A215292 -id:A215292 - cp's OEIS frontend

A215297 T(n,k) = number of permutations of 0..floor((n*k-2)/2) on odd squares of an n X k array such that each row and column of odd squares is increasing.

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 6, 6, 6, 1, 1, 10, 30, 30, 10, 1, 1, 20, 70, 280, 70, 20, 1, 1, 35, 420, 2100, 2100, 420, 35, 1, 1, 70, 1050, 23100, 23100, 23100, 1050, 70, 1, 1, 126, 6930, 210210, 1051050, 1051050, 210210, 6930, 126, 1, 1, 252, 18018, 2522520, 14294280

Offset: 1

R. H. Hardin, Aug 07 2012

Table starts
.1...1.....1........1...........1..............1.................1
.1...2.....3........6..........10.............20................35
.1...3.....6.......30..........70............420..............1050
.1...6....30......280........2100..........23100............210210
.1..10....70.....2100.......23100........1051050..........14294280
.1..20...420....23100.....1051050.......85765680........5703417720
.1..35..1050...210210....14294280.....5703417720......577185873264
.1..70..6930..2522520...814773960...577185873264...337653735859440
.1.126.18018.25729704.12547518984.48236247979920.43364386933948080
Even columns match A215292.
The first column is number of symmetric standard Young tableaux of shape (n), the second column is number of symmetric standard Young tableaux of shape (n,n) and the third column is number of symmetric standard Young tableaux of shape (n,n,n). - Ran Pan, May 21 2015

			Some solutions for n=5, k=4:
..x..0..x..4....x..0..x..1....x..1..x..3....x..0..x..6....x..0..x..1
..1..x..2..x....4..x..7..x....0..x..8..x....3..x..5..x....3..x..7..x
..x..3..x..8....x..2..x..3....x..2..x..5....x..1..x..7....x..2..x..5
..6..x..7..x....5..x..9..x....4..x..9..x....4..x..9..x....6..x..8..x
..x..5..x..9....x..6..x..8....x..6..x..7....x..2..x..8....x..4..x..9

R. H. Hardin, Table of n, a(n) for n = 1..1000
Hodge, Jonathan K.; Krines, Mark; Lahr, Jennifer, Preseparable extensions of multidimensional preferences, Order 26, No. 2, 125-147 (2009), Table 1.
Ran Pan, Exercise P, Problem 4, Project P.

Column 2 is A001405. Column 4 is A215288. Column 6 is A215290.

f1=floor(k/2), f2=floor((k+1)/2), f3=floor((n+1)/2), f4=floor(n/2);

T(n,k) = A060854(f1,f3)*A060854(f2,f4)*binomial(f1*f3+f2*f4,f1*f3).

A215288 Number of permutations of 0..floor((n*4-1)/2) on even squares of an n X 4 array such that each row and column of even squares is increasing.

Original entry on oeis.org

1, 6, 30, 280, 2100, 23100, 210210, 2522520, 25729704, 325909584, 3585005424, 47117214144, 546896235600, 7383099180600, 89212448432250, 1229149289511000, 15323394475903800, 214527522662653200, 2742051789669912720

Offset: 1

R. H. Hardin, Aug 07 2012

nonn

			Some solutions for n=5
..2..x..4..x....0..x..3..x....2..x..3..x....0..x..5..x....1..x..4..x
..x..0..x..1....x..2..x..5....x..0..x..4....x..1..x..3....x..0..x..6
..3..x..7..x....1..x..6..x....5..x..6..x....6..x..7..x....2..x..5..x
..x..6..x..9....x..4..x..7....x..1..x..8....x..2..x..4....x..3..x..8
..5..x..8..x....8..x..9..x....7..x..9..x....8..x..9..x....7..x..9..x

R. H. Hardin, Table of n, a(n) for n = 1..210

Column 4 of A215292.

f3=floor((n+1)/2),
f4=floor(n/2),
a(n) = A060854(2,f3)*A060854(2,f4)*binomial(2*f3+2*f4,2*f3).

A215287 Number of permutations of 0..floor((n*3-1)/2) on even squares of an n X 3 array such that each row and column of even squares is increasing.

