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A347487 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 = 4.

%I A347487 #14 Sep 15 2021 10:26:50
%S A347487 1,1,5,1,21,105,1,85,357,1785,8925,1,341,5797,28985,121737,608685,
%T A347487 3043425,1,1365,93093,376805,465465,7912905,33234201,39564525,
%U A347487 166171005,830855025,4154275125,1,5461,1490853,24208613,7454265,508380873,2057732105,8642474841
%N A347487 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 = 4.
%C A347487 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 A347487 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_4)^n with dimension increments (e_1,...,e_r).
%D A347487 R. P. Stanley, Enumerative Combinatorics (vol. 1), Cambridge University Press (1997), Section 1.3.
%H A347487 Álvar Ibeas, <a href="/A347487/b347487.txt">First 20 rows, flattened</a>
%F A347487 T(n, (n)) = 1. T(n, L) = A022168(n, e) * T(n - e, L \ {e}), if L is a partition of n and e < n is a part of L.
%e A347487 The number of subspace chains 0 < V_1 < V_2 < (F_4)^3 is 105 = T(3, (1, 1, 1)). There are 21 = A022168(3, 1) choices for a one-dimensional subspace V_1 and, for each of them, 5 = A022168(2, 1) extensions to a two-dimensional subspace V_2.
%e A347487 Triangle begins:
%e A347487   k:  1   2    3     4      5      6       7
%e A347487       --------------------------------------
%e A347487 n=1:  1
%e A347487 n=2:  1   5
%e A347487 n=3:  1  21  105
%e A347487 n=4:  1  85  357  1785   8925
%e A347487 n=5:  1 341 5795 28985 121737 608685 3043425
%Y A347487 Cf. A036038 (q = 1), A022168, A015002 (last entry in each row).
%K A347487 nonn,tabf
%O A347487 1,3
%A A347487 _Álvar Ibeas_, Sep 03 2021