OFFSET
0,3
COMMENTS
The number of m-dimensional isotropic subspaces of an n-dimensional symplectic space over Z/2.
LINKS
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.
FORMULA
T(n,m) = Product_{i=n-m+1..n} (2^(2i)-1)/Product_{i=1..m} (2^i-1).
EXAMPLE
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;
...
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 *)
PROG
(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)])
(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
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Simon Burton, Dec 27 2023
STATUS
approved