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A347491 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 = 9.

%I A347491 #14 Sep 15 2021 10:27:32
%S A347491 1,1,10,1,91,910,1,820,7462,74620,746200,1,7381,605242,6052420,
%T A347491 55077022,550770220,5507702200,1,66430,49031983,441826660,490319830,
%U A347491 40206226060,365876657146,402062260600,3658766571460,36587665714600,365876657146000,1,597871
%N A347491 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 = 9.
%C A347491 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 A347491 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_9)^n with dimension increments (e_1,...,e_r).
%D A347491 R. P. Stanley, Enumerative Combinatorics (vol. 1), Cambridge University Press (1997), Section 1.3.
%H A347491 Álvar Ibeas, <a href="/A347491/b347491.txt">First 20 rows, flattened</a>
%F A347491 T(n, (n)) = 1. T(n, L) = A022173(n, e) * T(n - e, L \ {e}), if L is a partition of n and e < n is a part of L.
%e A347491 The number of subspace chains 0 < V_1 < V_2 < (F_9)^3 is 910 = T(3, (1, 1, 1)). There are 91 = A022173(3, 1) choices for a one-dimensional subspace V_1 and, for each of them, 10 = A022173(2, 1) extensions to a two-dimensional subspace V_2.
%e A347491 Triangle begins:
%e A347491   k:  1   2    3     4      5
%e A347491       -----------------------
%e A347491 n=1:  1
%e A347491 n=2:  1  10
%e A347491 n=3:  1  91  910
%e A347491 n=4:  1 820 7462 74620 746200
%Y A347491 Cf. A036038 (q = 1), A022173, A015008 (last entry in each row).
%K A347491 nonn,tabf
%O A347491 1,3
%A A347491 _Álvar Ibeas_, Sep 03 2021