A368493 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.
1, 1, 3, 1, 15, 15, 1, 63, 315, 135, 1, 255, 5355, 11475, 2295, 1, 1023, 86955, 782595, 782595, 75735, 1, 4095, 1396395, 50868675, 213648435, 103378275, 4922775, 1, 16383, 22362795, 3268162755, 55558766835, 112909751955, 26883274275, 635037975
Offset: 0
Examples
Triangle begins: 1; 1, 3; 1, 15, 15; 1, 63, 315, 135; 1, 255, 5355, 11475, 2295; 1, 1023, 86955, 782595, 782595, 75735; 1, 4095, 1396395, 50868675, 213648435, 103378275, 4922775; ...
Links
- J. Baez, This Week's Finds in Mathematical Physics (Week 187)
- M. H. Poroch, Bounds on subspace codes based on totally isotropic subspace in symplectic spaces and extended symplectic spaces, Asian-European Journal of Mathematics, 12 (2019).
- Z. Wan, Notes on finite geometries and the construction of PBIB designs I, Some Anzahl theorems in symplectic geometry over finite fields, Acta Sci. 13 (1964) 515-516.
Programs
-
Mathematica
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 *) -
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
-
Python
from math import prod q = 2 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)])) print([N(n, m) for n in range(8) for m in range(n+1)])
Formula
T(n,m) = Product_{i=n-m+1..n} (2^(2i)-1)/Product_{i=1..m} (2^i-1).
Comments