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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).
0
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
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
KEYWORD
nonn,tabf
AUTHOR
Geoffrey Critzer, Nov 01 2022
STATUS
approved