A153255 Arises in enumerating orientable small covers over cubes.

1, 1, 1, 4, 43, 1156, 74581, 11226874, 3862830343, 2990426173816, 5144550664291081, 19470823356314891254, 160782837107861606438923, 2876650791540557329654540276, 110853465572134076561454447710221

Offset

1,4

Comments

A small cover, as defined by Davis and Januszkiewicz, is an n-dimensional closed smooth manifold M with a smooth action of standard real torus (Z_2)^2 action such that the action is locally isomorphic to a standard action of (Z_2)^2 on R^n and the orbit space M/(Z_2)^2 is a simple convex polytope. For instance, RP^n with a natural action of (Z_2)^2 is a small cover over an n-simplex. In general, real toric manifolds, the set of real points of a toric manifold, provide examples of small covers.

Hence we may think of small covers as a topological generalization of real toric manifolds in algebraic geometry. Small covers over hypercubes are known to be real Bott manifolds, which is obtained as iterated RP^1 bundles starting with a point, where each fibration is the projectivization of a Whitney sum of two real line bundles [Masuda and Panov]. Choi found the 1-1 correspondence between the set of real Bott manifolds and the set of acyclic digraphs in a previous work.

Links

Table of n, a(n) for n=1..15.

Suyoung Choi, The Number of orientable small covers over cubes, arXiv:0812.3861 [math.GT], 2008-2010.

Suyoung Choi, The number of small covers over cubes, Algebr. Geom. Topol. 8 (2008), no. 4, 2391-2399.

Michael W. Davis and Tadeusz Januszkiewicz, Convex polytopes, Coxeter orbifolds and torus actions, Duke Math. J., 62(2):417-451, 1991.

Mikiya Masuda and Taras E. Panov, Semifree circle actions, Bott towers and quasitoric manifolds, Sbornik Math., 199(8):1201-1223, 2008 (in Russian).

Formula

Sum_{n>0} a(n)*x^n/(2^(n-1)*x-1)^(n+1) = x/(1-x). [Vladeta Jovovic, Oct 24 2009]

Mathematica

r[0] = r[1] = 1; r[n_] := r[n] = Sum[(-1)^(k+1) Binomial[n, k] 2^(k(n-k)) r[n-k], {k, 1, n}];
a[1] = 1; a[n_] := Sum[(-1)^(k+1) Binomial[n-1, k] 2^((k-1)(n-1-k)) r[n-1-k], {k, 1, n-1}];
Array[a, 15] (* Jean-François Alcover, Feb 17 2019 *)

Crossrefs

Sequence in context: A277456 A317140 A152282 * A015084 A355081 A176827

Adjacent sequences: A153252 A153253 A153254 * A153256 A153257 A153258

Keyword

nonn

Author

Jonathan Vos Post, Dec 21 2008

Extensions

More terms from Vladeta Jovovic, Oct 24 2009

Status

approved