A215288 -id:A215288 - 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).

A342982 Triangle read by rows: T(n,k) is the number of tree-rooted planar maps with n edges and k+1 faces, n >= 0, k = 0..n.

Original entry on oeis.org

1, 1, 1, 2, 6, 2, 5, 30, 30, 5, 14, 140, 280, 140, 14, 42, 630, 2100, 2100, 630, 42, 132, 2772, 13860, 23100, 13860, 2772, 132, 429, 12012, 84084, 210210, 210210, 84084, 12012, 429, 1430, 51480, 480480, 1681680, 2522520, 1681680, 480480, 51480, 1430

Offset: 0

Andrew Howroyd, Apr 03 2021

The number of vertices is n + 1 - k.

A tree-rooted planar map is a planar map with a distinguished spanning tree.

			Triangle begins:
    1;
    1,     1;
    2,     6,     2;
    5,    30,    30,      5;
   14,   140,   280,    140,     14;
   42,   630,  2100,   2100,    630,    42;
  132,  2772, 13860,  23100,  13860,  2772,   132;
  429, 12012, 84084, 210210, 210210, 84084, 12012, 429;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
R. C. Mullin, On the Enumeration of Tree-Rooted Maps, Canadian Journal of Mathematics, Volume 19, 1967, pp. 174-183.

Columns k=0..2 are A000108, A002457, 2*A002803.
Row sums are A005568.
Central coefficients are A342983.
Cf. A001263, A215288, A269920, A342984.

Table[(2 n)!/(k!*(k + 1)!*(n - k)!*(n - k + 1)!), {n, 0, 8}, {k, 0, n}] // Flatten (* Michael De Vlieger, Apr 06 2021 *)

T(n,k) = {(2*n)!/(k!*(k+1)!*(n-k)!*(n-k+1)!)}
{ for(n=0, 10, print(vector(n+1, k, T(n,k-1)))) }

T(n,k) = (2*n)!/(k!*(k+1)!*(n-k)!*(n-k+1)!).
T(n,n-k) = T(n,k).
T(n, floor(n/2)) = A215288(n).
T(n,k) = A000108(n) * A001263(n+1,k+1). - Werner Schulte, Apr 04 2021

A342983 Number of tree-rooted planar maps with n+1 vertices and n+1 faces.

Original entry on oeis.org

1, 6, 280, 23100, 2522520, 325909584, 47117214144, 7383099180600, 1229149289511000, 214527522662653200, 38887279926227853120, 7271332144993605081120, 1395321310426879365566400, 273697641660657106322640000, 54708248601655917595233984000

Offset: 0

Andrew Howroyd, Apr 03 2021

nonn

The number of edges is 2*n.

Also, a(n) is the number of discrete walks that start and stop at the origin, never pass below the x-axis nor to the left of the y-axis, and, in any order, have n steps that increment x, n steps that decrement x, n steps that increment y, and n steps that decrement y. It is in this sense a way to generalize the 2n-step one-dimensional walks counted by A000108 to a count in two dimensions. Proof: There are A001448(n) ways to interleave two length-2n Dyck words (A000108(n)^2) - Lee A. Newberg, Nov 17 2023

Andrew Howroyd, Table of n, a(n) for n = 0..200

Central coefficients of A342982.
Even bisection of A215288.
Cf. A000108, A001246, A001448.

PARI
```
a(n) = {(4*n)!/(n!*(n+1)!)^2}
```

a(n) = (4*n)!/(n!*(n+1)!)^2.

a(n) = A000108(n)^2 * A001448(n) = A001246(n) * A001448(n). - Alois P. Heinz, Aug 02 2023

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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A342982 Triangle read by rows: T(n,k) is the number of tree-rooted planar maps with n edges and k+1 faces, n >= 0, k = 0..n.

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A342983 Number of tree-rooted planar maps with n+1 vertices and n+1 faces.

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