A355645 The number of regions in the G-Shi arrangement when G is the cycle graph C_n.

1, 3, 16, 61, 206, 659, 2052, 6297, 19162, 58015, 175088, 527333, 1586118, 4766571, 14316124, 42981169, 129009074, 387158327, 1161737160, 3485735805, 10458256030, 31376865283, 94134790196, 282412759241, 847255054986, 2541798719439, 7625463267232

Offset: 1

Author

Robin Truax, Jul 11 2022

Keywords

• nonn
• easy

Comments

The G-Shi arrangement of a graph G is the hyperplane arrangement given by hyperplanes x_i - x_j = 0 and x_i - x_j = 1 for each edge {i,j} of G with i < j.

a(n) is also the number of parking functions of length n with all cars preferring to park in the first 3 spots. - Jayden Thadani, May 20 2024

Links

• Paolo Xausa, Table of n, a(n) for n = 1..1000
• Douglas M. Chen, On the Structure of Permutation Invariant Parking, arXiv:2311.15699 [math.CO], 2023. See p. 16.
• Index entries for linear recurrences with constant coefficients, signature (7,-17,17,-6).

Crossrefs

Cf. A004146 (cycle graph labels).

Cf. A000244 (path graph regions), A001906 (path graph labels).

Programs

Mathematica

A355645[n_] := If[n == 1, 1, 3^n - 2^n - n]; Array[A355645, 50] (* or *)
LinearRecurrence[{7, -17, 17, -6}, {1, 3, 16, 61, 206}, 50] (* Paolo Xausa, May 24 2024 *)

Formula

a(n) = 3^n - 2^n - n for n >= 2.

G.f.: x*(1 - 4*x + 12*x^2 - 17*x^3 + 6*x^4)/((1 - x)^2*(1 - 2*x)*(1 - 3*x)). - Stefano Spezia, Jul 12 2022