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A347492 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 = 11.

%I A347492 #14 Sep 15 2021 10:27:51
%S A347492 1,1,12,1,133,1596,1,1464,16226,194712,2336544,1,16105,1964810,
%T A347492 23577720,261319730,3135836760,37630041120,1,177156,237758115,
%U A347492 2617126920,2853097380,348077880360,3857863173990,4176934564320,46294358087880,555532297054560,6666387564654720,1,1948717
%N A347492 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 = 11.
%C A347492 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 A347492 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_11)^n with dimension increments (e_1,...,e_r).
%D A347492 R. P. Stanley, Enumerative Combinatorics (vol. 1), Cambridge University Press (1997), Section 1.3.
%H A347492 Álvar Ibeas, <a href="/A347492/b347492.txt">First 20 rows, flattened</a>
%F A347492 T(n, (n)) = 1. T(n, L) = A022175(n, e) * T(n - e, L \ {e}), if L is a partition of n and e < n is a part of L.
%e A347492 The number of subspace chains 0 < V_1 < V_2 < (F_11)^3 is 1596 = T(3, (1, 1, 1)). There are 133 = A022175(3, 1) choices for a one-dimensional subspace V_1 and, for each of them, 12 = A022175(2, 1) extensions to a two-dimensional subspace V_2.
%e A347492 Triangle begins:
%e A347492   k:  1    2     3      4       5
%e A347492       ---------------------------
%e A347492 n=1:  1
%e A347492 n=2:  1   12
%e A347492 n=3:  1  133  1596
%e A347492 n=4:  1 1464 16226 194712 2336544
%Y A347492 Cf. A036038 (q = 1), A022175, A015011 (last entry in each row).
%K A347492 nonn,tabf
%O A347492 1,3
%A A347492 _Álvar Ibeas_, Sep 03 2021