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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.

%I A347490 #14 Sep 15 2021 10:27:24
%S A347490 1,1,9,1,73,657,1,585,4745,42705,384345,1,4681,304265,2738385,
%T A347490 22211345,199902105,1799118945,1,37449,19477641,156087945,175298769,
%U A347490 11394419985,92421406545,102549779865,831792658905,7486133930145,67375205371305,1,299593
%N 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.
%C A347490 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 A347490 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 A347490 R. P. Stanley, Enumerative Combinatorics (vol. 1), Cambridge University Press (1997), Section 1.3.
%H A347490 Álvar Ibeas, <a href="/A347490/b347490.txt">First 20 rows, flattened</a>
%F A347490 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 A347490 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 A347490 Triangle begins:
%e A347490   k:  1   2    3     4      5
%e A347490       -----------------------
%e A347490 n=1:  1
%e A347490 n=2:  1   9
%e A347490 n=3:  1  73  657
%e A347490 n=4:  1 585 4745 42705 384345
%Y A347490 Cf. A036038 (q = 1), A022172, A015007 (last entry in each row).
%K A347490 nonn,tabf
%O A347490 1,3
%A A347490 _Álvar Ibeas_, Sep 03 2021