A207868 -id:A207868 - cp's OEIS frontend

A207981 T(n,k)=Number of n X k nonnegative integer arrays with new values 0 upwards introduced in row major order and no element equal to any diagonal or antidiagonal neighbor (colorings ignoring permutations of colors).

1, 2, 2, 5, 7, 5, 15, 67, 67, 15, 52, 1080, 4192, 1080, 52, 203, 25287, 587208, 587208, 25287, 203, 877, 794545, 147512732, 953124784, 147512732, 794545, 877, 4140, 31858034, 58668937932, 3410310590912, 3410310590912, 58668937932, 31858034

Offset: 1

R. H. Hardin, Feb 22 2012

			Table starts
....1........2...........5............15............52...............203
....2........7..........67..........1080.........25287............794545
....5.......67........4192........587208.....147512732.......58668937932
...15.....1080......587208.....953124784.3410310590912.22727131144960768
...52....25287...147512732.3410310590912
..203...794545.58668937932
..877.31858034
.4140
...
Some solutions for n=4 k=3
..0..0..0....0..0..1....0..0..0....0..0..0....0..1..0....0..0..0....0..0..0
..1..1..1....1..2..1....1..1..1....1..1..1....2..1..0....1..2..1....1..1..1
..2..0..2....1..2..0....0..0..2....0..0..0....0..1..3....0..0..0....2..2..0
..1..0..1....0..2..0....1..1..1....1..1..1....0..4..0....2..1..1....0..1..0

Andrew Howroyd, Table of n, a(n) for n = 1..496 (terms 1..40 from R. H. Hardin).

Columns 1..4 are A000110, A207978, A207979, A207980.
Main diagonal is A361452.
Cf. A207868.

A208001 T(n,k)=Number of n X k nonnegative integer arrays with new values 0 upwards introduced in row major order and no element equal to any knight-move neighbor (colorings ignoring permutations of colors).

Original entry on oeis.org

1, 2, 2, 5, 15, 5, 15, 114, 114, 15, 52, 1657, 4141, 1657, 52, 203, 36401, 426422, 426422, 36401, 203, 877, 1094076, 86545486, 450288795, 86545486, 1094076, 877, 4140, 42436913, 29169661126, 1182700979380, 1182700979380, 29169661126, 42436913, 4140

Offset: 1

R. H. Hardin, Feb 22 2012

			Table starts
....1........2...........5............15............52.........203
....2.......15.........114..........1657.........36401.....1094076
....5......114........4141........426422......86545486.29169661126
...15.....1657......426422.....450288795.1182700979380
...52....36401....86545486.1182700979380
..203..1094076.29169661126
..877.42436913
.4140
...
Some solutions for n=4 k=3
..0..0..0....0..0..0....0..0..0....0..1..0....0..0..1....0..0..0....0..0..0
..1..0..1....1..2..1....1..0..1....1..0..1....0..0..1....1..0..1....1..0..1
..2..1..2....2..1..2....2..1..2....0..1..2....2..2..1....2..1..2....2..2..2
..1..0..3....3..0..0....1..0..1....1..2..1....2..3..1....1..2..1....1..0..1

Andrew Howroyd, Table of n, a(n) for n = 1..66 (terms 1..39 from R. H. Hardin).
Eric Weisstein's World of Mathematics, Knight Graph.
Eric Weisstein's World of Mathematics, Vertex Coloring.

Columns 1..4 are A000110, A207998, A207999, A208000.
Main diagonal is A361453.
Cf. A208434 (3 colorings), A208353 (4 colorings).
Cf. A207868 (grid graph).

A207864 Number of n X 2 nonnegative integer arrays with new values 0 upwards introduced in row major order and no element equal to any horizontal or vertical neighbor (colorings ignoring permutations of colors).

Original entry on oeis.org

1, 4, 34, 500, 10900, 322768, 12297768, 580849872, 33093252880, 2227152575552, 174131286983712, 15604440074084672, 1584856558077903168, 180712593036822482176, 22946861101272125055616, 3222156375409363475703040

Offset: 1

R. H. Hardin, Feb 21 2012

nonn

From Gus Wiseman, Mar 01 2019: (Start)
Also the number of stable partitions of the n-ladder graph. A stable partition of a graph is a set partition of the vertices where no edge has both ends in the same block. The n-ladder has 2n vertices and looks like:
  o-o-o-   -o
  | | | ... |
  o-o-o-   -o
(End)

			Some solutions for n=5:
  0 1   0 1   0 1   0 1   0 1   0 1   0 1   0 1   0 1   0 1
  1 0   1 0   1 2   1 2   1 0   1 0   1 2   1 0   1 0   1 0
  0 1   0 1   0 1   0 1   2 1   0 1   0 1   0 2   2 1   0 1
  1 2   1 0   1 0   1 3   3 0   2 0   3 2   2 1   1 0   1 2
  0 1   0 1   2 1   2 4   1 2   0 1   0 1   0 2   0 1   2 0

R. H. Hardin, Table of n, a(n) for n = 1..61

Column 2 of A207868.

