A288853 Triangle read by rows: T(n,k) is the number of surjective linear mappings from an n-dimensional vector space over F_2 onto a k-dimensional vector space, n>=0, 0<=k<=n.

1, 1, 1, 1, 3, 6, 1, 7, 42, 168, 1, 15, 210, 2520, 20160, 1, 31, 930, 26040, 624960, 9999360, 1, 63, 3906, 234360, 13124160, 629959680, 20158709760, 1, 127, 16002, 1984248, 238109760, 26668293120, 2560156139520, 163849992929280, 1, 255, 64770, 16322040, 4047865920, 971487820800, 217613271859200, 41781748196966400, 5348063769211699200

Offset: 0

Author

Geoffrey Critzer, Jun 18 2017

Keywords

• nonn
• tabl

Comments

The (q = 2) analog of A008279.

A022166(m,k)*T(n,k) is the number of m X n matrices over F_2 that have rank k.

a(n) is the number of n X n matrices over F_2 in Green's R class containing A where rank(A) = k. - Geoffrey Critzer, Oct 05 2022

Examples

			  1;
  1,  1;
  1,  3,   6;
  1,  7,  42,   168;
  1, 15, 210,  2520,  20160;
  1, 31, 930, 26040, 624960, 9999360;
  ...
		

Links

• Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
• Jeremy L. Martin, Lecture Notes on Algebraic Combinatorics, 2010-2023, Example 2.3.6.
• Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
• Wikipedia, Green's relations.

Crossrefs

Columns k=0-10 give: A000012, A000225, 6*A006095, 168*A006096, 20160*A006097, 9999360*A006110, 20158709760*A022189, 163849992929280*A022190, 5348063769211699200*A022191, 699612310033197642547200*A022192, 366440137299948128422802227200*A022193.

Main diagonal gives A002884.

Cf. A022166.

Programs

Mathematica

Table[Table[Product[q^n - q^i, {i, 0, k - 1}] /. q -> 2, {k, 0, n}], {n, 0,8}] // Grid

Formula

T(n,k) = Product_{j=0..k-1} (2^n - 2^j).

T(n,k) = A002884(k)*A022166(n,k).

Let g_m(x) = Sum_{n>=0} (2^m*x)^n/A005329(n) and e(x) = Sum_{n>=0} x^n/A005329(n). Then Sum_{k>=0} T(n,k)*x^k/A005329(k) = g_n(x)/e(x). - Geoffrey Critzer, Jun 01 2024