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A347489 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 = 7.

%I A347489 #14 Sep 15 2021 10:27:14
%S A347489 1,1,8,1,57,456,1,400,2850,22800,182400,1,2801,140050,1120400,7982850,
%T A347489 63862800,510902400,1,19608,6865251,48177200,54922008,2746100400,
%U A347489 19565965350,21968803200,156527722800,1252221782400,10017774259200,1,137257,336416907,16531644851,2691335256
%N A347489 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 = 7.
%C A347489 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 A347489 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_7)^n with dimension increments (e_1,...,e_r).
%D A347489 R. P. Stanley, Enumerative Combinatorics (vol. 1), Cambridge University Press (1997), Section 1.3.
%H A347489 Álvar Ibeas, <a href="/A347489/b347489.txt">First 20 rows, flattened</a>
%F A347489 T(n, (n)) = 1. T(n, L) = A022171(n, e) * T(n - e, L \ {e}), if L is a partition of n and e < n is a part of L.
%e A347489 The number of subspace chains 0 < V_1 < V_2 < (F_7)^3 is 456 = T(3, (1, 1, 1)). There are 57 = A022171(3, 1) choices for a one-dimensional subspace V_1 and, for each of them, 8 = A022171(2, 1) extensions to a two-dimensional subspace V_2.
%e A347489 Triangle begins:
%e A347489   k:  1   2    3     4      5
%e A347489       -----------------------
%e A347489 n=1:  1
%e A347489 n=2:  1   8
%e A347489 n=3:  1  57  456
%e A347489 n=4:  1 400 2850 22800 182400
%Y A347489 Cf. A036038 (q = 1), A022171, A015006 (last entry in each row).
%K A347489 nonn,tabf
%O A347489 1,3
%A A347489 _Álvar Ibeas_, Sep 03 2021