Cf. A000110, A000569, A109808, A229048, A240936, A321750, A321979, A321980, A321982, A322064.

Table[Expand[x*(x-1)*(x^2-3*x+3)^(n-1)]/.x^k_.->BellB[k],{n,20}] (* Gus Wiseman, Mar 01 2019 *)

It appears that the sequence terms are given by the Dobinski-type formula a(n+1) = (1/e) * Sum_{k>=0} (1+k+k^2)^n/k!. - Peter Bala, Mar 12 2012

Apply x^n -> B(n) to the polynomial chi(n) = x (x - 1) (x^2 - 3 x + 3)^(n - 1), where B = A000110. - Gus Wiseman, Mar 01 2019

A207863 Number of n X n nonnegative integer arrays with new values 0 upwards introduced in row major order and no element equal to any horizontal or vertical neighbor (colorings ignoring permutations of colors).

Original entry on oeis.org

1, 4, 2052, 455546040, 103204230192540988, 47890666906311937749901098886, 81523212441914346544073433818166292337910075, 839780322234263638551379612300530069754865868713324092842519030, 81184462760528172324981018065090176893681709356914242351556151349811667519060498973390

Offset: 1

R. H. Hardin, Feb 21 2012

nonn

			Some solutions for n=5
..0..1..0..1..0....0..1..0..1..0....0..1..2..0..1....0..1..0..1..0
..1..0..1..0..1....1..0..1..0..1....1..0..1..3..0....1..0..1..0..1
..0..1..0..1..0....2..3..0..1..2....0..1..0..1..2....0..1..0..1..0
..2..0..1..0..1....0..1..2..0..1....1..0..1..0..1....1..0..1..0..1
..0..1..0..1..0....1..0..1..4..0....0..1..0..1..0....0..1..0..1..0

Andrew Howroyd, Table of n, a(n) for n = 1..11

Diagonal of A207868.

Terms a(6) and beyond from Andrew Howroyd, Mar 11 2023

A207865 Number of n X 3 nonnegative integer arrays with new values 0 upwards introduced in row major order and no element equal to any horizontal or vertical neighbor (colorings ignoring permutations of colors).

Original entry on oeis.org

2, 34, 2052, 278982, 68162042, 26419793726, 15002771641712, 11807624578397830, 12361734324171045410, 16679505289611340233906, 28284296659644702241447604, 59055088260515144420128767278, 149249218670114328150978145481610

Offset: 1

R. H. Hardin, Feb 21 2012

nonn

			Some solutions for n=5
..0..1..2....0..1..0....0..1..0....0..1..0....0..1..0....0..1..0....0..1..0
..2..0..1....1..0..1....1..0..1....1..0..1....1..0..2....1..0..1....1..2..1
..0..1..0....0..1..0....0..1..0....0..1..0....0..1..3....0..1..0....2..0..2
..1..0..1....1..0..1....1..0..1....1..2..1....1..0..1....2..0..1....0..1..0
..0..1..0....0..2..0....2..1..0....0..1..0....0..1..2....0..1..0....1..0..1

Andrew Howroyd, Table of n, a(n) for n = 1..190 (terms 1..20 from R. H. Hardin)

Column 3 of A207868.

A207866 Number of n X 4 nonnegative integer arrays with new values 0 upwards introduced in row major order and no element equal to any horizontal or vertical neighbor (colorings ignoring permutations of colors).

Original entry on oeis.org

5, 500, 278982, 455546040, 1625686993918, 10764437129618296, 119010390034543491042, 2040368351234509987725212, 51350903376256822436382238230, 1818472780230512625778385363238944, 87604972777997602798029710458933219862, 5585379463993456459896660751935863285859008

Offset: 1

R. H. Hardin, Feb 21 2012

nonn

			Some solutions for n=5
..0..1..0..1....0..1..0..1....0..1..0..1....0..1..0..1....0..1..0..1
..1..0..1..0....1..0..1..0....1..2..3..2....1..0..1..0....1..2..1..0
..0..1..0..1....0..1..2..1....0..1..0..1....0..1..2..1....0..1..0..1
..1..2..1..0....1..0..1..0....1..0..1..0....1..0..1..0....1..2..1..0
..0..1..0..1....0..1..3..1....0..1..0..1....0..1..0..1....0..1..2..1

Andrew Howroyd, Table of n, a(n) for n = 1..140

Column 4 of A207868.

