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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

%I #17 Aug 02 2018 15:46:50

%S 1,2,1,5,7,3,16,50,56,21,67,446,1010,945,315,374,5395,22692,40455,

%T 32550,9765,2825,92881,704601,2167179,3193155,2255715,615195,29212,

%U 2350136,32061404,162602418,394534644,496062000,312519060,78129765,417199,89342600,2220570872,18194735010,68980503390,138302085600,151794972000

%N 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.

%H 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.

%F 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.

%e Triangle begins:

%e 1;

%e 2, 1;

%e 5, 7, 3;

%e 16, 50, 56, 21;

%e 67, 446, 1010, 945, 315;

%e 374, 5395, 22692, 40455, 32550, 9765;

%e ...

%t nn = 10; eq[z_] :=Sum[z^n/FunctionExpand[QFactorial[n, q]], {n, 0, nn}]; Grid[Map[Select[#, # > 0 &] &,

%t Table[FunctionExpand[QFactorial[n, q]] /. q -> 2, {n, 0,

%t nn}] CoefficientList[Series[ eq[z]^2/(1 - u (eq[z] - 1)) /. q -> 2, {z, 0, nn}], {z, u}]]]

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

%K nonn,tabl

%O 0,2

%A _Geoffrey Critzer_, Oct 17 2017