A270880 -id:A270880 - cp's OEIS frontend

A270881 Row sums of triangle A270880. Number of direct-sum decompositions of a finite vector space of n dimensions over GF(2).

1, 1, 4, 57, 2921, 540145, 364558049, 906918346689, 8394259686375297, 291375477821572448001, 38187935488350036891532801, 19005446750755761952317881973761, 36091267618694510017592440805677594625, 262587035725176662374187801686523815760228353, 7345273837043092730077580223639933697831592435638273

Offset: 0

Michel Marcus, Mar 25 2016

nonn

The generating function for these numbers was first derived in Bender & Goldman. My paper derives the direct formula for the numbers for any finite vector space over GF(q) so that when q = 1, the formula gives the Bell numbers--since a direct-sum decomposition is the vector space version of a set partition. This sequence gives the numbers for q = 2. - David P. Ellerman, Mar 26 2016

Edward A. Bender, and Jay R. Goldman, Enumerative Uses of Generating Functions, Indiana University Mathematics Journal 20 (8) (1971) 753-65.
Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
David Ellerman, The number of direct-sum decompositions of a finite vector space, arXiv:1603.07619 [math.CO], 2016.
David Ellerman, The Quantum Logic of Direct-Sum Decompositions, arXiv preprint arXiv:1604.01087 [quant-ph], 2016. See Section 7.5.

Cf. A270880.

g[n_] := q^Binomial[n, 2] * FunctionExpand[QFactorial[n, q]]*(q - 1)^n /. q -> 2; Table[Total[Table[Total[Map[g[n]/Apply[Times, g[#]]/Apply[Times, Table[Count[#, i], {i, 1, n}]!] &,IntegerPartitions[n, {m}]]], {m, 1, n}]], {n, 1, 15}] (* Geoffrey Critzer, May 18 2017 *)

Name extended by David P. Ellerman, Mar 26 2016

a(8)-a(14) from Geoffrey Critzer, May 18 2017

A270882 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.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 16, 12, 0, 1, 176, 560, 224, 0, 1, 3456, 40000, 53760, 13440, 0, 1, 128000, 5848832, 20951040, 15554560, 2666496, 0, 1, 9115648, 1934195712, 17826414592, 30398054400, 14335082496, 1791885312, 0, 1, 1259921408, 1510821195776, 37083513880576, 134908940386304, 133854174117888, 43693331447808, 4161269661696

Offset: 0

Michel Marcus, Mar 25 2016

			Triangle begins:
  1;
  0, 1;
  0, 1,      2;
  0, 1,     16,      12;
  0, 1,    176,     560,      224;
  0, 1,   3456,   40000,    53760,    13440;
  0, 1, 128000, 5848832, 20951040, 15554560, 2666496;
  ...

Jinyuan Wang, Rows n = 0..10 of triangle, flattened
David Ellerman, The number of direct-sum decompositions of a finite vector space, arXiv:1603.07619 [math.CO], 2016.
David Ellerman, The Quantum Logic of Direct-Sum Decompositions, arXiv preprint arXiv:1604.01087 [quant-ph], 2016. See Section 7.5.

Cf. A270880, A270883 (row sums).

The main diagonal appears to match A377642. - Nikita Babich, Nov 17 2024

(* 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 *)

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

Name extended by David P. Ellerman, Mar 26 2016

Row 8 from Geoffrey Critzer, May 20 2017

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).

Original entry on oeis.org

1, 1, 1, 3, 1, 28, 28, 1, 120, 280, 1680, 840, 1, 496, 9920, 29760, 138880, 277760, 83328, 1, 2016, 166656, 499968, 357120, 19998720, 19998720, 15554560, 139991040, 139991040, 27998208, 1, 8128, 2731008, 8193024, 48377856, 1354579968, 1354579968, 2902671360, 13545799680, 81274798080, 40637399040, 126427463680, 379282391040, 227569434624, 32509919232

Offset: 0

Geoffrey Critzer, Nov 01 2022

			Triangle begins:
  1;
  1;
  1,   3;
  1,  28,   28;
  1, 120,  280,  1680,    840;
  1, 496, 9920, 29760, 138880, 277760, 83328;
  ...
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.

David Ellerman, The number of direct-sum decompositions of a finite vector space, arXiv:1603.07619 [math.CO], 2016.
Wikipedia, Orderings of partitions

Cf. A270880, A270881 (row sums), A279038, A080575, A000041, A002884, A053601 (main diagonal).

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

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!.

A373536 Number of ways to form a direct sum decomposition of the vector space GF(2)^n and then choose a basis for each subspace in the decomposition.

Original entry on oeis.org

1, 1, 9, 364, 61320, 41747328, 113420740608, 1223445790457856, 52307167449899335680, 8861896666997422628536320, 5951934931285476447488997064704, 15857359709817958217841735837828513792, 167702614892018104786663957623269078052372480, 7044769706183185876455816992603242619680927682396160

Offset: 0

Geoffrey Critzer, Jun 08 2024

nonn

David Ellerman, The number of direct-sum decompositions of a finite vector space, arXiv:1603.07619 [math.CO], 2016.
R. P. Stanley, Exponential structures

Cf. A270880, A000262, A053601, A002884.

nn = 13; B[n_] := Product[q^n - q^i, {i, 0, n - 1}] /. q -> 2;
e[x_] := Sum[x^n/B[n], {n, 0, nn}]; f[x_] := Sum[x^n, {n, 0, nn}];
Table[B[n], {n, 0, nn}] CoefficientList[Series[Exp[f[x] - 1], {x, 0, nn}], x]

a(n) = A000262(n)*A053601(n).

Sum_{n>=0} a(n)*x^n/A002884(n) = exp(x/(1-x)).

cp's OEIS Frontend

A270881 Row sums of triangle A270880. Number of direct-sum decompositions of a finite vector space of n dimensions over GF(2).

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A270882 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.

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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).

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A373536 Number of ways to form a direct sum decomposition of the vector space GF(2)^n and then choose a basis for each subspace in the decomposition.

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Mathematica

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