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

0,3

Comments

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

Links

Table of n, a(n) for n=0..35.

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.

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

Main diagonal gives A028362.

Cf. A022166.

Sequence in context: A178657 A257490 A156289 * A095922 A263632 A284861

Adjacent sequences: A368490 A368491 A368492 * A368494 A368495 A368496

Keyword

nonn,tabl

Author

Simon Burton, Dec 27 2023

Status

approved