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A358165 Irregular triangular array read by rows. T(n,k) is the number of direct sum decompositions V_1 + V_2 + ... + V_m = GF(2)^n with the dimensions of the V_i corresponding to the k-th partition of n in canonical ordering, n >= 0, 1 <= k <= A000041(n).

%I A358165 #10 Apr 19 2024 17:49:30
%S A358165 1,1,1,3,1,28,28,1,120,280,1680,840,1,496,9920,29760,138880,277760,
%T A358165 83328,1,2016,166656,499968,357120,19998720,19998720,15554560,
%U A358165 139991040,139991040,27998208,1,8128,2731008,8193024,48377856,1354579968,1354579968,2902671360,13545799680,81274798080,40637399040,126427463680,379282391040,227569434624,32509919232
%N A358165 Irregular triangular array read by rows.  T(n,k) is the number of direct sum decompositions V_1 + V_2 + ... + V_m = GF(2)^n with the dimensions of the V_i corresponding to the k-th partition of n in canonical ordering, n >= 0, 1 <= k <= A000041(n).
%H A358165 David Ellerman, <a href="http://arxiv.org/abs/1603.07619">The number of direct-sum decompositions of a finite vector space</a>, arXiv:1603.07619 [math.CO], 2016.
%H A358165 Wikipedia, <a href="http://en.wikipedia.org/wiki/Orderings_of partitions">Orderings of partitions</a>
%F A358165 For i = 1,...,n let a_i be the number of parts of size i in the k-th partition of n in canonical ordering.  T(n,k) = A002884(n)/Product_{j=1..n} A002884(j)^a_j*a_j!.
%e A358165 Triangle begins:
%e A358165   1;
%e A358165   1;
%e A358165   1,   3;
%e A358165   1,  28,   28;
%e A358165   1, 120,  280,  1680,    840;
%e A358165   1, 496, 9920, 29760, 138880, 277760, 83328;
%e A358165   ...
%e A358165 T(4,3) = 280.  For n=4 the five partitions in canonical ordering are {4}, {3, 1}, {2, 2}, {2, 1, 1}, {1, 1, 1, 1}.  The third partition in this order is {2,2}.  So T(4,3) = A002884(4)/(A002884(2)^2*2!) = 280.
%t A358165 dsd2[n_, signature_] := Product[2^n - 2^i, {i, 0, n - 1}]/ Product[Product[2^k - 2^i, {i, 0, k - 1}]^signature[[k]]*signature[[k]]!, {k, 1, n}];Table[Map[dsd2[n, #] &,Map[Table[Count[#, i], {i, 1, n}] &, IntegerPartitions[n]]], {n, 0,6}] // Grid
%Y A358165 Cf. A270880, A270881 (row sums), A279038, A080575, A000041, A002884, A053601 (main diagonal).
%K A358165 nonn,tabf
%O A358165 0,4
%A A358165 _Geoffrey Critzer_, Nov 01 2022