cp's OEIS Frontend

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.

%I A288853 #40 Jun 03 2024 18:26:12
%S A288853 1,1,1,1,3,6,1,7,42,168,1,15,210,2520,20160,1,31,930,26040,624960,
%T A288853 9999360,1,63,3906,234360,13124160,629959680,20158709760,1,127,16002,
%U A288853 1984248,238109760,26668293120,2560156139520,163849992929280,1,255,64770,16322040,4047865920,971487820800,217613271859200,41781748196966400,5348063769211699200
%N 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.
%C A288853 The (q = 2) analog of A008279.
%C A288853 A022166(m,k)*T(n,k) is the number of m X n matrices over F_2 that have rank k.
%C A288853 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
%H A288853 Geoffrey Critzer, <a href="https://esirc.emporia.edu/handle/123456789/3595">Combinatorics of Vector Spaces over Finite Fields</a>, Master's thesis, Emporia State University, 2018.
%H A288853 Jeremy L. Martin, <a href="https://jlmartin.ku.edu/LectureNotes.pdf">Lecture Notes on Algebraic Combinatorics</a>, 2010-2023, Example 2.3.6.
%H A288853 Kent E. Morrison, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Morrison/morrison37.html">Integer Sequences and Matrices Over Finite Fields</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
%H A288853 Wikipedia, <a href="http://en.wikipedia.org/wiki/Green&#39;s_relations">Green's relations</a>.
%F A288853 T(n,k) = Product_{j=0..k-1} (2^n - 2^j).
%F A288853 T(n,k) = A002884(k)*A022166(n,k).
%F A288853 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
%e A288853   1;
%e A288853   1,  1;
%e A288853   1,  3,   6;
%e A288853   1,  7,  42,   168;
%e A288853   1, 15, 210,  2520,  20160;
%e A288853   1, 31, 930, 26040, 624960, 9999360;
%e A288853   ...
%t A288853 Table[Table[Product[q^n - q^i, {i, 0, k - 1}] /. q -> 2, {k, 0, n}], {n, 0,8}] // Grid
%Y A288853 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.
%Y A288853 Main diagonal gives A002884.
%Y A288853 Cf. A022166.
%K A288853 nonn,tabl
%O A288853 0,5
%A A288853 _Geoffrey Critzer_, Jun 18 2017