cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A334247 Number of acyclic orientations of the edges of an n-dimensional cube.

Original entry on oeis.org

1, 2, 14, 1862, 193270310, 47171704165698393638
Offset: 0

Views

Author

Matthew Scroggs, Apr 20 2020

Keywords

Comments

a(n) is the absolute value of the chromatic polynomial of the n-hypercube graph evaluated at -1.

Examples

			For n=2, there are 14 ways to orient the edges of a square without cycles (see links).

Links

Crossrefs

Cf. A334248 is the number of acyclic orientations with rotations and reflections of the same orientation excluded.
Cf. A140986, A158348, A296914, A334159, A334278.
Cf. A033815 (cross-polytope), A058809 (wheel), A338152 (demihypercube), A338153 (prism), A338154 (antiprism).

Programs

  • Maple
    with(GraphTheory): with(SpecialGraphs):
a:= n-> abs(ChromaticPolynomial(HypercubeGraph(n), -1)):
seq(a(n), n=0..4);  # Alois P. Heinz, Jan 14 2025

Formula

a(n) = Sum_{k=1..2^n} (-1)^(2^n-k) * k! * A334159(n, k). - Andrew Howroyd, Apr 21 2020
a(n) = |Sum_{k=0..2^n} (-1)^k * A334278(n, k)|. - Peter Kagey, Oct 13 2020

Extensions

a(5) from Andrew Howroyd, Apr 23 2020

A338154 a(n) is the number of acyclic orientations of the edges of the n-antiprism.

Original entry on oeis.org

426, 4968, 50640, 486930, 4547088, 41796168, 380789562, 3451622904, 31194607488, 281440825122, 2536622917920, 22848990484344, 205743704494026, 1852238413383048, 16673036119790640, 150072652217086770, 1350735146332489008, 12157047307392618408
Offset: 3

Views

Author

Peter Kagey, Oct 13 2020

Keywords

Comments

Conjectured linear recurrence and g.f. confirmed by Kagey's formula. - Ray Chandler, Mar 10 2024

Examples

			For n = 3, the 3-antiprism is the octahedron (3-dimensional cross-polytope), so a(3) = A033815(3) = 426.

Links

Crossrefs

Cf. A077263, A124352, A124353, A192742, A284699, A284700, A284701, A287988, A294152, A297384.
Cf. A033815 (cross-polytope), A058809 (wheel), A334247 (hypercube), A338152 (demihypercube), A338153 (prism).

Programs

  • Mathematica
    A338154[n_] := Round[-2^(1-n)*((7 - Sqrt[13])^n + (7 + Sqrt[13])^n) + 9^n + 5] (* Peter Kagey, Nov 15 2020 *)

Formula

Conjectures from Colin Barker, Oct 13 2020: (Start)
G.f.: 6*x^3*(71 - 379*x + 612*x^2 - 324*x^3) / ((1 - x)*(1 - 9*x)*(1 - 7*x + 9*x^2)).
a(n) = 17*a(n-1) - 88*a(n-2) + 153*a(n-3) - 81*a(n-4) for n>6.
(End)
a(n) = -2^(1-n)*((7-sqrt(13))^n + (7+sqrt(13))^n) + 9^n + 5. - Peter Kagey, Nov 15 2020

A338152 a(n) is the number of acyclic orientations of the edges of an n-dimensional demihypercube.

Original entry on oeis.org

1, 2, 24, 24024, 193270310, 767795414400
Offset: 1

Views

Author

Peter Kagey, Oct 13 2020

Keywords

Links

Crossrefs

Cf. A137888, A290060, A295174, A334280.
Cf. A033815 (cross-polytope), A058809 (wheel), A334247 (hypercube), A338153 (prism), A338154 (antiprism).

Programs

  • Mathematica
    Table[Abs[ChromaticPolynomial[GraphData[{"HalvedCube",n}]][-1]],{n,1,6}]

Formula

a(n) = |Sum_{k=0..2^(n-1)} (-1)^k * A334280(n, k)|.
Showing 1-3 of 3 results.

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