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%I #41 Feb 12 2020 02:15:22
%S 1,0,1,0,1,2,0,1,16,12,0,1,176,560,224,0,1,3456,40000,53760,13440,0,1,
%T 128000,5848832,20951040,15554560,2666496,0,1,9115648,1934195712,
%U 17826414592,30398054400,14335082496,1791885312,0,1,1259921408,1510821195776,37083513880576,134908940386304,133854174117888,43693331447808,4161269661696
%N Triangle read by rows: D*(n,m) is the number of direct-sum decompositions of a finite vector space of dimension n with m blocks over GF(2) with a block containing any given nonzero vector.
%H Jinyuan Wang, <a href="/A270882/b270882.txt">Rows n = 0..10 of triangle, flattened</a>
%H 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 David Ellerman, <a href="http://arxiv.org/abs/1604.01087">The Quantum Logic of Direct-Sum Decompositions</a>, arXiv preprint arXiv:1604.01087 [quant-ph], 2016. See Section 7.5.
%F Recurrence: a(n) = Sum_{k=0..n-1} q-binomial(n-1,k)*q^(n*(n-k))*D_q(k,m-1) where D_q(k,m-1) is given by A270880 for q = 2 and where the q-binomial for q = 2 is given by A022166. This formula is the q-analog of summation formula for the Stirling numbers of the second kind A008277 so when q = 1, it reduces to that formula. - _David P. Ellerman_, Mar 26 2016
%e Triangle begins:
%e 1;
%e 0, 1;
%e 0, 1, 2;
%e 0, 1, 16, 12;
%e 0, 1, 176, 560, 224;
%e 0, 1, 3456, 40000, 53760, 13440;
%e 0, 1, 128000, 5848832, 20951040, 15554560, 2666496;
%e ...
%t (* about 40 seconds on a laptop computer *) g[n_] := q^Binomial[n, 2] * FunctionExpand[QFactorial[n, q]]*(q - 1)^n /. q -> 2; d[k_, m_] :=Map[PadRight[#, 10] &,Table[Table[Total[Map[g[n]/Apply[Times, g[#]]/Apply[Times, Table[Count[#, i], {i, 1, n}]!] &,IntegerPartitions[n, {j}]]], {j, 1, n}], {n, 1, 10}]][[k, m]];d[0, m_] := If[m == 0, 1, 0]; d[k_, 0] := If[k == 0, 1, 0];s[n_, m_] :=Sum[FunctionExpand[QBinomial[n - 1, k, 2]]*2^(k (n - k))*d[k, m - 1], {k, 0, n - 1}]; Table[Table[s[n, m], {m, 1, n}], {n, 1,7}] (* _Geoffrey Critzer_, May 20 2017 *)
%Y Cf. A270880, A270883 (row sums).
%K nonn,tabl
%O 0,6
%A _Michel Marcus_, Mar 25 2016
%E Name extended by _David P. Ellerman_, Mar 26 2016
%E Row 8 from _Geoffrey Critzer_, May 20 2017
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