A060854 -id:A060854 - cp's OEIS frontend

A177848 Triangle, read by rows, T(n, k) = t(k, n-k+1) - t(1, n) + 1 where t(n, m) = (nm)!Beta(n, m).

1, 1, 1, 1, 3, 1, 1, 55, 55, 1, 1, 1993, 12073, 1993, 1, 1, 120841, 7983241, 7983241, 120841, 1, 1, 11404081, 12454040881, 149448498481, 12454040881, 11404081, 1, 1, 1556750161, 38109367290961, 8688935743482961, 8688935743482961, 38109367290961, 1556750161, 1

Offset: 1

Roger L. Bagula, May 14 2010

Row sums are {1, 2, 5, 112, 16061, 16208166, 174379388407, 17454093335048168, 27083470639271574245769, 421762213493139881153379087370, ...}.

			Triangle begins as:
  1;
  1,        1;
  1,        3,           1;
  1,       55,          55,            1;
  1,     1993,       12073,         1993,           1;
  1,   120841,     7983241,      7983241,      120841,        1;
  1, 11404081, 12454040881, 149448498481, 12454040881, 11404081, 1;

G. C. Greubel, Rows n = 1..30 of the triangle, flattened

Cf. A060854.

t[n_, k_]:= (n*k)!*Beta[n, k];
Table[t[k, n-k+1] - t[1, n] + 1, {n, 12}, {k, n}]//Flatten

def t(n, k): return factorial(n*k)*beta(n, k)
flatten([[t(k, n-k+1) - t(1,n) + 1 for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Feb 06 2021

Let t(n, k) = (n*k)!*Beta(n, k) then T(n, k) = t(k, n-k+1) - t(1, n) + 1.

Edited by G. C. Greubel, Feb 06 2021

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)

A215293 Number of permutations of 0..floor((n*n-2)/2) on odd squares of an n X n array such that each row and column of odd squares is increasing.

Original entry on oeis.org

1, 2, 6, 280, 23100, 85765680, 577185873264, 346915095471584640, 381134230556959188429120, 62144711688730139887005809020800, 18592619468814454675301397184588597886400, 1236552808693429892089668394551052130596983991526400, 151213938214745201135492692441902799026853717717324113365952000

Offset: 1

R. H. Hardin, Aug 07 2012

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

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

Diagonal of A215297.

Cf. A060854.

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

A215294 Number of permutations of 0..floor((n*3-2)/2) on odd squares of an n X 3 array such that each row and column of odd squares is increasing.

Original entry on oeis.org

1, 3, 6, 30, 70, 420, 1050, 6930, 18018, 126126, 336336, 2450448, 6651216, 49884120, 137181330, 1051723530, 2921454250, 22787343150, 63804560820, 504636071940, 1422156202740, 11377249621920, 32235540595440, 260363981732400

Offset: 1

R. H. Hardin, Aug 07 2012

nonn

a(n) is number of symmetric standard Young tableaux of shape (n,n,n). - Ran Pan, May 21 2015

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

R. H. Hardin, Table of n, a(n) for n = 1..210
Ran Pan, Problem 4, Project P.

Column 3 of A215297.

Cf. A060693.

a := n -> `if`(irem(n, 2) = 0, ((1/2)*n+1)*factorial((3/2)*n)/ (factorial((1/2)*n+1)^2*factorial((1/2)*n)), factorial((3/2)*n+3/2)/ (factorial((1/2)*n+1/2)^3*((9/2)*n+3/2))): # Peter Luschny, Sep 30 2018

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

a(n) = e(n) if n even otherwise o(n), where e(n) = 6*Gamma((3*n)/2)/((2 + n)*Gamma(1 + n/2)^2*Gamma(n/2)) and o(n) = (1 + n)*Gamma(1/2 + (3*n)/2)/(2*Gamma((3 + n)/2)^3). - Peter Luschny, Sep 30 2018

A215295 Number of permutations of 0..floor((n*5-2)/2) on odd squares of an nX5 array such that each row and column of odd squares is increasing.

Original entry on oeis.org

1, 10, 70, 2100, 23100, 1051050, 14294280, 814773960, 12547518984, 824551247520, 13781785137120, 999179422441200, 17699749768958400, 1379105502831342000, 25513451802379827000, 2100607531729272423000, 40191624107086745693400

Offset: 1

R. H. Hardin Aug 07 2012

nonn

Column 5 of A215297

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

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

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

A215296 Number of permutations of 0..floor((n*7-2)/2) on odd squares of an nX7 array such that each row and column of odd squares is increasing.

Original entry on oeis.org

1, 35, 1050, 210210, 14294280, 5703417720, 577185873264, 337653735859440, 43364386933948080, 32436561426593163840, 4886336289191784467040, 4340695403565368534887200, 733267473530864041786302000

Offset: 1

R. H. Hardin Aug 07 2012

nonn

Column 7 of A215297

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

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(4,f4)*binomial(3*f3+4*f4,3*f3)

A378173 Array read by antidiagonals: T(n,k) is the number of proper antichain partitions of the rectangular poset of size n X k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 5, 5, 1, 1, 14, 38, 14, 1, 1, 42, 372, 372, 42, 1, 1, 132, 4282, 14606, 4282, 132, 1, 1, 429, 55149

Offset: 1

Ludovic Schwob, Nov 18 2024

A partition of a poset into antichains is said to be proper if it does not contain two antichains A_1 and A_2, with x_1,y_1 in A_1 and x_2,y_2 in A_2, such that x_1y_2.

A proper antichain partition of a poset is endowed with an order relation, which is induced by the order relation of the poset. Let Y be a young diagram, and P the poset of shape Y. The number of linear extensions of P is the number of standard Young tableaux with shape Y. The sum over all proper antichain partitions of P, of the numbers of linear extensions of the induced orders, is equal to the number of normal generalized Young tableaux of shape Y with all rows and columns strictly increasing (cf. A299968).

			Array begins:
=====================================================================
n/k | 1     2      3      4      5      6 ...
----+----------------------------------------------------------------
  1 | 1     1      1      1      1      1 ...
  2 | 1     2      5     14     42    132 ...
  3 | 1     5     38    372   4282  55149 ...
  4 | 1    14    372  14606 ...
  5 | 1    42   4282 ...
  6 | 1   132  55149 ...

Cf. A060854, A299968, A374985.

T(n,k) = T(k,n).

T(n,2) = A000108(n).

cp's OEIS Frontend

A177848 Triangle, read by rows, T(n, k) = t(k, n-k+1) - t(1, n) + 1 where t(n, m) = (n*m)!*Beta(n, m).

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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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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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A215293 Number of permutations of 0..floor((n*n-2)/2) on odd squares of an n X n array such that each row and column of odd squares is increasing.

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A215294 Number of permutations of 0..floor((n*3-2)/2) on odd squares of an n X 3 array such that each row and column of odd squares is increasing.

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Programs

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A215295 Number of permutations of 0..floor((n*5-2)/2) on odd squares of an nX5 array such that each row and column of odd squares is increasing.

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A215296 Number of permutations of 0..floor((n*7-2)/2) on odd squares of an nX7 array such that each row and column of odd squares is increasing.

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Comments

Examples

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Formula

A378173 Array read by antidiagonals: T(n,k) is the number of proper antichain partitions of the rectangular poset of size n X k.

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Examples

A177848 Triangle, read by rows, T(n, k) = t(k, n-k+1) - t(1, n) + 1 where t(n, m) = (nm)!Beta(n, m).