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 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. 0
 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 (list; table; graph; refs; listen; history; text; internal format)
 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

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Last modified May 18 12:18 EDT 2024. Contains 372630 sequences. (Running on oeis4.)