Original entry on oeis.org

1, 3, 10, 30, 140, 420, 2310, 6930, 42042, 126126, 816816, 2450448, 16628040, 49884120, 350574510, 1051723530, 7595781050, 22787343150, 168212023980, 504636071940, 3792416540640, 11377249621920, 86787993910800, 260363981732400, 2011383287449200

Offset: 1

R. H. Hardin, Aug 07 2012

nonn

Also Schröder paths of length n having floor(n/2) peaks. - Peter Luschny, Sep 30 2018

			Some solutions for n=5:
  0 x 4   0 x 5   1 x 3   0 x 1   0 x 3   1 x 4   0 x 2
  x 3 x   x 1 x   x 0 x   x 4 x   x 2 x   x 0 x   x 1 x
  1 x 5   2 x 6   2 x 5   2 x 3   1 x 6   2 x 5   3 x 5
  x 7 x   x 3 x   x 6 x   x 6 x   x 5 x   x 6 x   x 6 x
  2 x 6   4 x 7   4 x 7   5 x 7   4 x 7   3 x 7   4 x 7

R. H. Hardin, Table of n, a(n) for n = 1..210

Column 3 of A215292.

Cf. A060854, A060693.

[(n-(n div 2)+1)*Factorial(2*n-(n div 2)) / (Factorial(n-(n div 2) +1)^2*Factorial((n div 2))): n in [1..30]]; // Vincenzo Librandi, Oct 01 2018

T := (n, k) -> (n-k+1)*(2*n-k)!/((n-k+1)!^2*k!):
a := n -> T(n, floor(n/2)): seq(a(n), n = 1..23); # Peter Luschny, Sep 30 2018

Table[(n - Floor[n/2] + 1) (2 n - Floor[n/2])! / ((n -Floor[n/2] + 1)!^2 Floor[n/2]!), {n, 1, 30}] (* Vincenzo Librandi, Oct 01 2018 *)

f3 = floor((n+1)/2); f4 = floor(n/2);
a(n) = A060854(2,f3)*A060854(1,f4)*binomial(2*f3+1*f4,2*f3).
a(n) = (n - f + 1)*(2*n - f)! / ((n - f + 1)!^2 * f!) where f = floor(n/2). - Peter Luschny, Sep 30 2018

A215290 Number of permutations of 0..floor((n*6-1)/2) on even squares of an nX6 array such that each row and column of even squares is increasing.

Original entry on oeis.org

1, 20, 420, 23100, 1051050, 85765680, 5703417720, 577185873264, 48236247979920, 5595404765670720, 545152292883918720, 69506917342699636800, 7562187114225380722800, 1033498905610802032116000

Offset: 1

R. H. Hardin Aug 07 2012

nonn

Column 6 of A215292

			Some solutions for n=4
..0..x..2..x..6..x....1..x..6..x..7..x....1..x..2..x..7..x....0..x..2..x..6..x
..x..1..x..3..x..5....x..0..x..2..x..4....x..0..x..4..x.10....x..3..x..5..x..7
..4..x..9..x.10..x....9..x.10..x.11..x....3..x..6..x..8..x....1..x..9..x.11..x
..x..7..x..8..x.11....x..3..x..5..x..8....x..5..x..9..x.11....x..4..x..8..x.10

R. H. Hardin, Table of n, a(n) for n = 1..210

f3=floor((n+1)/2)
f4=floor(n/2)
a(n) = A060854(3,f3)*A060854(3,f4)*binomial(3*f3+3*f4,3*f3)

A215286 Number of permutations of 0..floor((n*n-1)/2) on even squares of an n X n array such that each row and column of even squares is increasing.

Original entry on oeis.org

1, 2, 10, 280, 60060, 85765680, 2061378118800, 346915095471584640, 1736278161426147413954880, 62144711688730139887005809020800, 103104526145243794108489566205445861006400

Offset: 1

R. H. Hardin, Aug 07 2012

nonn

Diagonal of A215292.

