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 1-5 of 5 results.

A334279 Irregular table read by rows: T(n, k) is the coefficient of x^k in the chromatic polynomial of the 1-skeleton of the n-dimensional cross polytope, 0 <= k <= 2n.

Original entry on oeis.org

0, 0, 1, 0, -3, 6, -4, 1, 0, -64, 154, -137, 58, -12, 1, 0, -2790, 7467, -7852, 4300, -1346, 244, -24, 1, 0, -205056, 593016, -698250, 448015, -175004, 43608, -6990, 700, -40, 1, 0, -22852200, 70164670, -89812001, 64407806, -29113410, 8790285, -1822164, 260868, -25405, 1610, -60, 1
Offset: 1

Views

Author

Peter Kagey, Apr 21 2020

Keywords

Comments

A033815 is the number of acyclic orientations of the n-dimensional cross polytope, which is the absolute value of the chromatic polynomial evaluated at -1.
Sums of absolute values of entries in each row give A033815.
These graphs are chromatically unique, that is, there is no nonisomorphic graph with the same chromatic polynomial.
Conjectures from Peter Kagey, Apr 26 2020: (Start)
T(n,1) = -A161131(2n-1).
T(n,2n-2) = A212689(2n - 1).
T(n,2n-1) = A046092(n-1). (End)

Examples

			Table begins:
n/k| 0       1      2       3      4       5     6     7   8   9 10
---+---------------------------------------------------------------
  1| 0       0      1
  2| 0      -3      6      -4      1
  3| 0     -64    154    -137     58     -12     1
  4| 0   -2790   7467   -7852   4300   -1346   244   -24   1
  5| 0 -205056 593016 -698250 448015 -175004 43608 -6990 700 -40  1

Links

Crossrefs

A334278 is analogous for the n-dimensional hypercube.
Cf. A033815, A115400.
Cf. A161131, A212689, A046092.

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

A380589 Number of n-colorings of the Hypercube Graph Q5.

Original entry on oeis.org

0, 0, 2, 1185282, 130253748108, 2157531034816940, 7905235551766437150, 7365707045872206479742, 2337101560809838105414712, 327425229254999498091796728, 24489214732779742874109277530, 1119349138930999380736025706650, 34471067091433681765512048700932
Offset: 0

Views

Author

Alois P. Heinz, Jan 27 2025

Keywords

Comments

The Hypercube Graph Q5 has 32 vertices and 80 edges.
All terms are even.

Links

Crossrefs

Column k=5 of A342128.
Cf. A001477, A002378, A091940, A140986, A158348, A334278.

Programs

  • Maple
    a:= n-> (((((((((((((((((((((((((((((((n-80)*n+3160)*n-82080)*n+1575420)*n
    -23805776)*n+294640000)*n-3068289720)*n+27406254870)*n-212981036784)*n
    +1455643449120)*n-8822129447280)*n+47712047044920)*n-231347639674200)*n
    +1009138022379076)*n-3968583456247214)*n+14086095737441185)*n-45124968898112160)*n
    +130327084318442384)*n-338572422663483544)*n+788328935798745052)*n
    -1636781898149840504)*n+3009654466362869780)*n-4856773984500880124)*n
    +6797172300402030636)*n-8122089299204814072)*n+8114599308192145448)*n
    -6584797184952049568)*n+4160914137061367054)*n-1915734714629493936)*n
    +569711421560808713)*n-81768640551939777)*n:
seq(a(n), n=0..12);

Formula

a(n) = n^32 - 80*n^31 + 3160*n^30 - ... (see Maple program).

A342128 Table read by antidiagonals upwards: T(n,k) is the number of n-colorings of the vertices of the k-dimensional hypercube such that no two adjacent vertices have the same color. n >= 0, k >=0.

Original entry on oeis.org

0, 1, 0, 2, 0, 0, 3, 2, 0, 0, 4, 6, 2, 0, 0, 5, 12, 18, 2, 0, 0, 6, 20, 84, 114, 2, 0, 0, 7, 30, 260, 2652, 2970, 2, 0, 0, 8, 42, 630, 29660, 1321860, 1185282, 2, 0, 0, 9, 56, 1302, 198030, 187430900, 130253748108, 100301050602, 2, 0, 0, 10, 72, 2408, 932862, 10199069190, 2157531034816940
Offset: 0

Views

Author

Peter Kagey, Feb 28 2021

Keywords

Examples

			Table begins:
  n\k|  0   1     2         3                4                              5
  ---+-----------------------------------------------------------------------
   0 |  0   0     0         0                0                              0
   1 |  1   0     0         0                0                              0
   2 |  2   2     2         2                2                              2
   3 |  3   6    18       114             2970                        1185282
   4 |  4  12    84      2652          1321860                   130253748108
   5 |  5  20   260     29660        187430900               2157531034816940
   6 |  6  30   630    198030      10199069190            7905235551766437150
   7 |  7  42  1302    932862     269591166222         7365707045872206479742
   8 |  8  56  2408   3440024    4221404762120      2337101560809838105414712
   9 |  9  72  4104  10599192   44876701584360    327425229254999498091796728
  10 | 10  90  6570  28478970  355148098691850  24489214732779742874109277530

Links

Crossrefs

Columns and rows: A002378 (k=1), A091940 (k=2), A140986 (k=3), A158348 (k=4), A380589 (k=5), A307334 (n=3).
Cf. A334278, A342088 (analogous for cross-polytope).

Formula

T(n,k) = Sum_{i=0..2^k} A334278(k,i)*n^i.

A334280 Irregular table read by rows: T(n, k) is the coefficient of x^k in the chromatic polynomial of the 1-skeleton of the n-dimensional demihypercube, 0 <= k <= 2^(n-1).

Original entry on oeis.org

0, 1, 0, -1, 1, 0, -6, 11, -6, 1, 0, -2790, 7467, -7852, 4300, -1346, 244, -24, 1, 0, -36586695600, 132041735820, -208601259660, 194243767689, -120509323400, 53195294240, -17371817260, 4296667608, -815202340, 119111090, -13339000, 1127752, -69860, 3000, -80, 1
Offset: 1

Views

Author

Peter Kagey, Apr 22 2020

Keywords

Examples

			n\k| 0     1    2     3    4     5   6   7 8
---+----------------------------------------
  1| 0     1
  2| 0    -1    1
  3| 0    -6   11    -6    1
  4| 0 -2790 7467 -7852 4300 -1346 244 -24 1

Links

Crossrefs

Cf. A048994 (simplex), A334278 (hypercube), A334279 (cross-polytope).

Programs

  • Mathematica
    T[n_, k_] := Coefficient[ChromaticPolynomial[GraphData[{"HalvedCube", n}], x], x, k]
Showing 1-5 of 5 results.

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