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A347485 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 = 2.

%I A347485 #14 Sep 15 2021 10:26:31
%S A347485 1,1,3,1,7,21,1,15,35,105,315,1,31,155,465,1085,3255,9765,1,63,651,
%T A347485 1395,1953,9765,22785,29295,68355,205065,615195,1,127,2667,11811,8001,
%U A347485 82677,177165,413385,248031,1240155,2893695,3720465,8681085,26043255,78129765
%N A347485 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 = 2.
%C A347485 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 A347485 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_2)^n with dimension increments (e_1,...,e_r).
%D A347485 R. P. Stanley, Enumerative Combinatorics (vol. 1), Cambridge University Press (1997), Section 1.3.
%H A347485 Álvar Ibeas, <a href="/A347485/b347485.txt">First 20 rows, flattened</a>
%F A347485 T(n, (n)) = 1. T(n, L) = A022166(n, e) * T(n - e, L \ {e}), if L is a partition of n and e < n is a part of L.
%e A347485 The number of subspace chains 0 < V_1 < V_2 < (F_2)^3 is 21 = T(3, (1, 1, 1)). There are 7 = A022166(3, 1) choices for a one-dimensional subspace V_1 and, for each of them, 3 = A022166(2, 1) extensions to a two-dimensional subspace V_2.
%e A347485 Triangle begins:
%e A347485   k:  1  2   3   4    5    6    7
%e A347485       ---------------------------
%e A347485 n=1:  1
%e A347485 n=2:  1  3
%e A347485 n=3:  1  7  21
%e A347485 n=4:  1 15  35 105  315
%e A347485 n=5:  1 31 155 465 1085 3255 9765
%Y A347485 Cf. A036038 (q = 1), A022166, A005329 (last entry in each row).
%K A347485 nonn,tabf
%O A347485 1,3
%A A347485 _Álvar Ibeas_, Sep 03 2021