A347492 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 = 11.

1, 1, 12, 1, 133, 1596, 1, 1464, 16226, 194712, 2336544, 1, 16105, 1964810, 23577720, 261319730, 3135836760, 37630041120, 1, 177156, 237758115, 2617126920, 2853097380, 348077880360, 3857863173990, 4176934564320, 46294358087880, 555532297054560, 6666387564654720, 1, 1948717

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_11)^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) = A022175(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_11)^3 is 1596 = T(3, (1, 1, 1)). There are 133 = A022175(3, 1) choices for a one-dimensional subspace V_1 and, for each of them, 12 = A022175(2, 1) extensions to a two-dimensional subspace V_2.
Triangle begins:
  k:  1    2     3      4       5
      ---------------------------
n=1:  1
n=2:  1   12
n=3:  1  133  1596
n=4:  1 1464 16226 194712 2336544

Crossrefs

Cf. A036038 (q = 1), A022175, A015011 (last entry in each row).

Sequence in context: A280975 A075072 A265825 * A038327 A225750 A157780

Adjacent sequences: A347489 A347490 A347491 * A347493 A347494 A347495

Keyword

nonn,tabf

Author

Álvar Ibeas, Sep 03 2021

Status

approved