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A347488 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 = 5.

%I A347488 #14 Sep 15 2021 10:26:17
%S A347488 1,1,6,1,31,186,1,156,806,4836,29016,1,781,20306,121836,629486,
%T A347488 3776916,22661496,1,3906,508431,2558556,3050586,79315236,409795386,
%U A347488 475891416,2458772316,14752633896,88515803376,1,19531,12714681,320327931,76288086
%N A347488 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 = 5.
%C A347488 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 A347488 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_5)^n with dimension increments (e_1,...,e_r).
%D A347488 R. P. Stanley, Enumerative Combinatorics (vol. 1), Cambridge University Press (1997), Section 1.3.
%H A347488 Álvar Ibeas, <a href="/A347488/b347488.txt">First 20 rows, flattened</a>
%F A347488 T(n, (n)) = 1. T(n, L) = A022169(n, e) * T(n - e, L \ {e}), if L is a partition of n and e < n is a part of L.
%e A347488 The number of subspace chains 0 < V_1 < V_2 < (F_5)^3 is 186 = T(3, (1, 1, 1)). There are 31 = A022169(3, 1) choices for a one-dimensional subspace V_1 and, for each of them, 6 = A022169(2, 1) extensions to a two-dimensional subspace V_2.
%e A347488 Triangle begins:
%e A347488   k:  1   2    3     4      5
%e A347488       -----------------------
%e A347488 n=1:  1
%e A347488 n=2:  1   6
%e A347488 n=3:  1  31  186
%e A347488 n=4:  1 156  806  4836  29016
%Y A347488 Cf. A036038 (q = 1), A022169, A015004 (last entry in each row).
%K A347488 nonn,tabf
%O A347488 1,3
%A A347488 _Álvar Ibeas_, Sep 03 2021