Terms a(11) and beyond from Andrew Howroyd, Mar 11 2023

A207867 Number of n X 5 nonnegative integer arrays with new values 0 upwards introduced in row major order and no element equal to any horizontal or vertical neighbor (colorings ignoring permutations of colors).

Original entry on oeis.org

15, 10900, 68162042, 1625686993918, 103204230192540988, 14231913620170652461908, 3737733535781386231895815398, 1704434626479905430949721771156450, 1259921002642906004433313180012192804586, 1431587436371216664854796169984729275373681588

Offset: 1

R. H. Hardin, Feb 21 2012

nonn

			Some solutions for n=5
..0..1..0..1..0....0..1..0..1..0....0..1..0..1..0....0..1..0..1..0
..1..0..1..0..1....1..0..1..0..1....2..0..1..0..1....1..0..1..0..1
..0..1..0..1..0....0..1..0..1..0....0..1..0..1..0....0..1..0..1..0
..1..0..1..0..1....2..0..1..0..1....1..0..1..2..1....1..2..1..0..1
..0..1..0..1..0....0..1..0..1..0....0..1..0..1..0....0..1..0..1..0

Andrew Howroyd, Table of n, a(n) for n = 1..100

Column 5 of A207868.

Terms a(6) and beyond from Andrew Howroyd, Mar 11 2023

A360922 Array read by antidiagonals: T(m,n) is the number of acyclic orientations in the grid graph P_m X P_n.

Original entry on oeis.org

1, 2, 2, 4, 14, 4, 8, 98, 98, 8, 16, 686, 2398, 686, 16, 32, 4802, 58670, 58670, 4802, 32, 64, 33614, 1435414, 5015972, 1435414, 33614, 64, 128, 235298, 35118638, 428816558, 428816558, 35118638, 235298, 128, 256, 1647086, 859207558, 36659327366, 128091434266, 36659327366, 859207558, 1647086, 256

Offset: 1

Andrew Howroyd, Mar 07 2023

			Array begins:
=====================================================
m\n|  1     2        3           4              5 ...
---+-------------------------------------------------
1  |  1     2        4           8             16 ...
2  |  2    14       98         686           4802 ...
3  |  4    98     2398       58670        1435414 ...
4  |  8   686    58670     5015972      428816558 ...
5  | 16  4802  1435414   428816558   128091434266 ...
6  | 32 33614 35118638 36659327366 38261306901842 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..253 (first 22 antidiagonals).
Eric Weisstein's World of Mathematics, Grid Graph.
Wikipedia, Acyclic orientation.

Main diagonal is A080690.
Rows 1..2 are A000079(n-1), A109808.
Cf. A116469 (spanning trees), A178435, A207868 (unlabeled colorings).

T(m,n) = T(n,m).

cp's OEIS Frontend

A207981 T(n,k)=Number of n X k nonnegative integer arrays with new values 0 upwards introduced in row major order and no element equal to any diagonal or antidiagonal neighbor (colorings ignoring permutations of colors).

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A208001 T(n,k)=Number of n X k nonnegative integer arrays with new values 0 upwards introduced in row major order and no element equal to any knight-move neighbor (colorings ignoring permutations of colors).

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A207864 Number of n X 2 nonnegative integer arrays with new values 0 upwards introduced in row major order and no element equal to any horizontal or vertical neighbor (colorings ignoring permutations of colors).

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A207863 Number of n X n nonnegative integer arrays with new values 0 upwards introduced in row major order and no element equal to any horizontal or vertical neighbor (colorings ignoring permutations of colors).

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A207865 Number of n X 3 nonnegative integer arrays with new values 0 upwards introduced in row major order and no element equal to any horizontal or vertical neighbor (colorings ignoring permutations of colors).

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A207866 Number of n X 4 nonnegative integer arrays with new values 0 upwards introduced in row major order and no element equal to any horizontal or vertical neighbor (colorings ignoring permutations of colors).

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A207867 Number of n X 5 nonnegative integer arrays with new values 0 upwards introduced in row major order and no element equal to any horizontal or vertical neighbor (colorings ignoring permutations of colors).

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A360922 Array read by antidiagonals: T(m,n) is the number of acyclic orientations in the grid graph P_m X P_n.

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