A270881 -id:A270881 - cp's OEIS frontend

A270880 Triangle read by rows: T(n,m) is the number of direct-sum decompositions of a finite vector space of dimension n with m blocks over GF(2).

1, 0, 1, 0, 1, 3, 0, 1, 28, 28, 0, 1, 400, 1680, 840, 0, 1, 10416, 168640, 277760, 83328, 0, 1, 525792, 36053248, 159989760, 139991040, 27998208, 0, 1, 51116992, 17811244032, 209056841728, 419919790080, 227569434624, 32509919232

Offset: 0

Michel Marcus, Mar 25 2016

			Triangle begins:
1;
0, 1;
0, 1, 3;
0, 1, 28, 28;
0, 1, 400, 1680, 840;
0, 1, 10416, 168640, 277760, 83328;
...

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. A053601 (right diagonal), A270881 (row sums), A270882.

g[n_] := q^Binomial[n, 2] *FunctionExpand[QFactorial[n, q]]*(q - 1)^n /. q -> 2;Table[Table[Total[Map[g[n]/Apply[Times, g[#]]/Apply[Times, Table[Count[#, i], {i, 1, n}]!] &,IntegerPartitions[n, {m}]]], {m, 1, n}], {n, 1, 6}] // Grid (* Geoffrey Critzer, May 18 2017 *)

T(n,m) = Sum_ g(n)/(g(n_1)*g(n_2)***g(n_m))/(a_1!*a_2!***a_n!) where the sum is over all partitions of n into m parts and a_1,a_2,...,a_n is the part count signature of the partition and g(n) = A002884(n). - Geoffrey Critzer, May 18 2017 (after formula given in first Ellerman link above).

A270883 Row sums of triangle A270882. Number of direct-sum decompositions of an n-dimensional vector space over GF(2) with any given nonzero vector in a block.

Original entry on oeis.org

1, 1, 3, 29, 961, 110657, 45148929, 66294748161, 355213310611457, 7025248750804353025, 517789725632146766102529, 143350189472963401121415823361, 150053549525040193876302690826321921, 597137918840965720442548744290289324130305, 9075744511279922489436849557317778793074029232129

Offset: 0

Michel Marcus, Mar 25 2016

nonn

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. A270881, A270882.

Recurrence: a(n) = Sum_{k=0,...,n-1} q-binomial(n-1,k)*q^(n*(n-k))*D_q(k) where D_q(k) is given by A270881 for q = 2 and where the q-binomial for q = 2 is given by A022166. This summation formula is the q-analog of the summation formula for the Bell numbers A000110 when q = 1. - David P. Ellerman, Mar 26 2016

Name edited by David P. Ellerman, Mar 26 2016

a(8)-a(14) from Geoffrey Critzer, May 21 2017

A289543 Number of direct sum decompositions of GF(2)^n that do not contain any subspaces of dimension 1.

Original entry on oeis.org

1, 0, 1, 1, 281, 9921, 16078337, 13596908545, 191426147495937, 3273234077014474753, 497324772153177747947521, 154709087482207635347155451905, 291534668371237082293312814285062145, 1534814232386517133354150755522868689240065, 39269743760371912650589750432327799926355436503041, 3338607968166762847572429548161284663670177988768356630529

Offset: 0

Geoffrey Critzer, Jul 19 2017

nonn

q-analog of A000296.

David Ellerman, The number of direct-sum decompositions of a finite vector space, arXiv:1603.07619 [math.CO], 2016.
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1

Cf. A270881, A287406.

nn = 15; q := 2; g[n_] := (q - 1)^n  q^Binomial[n, 2] FunctionExpand[QFactorial[n, q]]; G[z_] :=Sum[z^k/g[k], {k, 1, nn}];Table[g[n], {n, 0, nn}] CoefficientList[
  Series[Exp[G[z] - z], {z, 0, nn}], z]

a(n)/A002884(n) is the coefficient of x^n in the expansion of exp(Sum_{k>1}x^k/A002884(k)).

A298561 Triangle read by rows. T(n,k) is the number of direct sum decompositions of GF(2)^n into subspaces of dimension at most k, 1<=k<=n.

Original entry on oeis.org

1, 3, 4, 28, 56, 57, 840, 2800, 2920, 2921, 83328, 499968, 539648, 540144, 540145, 27998208, 323534848, 363889408, 364556032, 364558048, 364558049, 32509919232, 765789208576, 904149876736, 906907414528, 906918338560, 906918346688, 906918346689

Offset: 1

Geoffrey Critzer, Jan 21 2018

			  1
  3,     4,
  28,    56,     57,
  840,   2800,   2920,   2921,
  83328, 499968, 539648, 540144, 540145,

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.

Cf. A270881 (main diagonal), A053601 (column 1), A298339.

nn = 7; \[Gamma][n_] := (q - 1)^n  q^Binomial[n, 2] FunctionExpand[QFactorial[n, q]] /. q -> 2; Flatten[Table[Table[Transpose[
     Map[Drop[#, 1] &,Table[Table[\[Gamma][n], {n, 0, nn}] CoefficientList[Series[Exp[Sum[z^i/\[Gamma][i], {i, 1, k}]], {z, 0, nn}],z], {k, 1, nn}]]][[j, k]], {k, 1, j}], {j, 1, nn}]]

exp(Sum_{j=0...k} x^j/A002884(j)) = Sum_{n>=0} T(n,k)/A002884(n)*x^n.

A303533 Number of ordered direct sum decompositions of the vector space GF(2)^n.

