A270882 -id:A270882 - cp's OEIS frontend

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.

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

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).

Original entry on oeis.org

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).

A377642 a(n) = (1/(n-1)!) * Product_{i=1..n-1} (2^n-2^i).

Original entry on oeis.org

1, 2, 12, 224, 13440, 2666496, 1791885312, 4161269661696, 33955960439439360, 987107315743488737280, 103404624282172311371513856, 39408968779516596852827017445376, 55084280201257118417007491904448757760, 284322478318511376197290687371005495020093440

Offset: 1

Nikita Babich, Nov 05 2024

Nikita Babich, Table of n, a(n) for n = 1..100

Appears to be main diagonal of A270882.

Cf. A000079, A000142, A002884, A028365, A048651, A053601.

Table[Product[2^n - 2^i, {i, 1, n - 1}]/Factorial[n - 1], {n, 1, 20}]

a(n)=prod(i=1, n-1, 2^n-2^i)/(n-1)! \\ Andrew Howroyd, Nov 10 2024

a(n) = (Product_{i=1..n-1}(2^n-2^i))/((n-1)!).
a(n) = A028365(n-1)/A000142(n-1).
a(n) = A000079(n-1) * A053601(n-1).
a(n) ~ A048651 * 2^(n*(n-1)) / (n-1)!. - Vaclav Kotesovec, Nov 13 2024

cp's OEIS Frontend

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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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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A377642 a(n) = (1/(n-1)!) * Product_{i=1..n-1} (2^n-2^i).

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