A007846 -id:A007846 - cp's OEIS frontend

A087783 Array T(n,k) (n >= 1, k >= 1) read by antidiagonals, giving number of ways of arranging the numbers 1 ... mn into an m X n array so there is exactly one local maximum.

1, 2, 2, 4, 16, 4, 8, 208, 208, 8, 16, 3584, 29568, 3584, 16, 32, 76544, 7452704, 7452704, 76544, 32, 64, 1947648, 2941306368, 35704394880, 2941306368, 1947648, 64, 128, 57477120, 1683453629440, 331333877743200, 331333877743200, 1683453629440, 57477120, 128, 256, 1929117696, 1323082429842432, 5338455334819710720, 88366736882654697600, 5338455334819710720, 1323082429842432, 1929117696, 256, 512, 72545402880, 1370418864769445888, 137813651152462288749440

Offset: 1

R. H. Hardin, Oct 25 2003

			Array begins:
1,2,4,8,16,32,64,128,256,512,1024,2048,4096,8192,...
2,16,208,3584,76544,1947648,57477120,1929117696...
4,208,29568,7452704,2941306368,1683453629440...
8,3584,7452704,35704394880,331333877743200,...
16,76544,2941306368,331333877743200,...

Cf. A000079 (k = 1 or n = 1), A087518 (n=k). A087923-A087932 give rows 2 through 11. Cf. A007846.

T(n, k) = T(k, n). T(n, 1) = 2^(n-1) (see A000079).

A087518 Number of arrangements of 1..n^2 in n X n array with exactly one local maximum.

Original entry on oeis.org

1, 16, 29568, 35704394880, 88366736882654697600

Offset: 1

R. H. Hardin, Oct 23 2003

Neighbors of a cell are only considered in the X and Y directions.
Number of ways to dig an n X n hole where each cell is at a different depth and water will collect in just one puddle.
Computed using a C program, brute force enumeration except using a cache of old results to cut off the search tree. For many related problems however, R. H. Hardin has coded them in S/R (which is COSPAN's language).

			For n=2, 12 has only one local max., whereas 13 has two and is excluded.
........ 34 ................................ 42

For the same problem with an n X 1 array see A000079. With no shape constraint, see A007846. Cf. A087519. Main diagonal of A087783.

A087923 Number of ways of arranging the numbers 1 ... 2n into a 2 X n array so there is exactly one local maximum.

Original entry on oeis.org

2, 16, 208, 3584, 76544, 1947648, 57477120, 1929117696, 72545402880, 3020819005440, 137959904378880, 6855868809216000, 368270708268072960, 21262037565623500800, 1312956239068318924800, 86347473137975269785600, 6025205587810776514560000, 444600907757468888806195200

Offset: 1

R. H. Hardin, Oct 27 2003

nonn

Also the number of random walk labelings of the grid graph P_2 X P_n. - Sela Fried, Apr 14 2023

Andrew Howroyd, Table of n, a(n) for n = 1..200
Sela Fried and Toufik Mansour, Graph labelings obtainable by random walks, arXiv:2304.05728 [math.CO], 2023.

Row 2 of A087783.

Cf. A007846.

a := n -> 2*((2*n - 2)! / doublefactorial(2*n - 1)) * add((2*k*(n - k + 1) - 1) * binomial(2*n, 2*k) / binomial(n, k), k = 1..n):
seq(a(n), n = 1..18); # Peter Luschny, Apr 17 2023

a(n)={2*sum(k=1, n, (2*n-2)!*(2*k*(n-k+1)-1)*2^n*k!*(n-k)!/((2*k)!*(2*n-2*k)!))} \\ Andrew Howroyd, Feb 26 2020

a(n) = 2*Sum_{k=1..n} (2*n-2)!*(2*k*(n-k+1)-1)/((2*k-1)!!*(2*n-2*k-1)!!). - Maximilian Göbel, Feb 26 2020
From Sela Fried, Apr 13 2023: (Start)
a(n) = 2^(n - 1)*(n - 1)!*Sum_{k=0..n-1} (n*binomial(2*(n - 1), 2*k) + binomial(2*n - 1, 2*k))/binomial(n - 1,k).
E.g.f.: ((1 - 2*x)^2*arctan(2*x/sqrt(1 - 4*x)) + 2*x*sqrt(1 - 4*x))/(2*(sqrt(1 - 4*x))^3).
(End)
a(n) ~ Pi * 2^(2*n - 5/2) * n^(n+1) / exp(n). - Vaclav Kotesovec, Apr 13 2023

Terms a(16) and beyond from Andrew Howroyd, Feb 26 2020

A087932 Number of ways of arranging the numbers 1 ... 11n into an 11 X n array so there is exactly one local maximum.

