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A347486 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 = 3.

%I A347486 #14 Sep 15 2021 10:26:42
%S A347486 1,1,4,1,13,52,1,40,130,520,2080,1,121,1210,4840,15730,62920,251680,1,
%T A347486 364,11011,33880,44044,440440,1431430,1761760,5725720,22902880,
%U A347486 91611520,1,1093,99463,925771,397852,12035023,37030840,120350230,48140092,481400920,1564552990
%N A347486 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 = 3.
%C A347486 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 A347486 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_3)^n with dimension increments (e_1,...,e_r).
%D A347486 R. P. Stanley, Enumerative Combinatorics (vol. 1), Cambridge University Press (1997), Section 1.3.
%H A347486 Álvar Ibeas, <a href="/A347486/b347486.txt">First 20 rows, flattened</a>
%F A347486 T(n, (n)) = 1. T(n, L) = A022167(n, e) * T(n - e, L \ {e}), if L is a partition of n and e < n is a part of L.
%e A347486 The number of subspace chains 0 < V_1 < V_2 < (F_3)^3 is 52 = T(3, (1, 1, 1)). There are 13 = A022167(3, 1) choices for a one-dimensional subspace V_1 and, for each of them, 4 = A022167(2, 1) extensions to a two-dimensional subspace V_2.
%e A347486 Triangle begins:
%e A347486   k:  1   2    3    4     5     6      7
%e A347486       ----------------------------------
%e A347486 n=1:  1
%e A347486 n=2:  1   4
%e A347486 n=3:  1  13   52
%e A347486 n=4:  1  40  130  520  2080
%e A347486 n=5:  1 121 1210 4840 15730 62920 251680
%Y A347486 Cf. A036038 (q = 1), A022167, A015001 (last entry in each row).
%K A347486 nonn,tabf
%O A347486 1,3
%A A347486 _Álvar Ibeas_, Sep 03 2021