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.

A219471 T(n,k)=Number of nXk arrays of the minimum value of corresponding elements and their horizontal or antidiagonal neighbors in a random, but sorted with lexicographically nondecreasing rows and columns, 0..3 nXk array.

Original entry on oeis.org

4, 4, 10, 10, 23, 20, 20, 68, 82, 35, 35, 211, 673, 239, 56, 56, 547, 4838, 5040, 619, 84, 84, 1248, 28159, 87083, 32229, 1471, 120, 120, 2663, 143718, 1227441, 1306891, 185800, 3259, 165, 165, 5432, 674954, 14696377, 43792361, 17399442, 982456, 6800
Offset: 1

Views

Author

R. H. Hardin Nov 20 2012

Keywords

Comments

Table starts
...4.....4.......10..........20...........35............56............84
..10....23.......68.........211..........547..........1248..........2663
..20....82......673........4838........28159........143718........674954
..35...239.....5040.......87083......1227441......14696377.....157191488
..56...619....32229.....1306891.....43792361....1222433026...29873644925
..84..1471...185800....17399442...1357658929...86755752059.4758315482104
.120..3259...982456...208345624..37552353044.5432184254977
.165..6800..4815782..2285009931.947150536824
.220.13464.22059734.23253970439
.286.25453.95049799
.364.46178
.455

Examples

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

Links

Crossrefs

Column 1 is A000292(n+1)
Row 1 is A000292 for n>1
Diagonal is A216678

A216800 T(n,k)=Number of permutations of an nXk array with each element moving exactly one horizontally or vertically and without 2-loops.

Original entry on oeis.org

0, 0, 0, 0, 2, 0, 0, 2, 2, 0, 0, 6, 0, 6, 0, 0, 10, 8, 8, 10, 0, 0, 22, 0, 88, 0, 22, 0, 0, 42, 32, 292, 292, 32, 42, 0, 0, 86, 0, 1774, 0, 1774, 0, 86, 0, 0, 170, 128, 7676, 10140, 10140, 7676, 128, 170, 0, 0, 342, 0, 39844, 0, 207408, 0, 39844, 0, 342, 0, 0, 682, 512, 186996
Offset: 1

Views

Author

R. H. Hardin Sep 17 2012

Keywords

Comments

Table starts
.0....0....0........0.........0............0..............0.................0
.0....2....2........6........10...........22.............42................86
.0....2....0........8.........0...........32..............0...............128
.0....6....8.......88.......292.........1774...........7676.............39844
.0...10....0......292.........0........10140..............0............361200
.0...22...32.....1774.....10140.......207408........1879040..........27918806
.0...42....0.....7676.........0......1879040..............0.........489300384
.0...86..128....39844....361200.....27918806......489300384.......22902801416
.0..170....0...186996.........0....302667484..............0......539812977524
.0..342..512...927134..12911864...3991662992...129127695440....20296105418276
.0..682....0..4460016.........0..46492284664..............0...550843711073468
.0.1366.2048.21812696.461788640.585384941184.34175930646380.18708387030790218

Examples

			Some solutions for n=4 k=4
..1..5..3..7....1..5..3..7....4..0..1..2....1..5..3..7....4..0..1..2
..0..9..2..6....0..4..2..6....8..6..7..3....0..4..2..6....5..6..7..3
..4.13.11.15....9.13.11.15...12..5..9.10...12..8.14.10...12..8..9.10
..8.12.10.14....8.12.10.14...13.14.15.11...13..9.15.11...13.14.15.11

Links

Crossrefs

Column 2 is A014113(n-1)
Diagonal is A216678

A216675 Number of ways one can draw arrows between adjacent nodes of an n X n grid such that each node has one outgoing and one incoming arrow.

Original entry on oeis.org

0, 4, 0, 1296, 0, 45265984, 0, 168709341081856, 0, 66865709036047973991424, 0, 2815414274858422422282241600000000, 0, 12589335654221209921194197564847684000000000000, 0, 5977481098898922857923760209743284068237948337696882106105856, 0
Offset: 1

Views

Author

M. F. Hasler, Sep 13 2012

Keywords

Comments

"Adjacent" is meant in the sense of von Neumann neighborhoods (4 neighbors for "interior" nodes, 3 resp. 2 for nodes on the borders resp. in the corners).
Alternate definition: Number of permutations of an n X n array with each element moving exactly one step horizontally or vertically. (Suggested by R. H. Hardin.)
From Adam P. Goucher, Aug 01 2013: (Start)
Also the permanent of the adjacency matrix of the n X n grid graph, which is the determinant of the modified adjacency matrix where vertical and horizontal edges have weights of 1 and i, respectively.
Also the square of the number of domino tilings of an n X n chessboard.
(End)

Examples

			For a 1 X 1 grid, there is no such possibility.
For a 2 X 2 grid, on can draw arrows between 2 pairs of nodes in horizontal or vertical sense, and the clockwise and counterclockwise cyclic "permutation" of the 4 nodes.
For a 3 X 3 grid, there is no possibility, neither for a 5 X 5 grid.

Links

Crossrefs

See A216678 for the same problem with an additional constraint ("no 2-loops").
Cf. A216796-A216800 for more general n X k grids.

Programs

  • Mathematica
    Table[If[Mod[n,2]==0,Det[MapIndexed[(#1 I^Mod[Total[#2],2])&, Normal[AdjacencyMatrix[GridGraph[{n,n}]]],{2}]],0],{n,1,20}] (* Adam P. Goucher, Aug 01 2013 *)
  • Python
    from sympy.abc import x
from sympy import resultant, chebyshevu, I
def A216675(n): return 0 if n&1 else resultant(chebyshevu(n,x/2),chebyshevu(n,I*x/2)) # Chai Wah Wu, Nov 07 2023

Formula

a(2n) = A004003(n)^2; a(2n + 1) = 0. - Adam P. Goucher, Aug 01 2013

Extensions

Terms beyond a(5) from R. H. Hardin, Sep 15 2012
Terms beyond a(8) from Adam P. Goucher, Aug 01 2013
Showing 1-3 of 3 results.

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