Original entry on oeis.org

1, 1, 7, 225, 31041, 17698273, 41014759873, 383214694567809, 14378402336340492033, 2162169920997910948019713, 1301828396408136687071569640449, 3136821919822089791220365613645953025, 30240714417270288646830264781681630189187073

Offset: 0

Geoffrey Critzer, Apr 25 2018

nonn

Cf. A270881.

nn = 12; \[Gamma][n_] := (q - 1)^n  q^Binomial[n, 2] FunctionExpand[QFactorial[n, q]] /. q -> 2; \[CapitalGamma][z_] :=
Sum[z^k/\[Gamma][k], {k, 0, nn}];Table[\[Gamma][n], {n, 0, nn}] CoefficientList[Series[1/(1 - (\[CapitalGamma][z] - 1)), {z, 0, nn}], z]

Sum_{n>=0}a(n)x^n/g(n) = 1/(2-(Sum_{n>=0}x^n/g(n))) where g(n) = A002884(n).

a(n) ~ c * d^n * 2^(n^2), where d = 1.149524744759658194895953141071829185374022882216951573931... and c = 0.2546517972696293457891304601766804587838159436304512... - Vaclav Kotesovec, May 06 2018

A287206 Triangle read by rows: T(n,k) is the number of direct sum decompositions of a finite vector space of n dimensions over GF(2) that have exactly k subspaces of dimension 1, n>=0, 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 1, 0, 3, 1, 28, 0, 28, 281, 120, 1680, 0, 840, 9921, 139376, 29760, 277760, 0, 83328, 16078337, 20000736, 140491008, 19998720, 139991040, 0, 27998208, 13596908545, 130684723136, 81282991104, 380636971008, 40637399040, 227569434624, 0, 32509919232, 191426147495937, 443803094908800, 2132774681579520, 884358943211520, 3105997683425280, 265280940933120, 1237977724354560, 0, 132640470466560

Offset: 0

Geoffrey Critzer, May 21 2017

			Triangle T(n,k) begins:
     1;
     0,      1;
     1,      0,     3;
     1,     28,     0,     28,
   281,    120,  1680,      0,  840;
  9921, 139376, 29760, 277760,    0, 83328;
  ...

David Ellerman, The number of direct-sum decompositions of a finite vector space, arXiv:1603.07619 [math.CO], 2016.
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1

Cf. A270881 (row sums), A053601 (main diagonal), A289543 (column k=0).

nn = 8; g[n_] := QFactorial[n, q]*(q - 1)^n*q^Binomial[n, 2] /. q -> 2; e[u_] := Sum[u^r/g[r], {r, 0, nn}];
Table[Table[(Table[g[n], {n, 0, nn}] CoefficientList[  Series[Exp[e[u] - 1 - u + u t], {u, 0, nn}], {u, t}])[[n,
     k]], {k, 1, n}], {n, 1, nn + 1}] // Grid

Sum_{n>=0} T(n,k)*u^n/g(n)*t^k = exp(Sum_{r>=0} u^r/g(r) - 1 - u + t*u) where g = A002884.

A298399 Triangle read by rows: T(n,k) is the number of direct sum decompositions of GF(2)^n whose maximal subspace has dimension k, 1<=k<=n, n>=1.

Original entry on oeis.org

1, 3, 1, 28, 28, 1, 840, 1960, 120, 1, 83328, 416640, 39680, 496, 1, 27998208, 295536640, 40354560, 666624, 2016, 1, 32509919232, 733279289344, 138360668160, 2757537792, 10924032, 8128, 1, 132640470466560, 6568159593103360, 1654847774392320, 38430207737856, 181463777280, 176865280, 32640, 1

Offset: 1

Geoffrey Critzer, Jan 18 2018

			Triangle begins:
      1;
      3,      1;
     28,     28,     1;
    840,   1960,   120,   1;
  83328, 416640, 39680, 496,  1;
  ...

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.

Cf. A053601 (column 1), A270881 (row sums), A298561.

nn = 7; \[Gamma][n_] := (q - 1)^n  q^Binomial[n, 2] FunctionExpand[ QFactorial[n, q]] /. q -> 2; Grid[Map[Select[#, # > 0 &] &,
  Drop[Transpose[Table[Table[\[Gamma][n], {n, 0, nn}] CoefficientList[Series[Exp[Sum[z^i/\[Gamma][i], {i, 1, k + 1}]] -
Exp[Sum[z^i/\[Gamma][i], {i, 1, k}]], {z, 0, nn}], z], {k, 0, 4}]], 1]]]

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!.

cp's OEIS Frontend

A270880 Triangle read by rows: T(n,m) is the number of direct-sum decompositions of a finite vector space of dimension n with m blocks over GF(2).

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A270883 Row sums of triangle A270882. Number of direct-sum decompositions of an n-dimensional vector space over GF(2) with any given nonzero vector in a block.

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A289543 Number of direct sum decompositions of GF(2)^n that do not contain any subspaces of dimension 1.

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A298561 Triangle read by rows. T(n,k) is the number of direct sum decompositions of GF(2)^n into subspaces of dimension at most k, 1<=k<=n.

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A303533 Number of ordered direct sum decompositions of the vector space GF(2)^n.

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A287206 Triangle read by rows: T(n,k) is the number of direct sum decompositions of a finite vector space of n dimensions over GF(2) that have exactly k subspaces of dimension 1, n>=0, 0<=k<=n.

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A298399 Triangle read by rows: T(n,k) is the number of direct sum decompositions of GF(2)^n whose maximal subspace has dimension k, 1<=k<=n, n>=1.

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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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