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