A347490 - OEIS

%I #14 Sep 15 2021 10:27:24

%S 1,1,9,1,73,657,1,585,4745,42705,384345,1,4681,304265,2738385,

%T 22211345,199902105,1799118945,1,37449,19477641,156087945,175298769,

%U 11394419985,92421406545,102549779865,831792658905,7486133930145,67375205371305,1,299593

%N 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.

%C 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.

%C 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).

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

%H Álvar Ibeas, <a href="/A347490/b347490.txt">First 20 rows, flattened</a>

%F 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.

%e 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.

%e Triangle begins:

%e   k:  1   2    3     4      5

%e       -----------------------

%e n=1:  1

%e n=2:  1   9

%e n=3:  1  73  657

%e n=4:  1 585 4745 42705 384345

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

%K nonn,tabf

%O 1,3

%A _Álvar Ibeas_, Sep 03 2021