A293845 Triangle read by rows: T(n,k) is the number of chains of length k in the partially ordered (by subspace inclusion) set of all subspaces of GF(2)^n, n>=0, 0<=k<=n.

1, 2, 1, 5, 7, 3, 16, 50, 56, 21, 67, 446, 1010, 945, 315, 374, 5395, 22692, 40455, 32550, 9765, 2825, 92881, 704601, 2167179, 3193155, 2255715, 615195, 29212, 2350136, 32061404, 162602418, 394534644, 496062000, 312519060, 78129765, 417199, 89342600, 2220570872, 18194735010, 68980503390, 138302085600, 151794972000

Offset

0,2

Links

Table of n, a(n) for n=0..42.

Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.

Formula

T(n,k)/A005329(n) is the coefficient of y^k*x^n in eq(x)^2/(1 - y (eq(x) - 1)) where eq(x) is the q-exponential function.

Example

Triangle begins:
1;
2, 1;
5, 7, 3;
16, 50, 56, 21;
67, 446, 1010, 945, 315;
374, 5395, 22692, 40455, 32550, 9765;
...

Mathematica

nn = 10; eq[z_] :=Sum[z^n/FunctionExpand[QFactorial[n, q]], {n, 0, nn}]; Grid[Map[Select[#, # > 0 &] &,
  Table[FunctionExpand[QFactorial[n, q]] /. q -> 2, {n, 0,
     nn}] CoefficientList[Series[ eq[z]^2/(1 - u (eq[z] - 1)) /. q -> 2, {z, 0, nn}], {z, u}]]]

Crossrefs

Cf. A289546, A293844 (row sums), A005329 (main diagonal), A006116 (column k = 0).

Sequence in context: A085240 A002251 A093545 * A249265 A259972 A193762

Adjacent sequences: A293842 A293843 A293844 * A293846 A293847 A293848

Keyword

nonn,tabl

Author

Geoffrey Critzer, Oct 17 2017

Status

approved