P. Stanley, Enumerative Combinatorics (vol. 1), Cambridge University Press (1997), Section 1.3.

Links

Álvar Ibeas, First 20 rows, flattened

Formula

T(n, (n)) = 1. T(n, L) = A022172(n, e) * T(n - e, L \ {e}), if L is a partition of n and e < n is a part of L.

Example

The number of subspace chains 0 < V_1 < V_2 < (F_8)^3 is 657 = T(3, (1, 1, 1)). There are 73 = A022172(3, 1) choices for a one-dimensional subspace V_1 and, for each of them, 9 = A022172(2, 1) extensions to a two-dimensional subspace V_2.
Triangle begins:
  k:  1   2    3     4      5
      -----------------------
n=1:  1
n=2:  1   9
n=3:  1  73  657
n=4:  1 585 4745 42705 384345

Crossrefs

Cf. A036038 (q = 1), A022172, A015007 (last entry in each row).

Sequence in context: A283082 A376634 A318935 * A038291 A075504 A373628

Adjacent sequences: A347487 A347488 A347489 * A347491 A347492 A347493

Keyword

nonn,tabf

Author

Álvar Ibeas, Sep 03 2021

Status

approved