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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. 2

%I #14 Sep 15 2021 10:26:31

%S 1,1,3,1,7,21,1,15,35,105,315,1,31,155,465,1085,3255,9765,1,63,651,

%T 1395,1953,9765,22785,29295,68355,205065,615195,1,127,2667,11811,8001,

%U 82677,177165,413385,248031,1240155,2893695,3720465,8681085,26043255,78129765

%N 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 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 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 R. P. Stanley, Enumerative Combinatorics (vol. 1), Cambridge University Press (1997), Section 1.3.

%H Álvar Ibeas, <a href="/A347485/b347485.txt">First 20 rows, flattened</a>

%F 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 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 Triangle begins:

%e k: 1 2 3 4 5 6 7

%e ---------------------------

%e n=1: 1

%e n=2: 1 3

%e n=3: 1 7 21

%e n=4: 1 15 35 105 315

%e n=5: 1 31 155 465 1085 3255 9765

%Y Cf. A036038 (q = 1), A022166, A005329 (last entry in each row).

%K nonn,tabf

%O 1,3

%A _Álvar Ibeas_, Sep 03 2021

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Last modified April 20 16:37 EDT 2024. Contains 371845 sequences. (Running on oeis4.)