			Some solutions for n=5
..0..x..1..x..2....0..x..2..x..6....0..x..2..x..4....1..x..2..x..6
..x..4..x..7..x....x..1..x..3..x....x..3..x..5..x....x..0..x..8..x
..3..x..6..x..9....5..x..8..x.11....1..x..8..x..9....3..x..5..x.10
..x..5..x.11..x....x..4..x..9..x....x..7..x.10..x....x..9..x.12..x
..8..x.10..x.12....7..x.10..x.12....6..x.11..x.12....4..x..7..x.11

R. H. Hardin, Table of n, a(n) for n = 1..40

Cf. A060854, A215292.

f1 = floor((n+1)/2)
f2 = floor(n/2)
T(n,k) = A060854(f1,f1)*A060854(f2,f2)*binomial(f1*f1+f2*f2,f1*f1).

A215289 Number of permutations of 0..floor((n*5-1)/2) on even squares of an n X 5 array such that each row and column of even squares is increasing.

Original entry on oeis.org

1, 10, 140, 2100, 60060, 1051050, 42882840, 814773960, 41227562376, 824551247520, 48236247979920, 999179422441200, 64899082486180800, 1379105502831342000, 96951116849043342600, 2100607531729272423000, 157112712418611824074200, 3456479673209460129632400, 271742147399231010918736320

Offset: 1

R. H. Hardin, Aug 07 2012

			Some solutions for n=5:
..0..x..2..x..6....1..x..2..x..6....0..x..3..x..9....0..x..1..x..8
..x..3..x..4..x....x..0..x..4..x....x..2..x..4..x....x..3..x..6..x
..1..x..7..x.10....7..x..9..x.11....1..x..7..x.11....2..x..4..x..9
..x..5..x.11..x....x..3..x..5..x....x..5..x..8..x....x..7..x.10..x
..8..x..9..x.12....8..x.10..x.12....6..x.10..x.12....5..x.11..x.12

R. H. Hardin, Table of n, a(n) for n = 1..210

Column 5 of A215292.

a(n) = A060854(3,f3)*A060854(2,f4)*binomial(3*f3+2*f4,3*f3), where f3 = floor((n+1)/2) and f4 = floor(n/2).

A215291 Number of permutations of 0..floor((n*7-1)/2) on even squares of an nX7 array such that each row and column of even squares is increasing.

Original entry on oeis.org

1, 35, 2310, 210210, 42882840, 5703417720, 2061378118800, 337653735859440, 173457547735792320, 32436561426593163840, 21174123919831066023840, 4340695403565368534887200, 3373030378241974592216989200

Offset: 1

R. H. Hardin Aug 07 2012

nonn

Column 7 of A215292

			Some solutions for n=3
..0..x..3..x..5..x..6....0..x..1..x..2..x..3....1..x..2..x..3..x..5
..x..1..x..8..x.10..x....x..5..x..8..x.10..x....x..0..x..6..x.10..x
..2..x..4..x..7..x..9....4..x..6..x..7..x..9....4..x..7..x..8..x..9

R. H. Hardin, Table of n, a(n) for n = 1..210

f3=floor((n+1)/2)
f4=floor(n/2)
a(n) = A060854(4,f3)*A060854(3,f4)*binomial(4*f3+3*f4,4*f3)

cp's OEIS Frontend

A215297 T(n,k) = number of permutations of 0..floor((n*k-2)/2) on odd squares of an n X k array such that each row and column of odd squares is increasing.

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A215288 Number of permutations of 0..floor((n*4-1)/2) on even squares of an n X 4 array such that each row and column of even squares is increasing.

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A215287 Number of permutations of 0..floor((n*3-1)/2) on even squares of an n X 3 array such that each row and column of even squares is increasing.

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A215290 Number of permutations of 0..floor((n*6-1)/2) on even squares of an nX6 array such that each row and column of even squares is increasing.

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A215286 Number of permutations of 0..floor((n*n-1)/2) on even squares of an n X n array such that each row and column of even squares is increasing.

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A215289 Number of permutations of 0..floor((n*5-1)/2) on even squares of an n X 5 array such that each row and column of even squares is increasing.

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Formula

A215291 Number of permutations of 0..floor((n*7-1)/2) on even squares of an nX7 array such that each row and column of even squares is increasing.

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Comments

Examples

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Formula