Original entry on oeis.org

1024, 137959904378880, 5972307516078119199647692800

Offset: 1

R. H. Hardin, Oct 27 2003

A row of the array in A087783. Cf. A007846.

A261834 Number of n-step adjacent expansions on the hexagonal (honeycomb) lattice. Holes allowed.

Original entry on oeis.org

1, 6, 48, 468, 5328, 68928, 994464, 15781920, 272594160, 5081825664

Offset: 0

Francois Alcover, Mar 24 2016

Initially only one cell C[0] is occupied on the lattice.
Then, for each i of (1..n), C[i] is chosen among the free cells adjacent to at least one of (C[0],...,C[i-1]).
a(n) is the number of distinct (C[1],...,C[n]).

			a(1) = 6 because a point has 6 neighbors on the hexagonal grid.
a(2) = 48 = a(1) * 8 because a two-cell group has 8 free neighbors.

Francois Alcover, tree.

Cf. A007846 (same principle but on the rectangular lattice).

Cf. A001334.

More terms from Francois Alcover, Apr 29 2016
Rephrasing and culling comments from Francois Alcover, Apr 29 2016
Added crossref to A007846 from Francois Alcover, May 01 2016

A087924 Number of ways of arranging the numbers 1 ... 3n into a 3 X n array so there is exactly one local maximum.

Original entry on oeis.org

4, 208, 29568, 7452704, 2941306368, 1683453629440, 1323082429842432, 1370418864769445888, 1811764909105311267840, 2980039854936078219167744, 5972307516078119199647692800

Offset: 1

R. H. Hardin, Oct 27 2003

nonn

A row of the array in A087783. Cf. A007846.

A087925 Number of ways of arranging the numbers 1 ... 4n into a 4 X n array so there is exactly one local maximum.

Original entry on oeis.org

8, 3584, 7452704, 35704394880, 331333877743200, 5338455334819710720, 137813651152462288749440, 5359457796069778002771306240, 299167849199075018703047460201600

Offset: 1

R. H. Hardin, Oct 27 2003

nonn

A row of the array in A087783. Cf. A007846.

A087926 Number of ways of arranging the numbers 1 ... 5n into a 5 X n array so there is exactly one local maximum.

Original entry on oeis.org

16, 76544, 2941306368, 331333877743200, 88366736882654697600, 48653237600359922073441120, 49992325433734834899948931370880

Offset: 1

R. H. Hardin, Oct 27 2003

nonn

A row of the array in A087783. Cf. A007846.

A087927 Number of ways of arranging the numbers 1 ... 6n into a 6 X n array so there is exactly one local maximum.

Original entry on oeis.org

32, 1947648, 1683453629440, 5338455334819710720, 48653237600359922073441120, 1080582891996915763252912268789760

Offset: 1

R. H. Hardin, Oct 27 2003

nonn

A row of the array in A087783. Cf. A007846.

A087928 Number of ways of arranging the numbers 1 ... 7n into a 7 X n array so there is exactly one local maximum.

Original entry on oeis.org

64, 57477120, 1323082429842432, 137813651152462288749440, 49992325433734834899948931370880

Offset: 1

R. H. Hardin, Oct 27 2003

nonn

A row of the array in A087783. Cf. A007846.

cp's OEIS Frontend

A087783 Array T(n,k) (n >= 1, k >= 1) read by antidiagonals, giving number of ways of arranging the numbers 1 ... mn into an m X n array so there is exactly one local maximum.

Views

Author

Keywords

Examples

Crossrefs

Formula

A087518 Number of arrangements of 1..n^2 in n X n array with exactly one local maximum.

Views

Author

Keywords

Comments

Examples

Crossrefs

A087923 Number of ways of arranging the numbers 1 ... 2n into a 2 X n array so there is exactly one local maximum.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

PARI

Formula

Extensions

A087932 Number of ways of arranging the numbers 1 ... 11n into an 11 X n array so there is exactly one local maximum.

Views

Author

Keywords

Crossrefs

A261834 Number of n-step adjacent expansions on the hexagonal (honeycomb) lattice. Holes allowed.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A087924 Number of ways of arranging the numbers 1 ... 3n into a 3 X n array so there is exactly one local maximum.

Views

Author

Keywords

Crossrefs

A087925 Number of ways of arranging the numbers 1 ... 4n into a 4 X n array so there is exactly one local maximum.

Views

Author

Keywords

Crossrefs

A087926 Number of ways of arranging the numbers 1 ... 5n into a 5 X n array so there is exactly one local maximum.

Views

Author

Keywords

Crossrefs

A087927 Number of ways of arranging the numbers 1 ... 6n into a 6 X n array so there is exactly one local maximum.

Views

Author

Keywords

Crossrefs

A087928 Number of ways of arranging the numbers 1 ... 7n into a 7 X n array so there is exactly one local maximum.

Views

Author

Keywords

Crossrefs