A289545 Number of flags in an n-dimensional vector space over GF(2).

1, 1, 4, 36, 696, 27808, 2257888, 369572160, 121459776768, 79991977040128, 105466641591287296, 278244130564826548224, 1468496684404408240109568, 15502543140842029367582248960, 327332729703063815298568073396224, 13823536566775628445052117519260598272

Offset: 0

Author

Geoffrey Critzer, Jul 28 2017

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..80
• Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
• Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

Crossrefs

Cf. A381299.

Column k=2 of A381426.

Programs

Maple

b:= proc(o, u, t) option remember; `if`(u+o=0, 1, `if`(t=1,
      b(u+o, 0$2), 0)+add(2^(u+j-1)*b(o-j, u+j-1, 1), j=1..o))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..16);  # Alois P. Heinz, Feb 21 2025

Mathematica

nn = 15; eq[z_] :=Sum[z^n/FunctionExpand[QFactorial[n, q]], {n, 0, nn}];Table[FunctionExpand[QFactorial[n, q]] /. q -> 2, {n, 0,
   nn}] CoefficientList[Series[ 1/(1 - (eq[z] - 1)) /. q -> 2, {z, 0, nn}], z]

Formula

a(n) = Sum A005329(n)/( A005329(n_1)*A005329(n_2)*...*A005329(n_k) ) where the sum is over all compositions of n = n_1 + n_2 + ... + n_k.

G.f. a(n)/A005329(n) is the coefficient of x^n in 1/(2 - eq(x)) where eq(x) is the 2-exponential function.

a(n) = Sum_{k=0..binomial(n,2)} 2^k * A381299(n,k). - Alois P. Heinz, Feb 21 2025