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).

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

0,4

Links

Table of n, a(n) for n=0..44.

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

Wikipedia, Orderings of partitions

Formula

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!.

Example

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.

Mathematica

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

Crossrefs

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

Sequence in context: A287206 A288517 A188110 * A298399 A365087 A120066

Adjacent sequences: A358162 A358163 A358164 * A358166 A358167 A358168

Keyword

nonn,tabf

Author

Geoffrey Critzer, Nov 01 2022

Status

approved