A341769 Number of connected components of Euclidean n-space when the hyperplanes x_i+x_j=1, x_i=0, x_i=1 (1 <= i < j <= n) are deleted.

3, 12, 64, 436, 3624, 35516, 400544, 5106180, 72574936, 1137563980, 19489399824, 362279121044, 7261032943688, 156078126597084, 3581487541784704, 87378336982197028, 2258453972652164280, 61646205047945592428, 1771962416919392083184, 53498826047517147678132

Offset

1,1

Comments

a(n) is also the number of labeled colored threshold graphs on n vertices. Threshold graphs are constructed recursively such that each vertex added is either adjacent to all previous vertices (called dominating vertex) or an isolated vertex. Threshold graphs where either no vertex is colored, or from some vertex onwards in the construction, all dominating vertices are colored red and all isolated vertices are colored blue are called colored threshold graphs. Labeled colored threshold graphs are those with n vertices labeled distinctly using {1,...,n}.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..423

P. Deshpande, K. Menon, and A. Singh, Counting regions of the boxed threshold arrangement, arXiv:2101.12060 [math.CO], 2021.

Formula

a(n) = 4*A000670(n) + Sum_{k=1..n} 4*(k!-(k-1)!)*(k*A008277(n,k) - n*A008277(n-1,k-1)) for n >= 2.

E.g.f.: (1 - x) * e^(2*x) / (2 - e^x)^2.

Mathematica

MapAt[# - 1 &, Array[2 (PolyLog[-#, 1/2] + KroneckerDelta[#]) + Sum[4 (k! - (k - 1)!) (k StirlingS2[#, k] - # StirlingS2[# - 1, k - 1]), {k, #}] &, 20], 1] (* Michael De Vlieger, May 07 2021 *)

Crossrefs

Cf. A000670, A008277.

Sequence in context: A207557 A235129 A222033 * A302195 A359660 A196559

Adjacent sequences: A341766 A341767 A341768 * A341770 A341771 A341772

Keyword

nonn

Author

Krishna Menon, Feb 19 2021

Status

approved