A368493 - OEIS

%I #40 Dec 30 2023 03:08:50

%S 1,1,3,1,15,15,1,63,315,135,1,255,5355,11475,2295,1,1023,86955,782595,

%T 782595,75735,1,4095,1396395,50868675,213648435,103378275,4922775,1,

%U 16383,22362795,3268162755,55558766835,112909751955,26883274275,635037975

%N T(n,m) is the number of m-dimensional isotropic subspaces of a 2n-dimensional symplectic space over Z/2, n >= 0 and 0 <= m <= n.

%C The number of m-dimensional isotropic subspaces of an n-dimensional symplectic space over Z/2.

%H J. Baez, <a href="https://math.ucr.edu/home/baez/week187.html">This Week's Finds in Mathematical Physics (Week 187)</a>

%H M. H. Poroch, <a href="https://doi.org/10.1142/S1793557119500694">Bounds on subspace codes based on totally isotropic subspace in symplectic spaces and extended symplectic spaces</a>, Asian-European Journal of Mathematics, 12 (2019).

%H Z. Wan, <a href="https://www.sciengine.com/Math%20A0/doi/10.1360/ya1964-13-3-515">Notes on finite geometries and the construction of PBIB designs I, Some Anzahl theorems in symplectic geometry over finite fields</a>, Acta Sci. 13 (1964) 515-516.

%F T(n,m) = Product_{i=n-m+1..n} (2^(2i)-1)/Product_{i=1..m} (2^i-1).

%e Triangle begins:

%e   1;

%e   1,    3;

%e   1,   15,      15;

%e   1,   63,     315,      135;

%e   1,  255,    5355,    11475,      2295;

%e   1, 1023,   86955,   782595,    782595,     75735;

%e   1, 4095, 1396395, 50868675, 213648435, 103378275, 4922775;

%e   ...

%t T[n_,m_]:=Product[(2^(2i)-1),{i,n-m+1,n}]/Product[(2^i-1),{i,1,m}]; Table[T[n,m],{n,0,7},{m,0,n}] (* _Stefano Spezia_, Dec 28 2023 *)

%o (Python)

%o from math import prod

%o q = 2

%o N = lambda n, m : (prod([q**(2*i)-1 for i in range(n-m+1, n+1)])//prod([q**i-1 for i in range(1, m+1)]))

%o print([N(n, m) for n in range(8) for m in range(n+1)])

%o (PARI) T(n,m) = prod(i=n-m+1, n, 2^(2*i)-1)/prod(i=1, m, 2^i-1); \\ _Michel Marcus_, Dec 27 2023

%Y Main diagonal gives A028362.

%Y Cf. A022166.

%K nonn,tabl

%O 0,3

%A _Simon Burton_, Dec 27 2023