A264656 -id:A264656 - cp's OEIS frontend

A275062 Number A(n,k) of permutations p of [n] such that p(i)-i is a multiple of k for all i in [n]; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 6, 1, 1, 1, 1, 2, 24, 1, 1, 1, 1, 1, 4, 120, 1, 1, 1, 1, 1, 2, 12, 720, 1, 1, 1, 1, 1, 1, 4, 36, 5040, 1, 1, 1, 1, 1, 1, 2, 8, 144, 40320, 1, 1, 1, 1, 1, 1, 1, 4, 24, 576, 362880, 1, 1, 1, 1, 1, 1, 1, 2, 8, 72, 2880, 3628800, 1

Offset: 0

Alois P. Heinz, Jul 15 2016

			A(5,0) = A(5,5) = 1: 12345.
A(5,1) = 5! = 120: all permutations of {1,2,3,4,5}.
A(5,2) = 12: 12345, 12543, 14325, 14523, 32145, 32541, 34125, 34521, 52143, 52341, 54123, 54321.
A(5,3) = 4: 12345, 15342, 42315, 45312.
A(5,4) = 2: 12345, 52341.
A(7,4) = 8: 1234567, 1274563, 1634527, 1674523, 5234167, 5274163, 5634127, 5674123.
Square array A(n,k) begins:
  1,       1,     1,   1,   1,  1,  1, 1, 1, 1, 1, ...
  1,       1,     1,   1,   1,  1,  1, 1, 1, 1, 1, ...
  1,       2,     1,   1,   1,  1,  1, 1, 1, 1, 1, ...
  1,       6,     2,   1,   1,  1,  1, 1, 1, 1, 1, ...
  1,      24,     4,   2,   1,  1,  1, 1, 1, 1, 1, ...
  1,     120,    12,   4,   2,  1,  1, 1, 1, 1, 1, ...
  1,     720,    36,   8,   4,  2,  1, 1, 1, 1, 1, ...
  1,    5040,   144,  24,   8,  4,  2, 1, 1, 1, 1, ...
  1,   40320,   576,  72,  16,  8,  4, 2, 1, 1, 1, ...
  1,  362880,  2880, 216,  48, 16,  8, 4, 2, 1, 1, ...
  1, 3628800, 14400, 864, 144, 32, 16, 8, 4, 2, 1, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-10 give: A000012, A000142, A010551, A264557, A264635, A264656, A264701, A264791, A275063, A275064, A275065.
A(k*n,n) for k=1..4 give: A000012, A000079, A000400, A009968.
Cf. A225816.

A:= (n, k)-> mul(floor((n+i)/k)!, i=0..k-1):
seq(seq(A(n, d-n), n=0..d), d=0..14);

A[n_, k_] := Product[Floor[(n+i)/k]!, {i, 0, k-1}];
Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, May 26 2019, from Maple *)

A(n,k) = Product_{i=0..k-1} floor((n+i)/k)!.
A(k*n,k) = (n!)^k = A225816(k,n).
For k > 0, A(n, k) ~ (2*Pi*n)^((k - 1)/2) * n! / k^(n + k/2). - Vaclav Kotesovec, Oct 02 2018

A264692 T(n,k)=Number of nXk arrays of permutations of 0..n*k-1 with rows nondecreasing modulo 3 and columns nondecreasing modulo 5.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 7, 6, 1, 4, 56, 80, 23, 1, 8, 348, 3196, 2051, 84, 2, 24, 4672, 171962, 465584, 59845, 897, 4, 72, 93864, 17647792, 207022624, 110630784, 3406328, 9292, 8, 216, 1776576, 2685878592

Offset: 1

R. H. Hardin, Nov 21 2015

Table starts
.1....1.......1.........2.........4........8.........24......72.216
.1....1.......7........56.......348.....4672......93864.1776576
.1....6......80......3196....171962.17647792.2685878592
.1...23....2051....465584.207022624
.1...84...59845.110630784
.2..897.3406328
.4.9292

			Some solutions for n=3 k=4
..6..0.10..5....0.10..1.11....0..3..1.10...10..2..5.11....0..7.10.11
..3..1.11..2....6..5..8..2....6..9.11..5....0..4..1..7....6..3..5..2
..9..7..4..8....3..9..4..7....7..4..2..8....3..9..6..8....9..4..1..8

Column 1 is A264656.

Row 1 is A264557.

A264698 T(n,k)=Number of nXk arrays of permutations of 0..n*k-1 with rows nondecreasing modulo 4 and columns nondecreasing modulo 5.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 5, 3, 1, 2, 28, 68, 22, 1, 4, 159, 1541, 1378, 93, 2, 8, 1383, 37595, 102573, 26662, 790, 4, 16, 16818, 1947196, 20144706, 15357579, 975419, 7202, 8, 48, 168974, 148906136

Offset: 1

R. H. Hardin, Nov 21 2015

Table starts
.1....1......1........1........2.......4.........8.....16.48
.1....2......5.......28......159....1383.....16818.168974
.1....3.....68.....1541....37595.1947196.148906136
.1...22...1378...102573.20144706
.1...93..26662.15357579
.2..790.975419
.4.7202

			Some solutions for n=4 k=4
.10.15.11..7....5..6.10.15....1.10..2..6....1..5.10..6....0..5.15.11
..5.13..1..2....0.13.11..7....3.15..7.11...13.15..7.11...12..6.10..2
..0..9..6..3...12..9..1..3....8..0.12.14....4..0..8.12....4..8..1.13
.12..4..8.14....8..4..2.14....4..5.13..9....9..2.14..3....9.14..7..3

Column 1 is A264656.

Row 1 is A264635.

A264714 T(n,k)=Number of nXk arrays of permutations of 0..n*k-1 with rows nondecreasing modulo 5 and columns nondecreasing modulo 6.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 18, 40, 11, 1, 2, 116, 803, 815, 94, 1, 4, 627, 23775, 74070, 22320, 365, 2, 8, 6211, 468540, 3908105, 3432012, 352181, 2891, 4, 16, 59011, 17266449, 530614763

Offset: 1

R. H. Hardin, Nov 21 2015

Table starts
.1....1......1.......1.......1.........2........4.....8.16
.1....2......2......18.....116.......627.....6211.59011
.1....2.....40.....803...23775....468540.17266449
.1...11....815...74070.3908105.530614763
.1...94..22320.3432012
.1..365.352181
.2.2891

			Some solutions for n=4 k=4
.15..0.13..8....0..6..8.13....0..1.13.14....0..1..7.13....0..6..7..2
.10.12..7..2....1.12..2..7...12..7..2..8...12..2..3.14....1.12..3..8
.11..6..1.14...10..3..4.14....6..3..9..4....6..8..9..4...13.14..9..4
..5..3..4..9....5.15.11..9....5.15.10.11...15..5.10.11...15..5.10.11

Column 1 is A264701.

Row 1 is A264656.

A264837 T(n,k)=Number of nXk arrays of permutations of 0..n*k-1 with rows nondecreasing modulo 5 and columns nondecreasing modulo 7.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 15, 22, 8, 1, 2, 68, 409, 271, 46, 1, 4, 401, 4800, 13814, 2703, 182, 1, 8, 2179, 147550, 1059838, 742363, 61692, 701, 2, 16, 23229, 5692511, 243485517

Offset: 1

R. H. Hardin, Nov 26 2015

Table starts
.1...1.....1......1.......1.........2.......4.....8.16
.1...2.....2.....15......68.......401....2179.23229
.1...2....22....409....4800....147550.5692511
.1...8...271..13814.1059838.243485517
.1..46..2703.742363
.1.182.61692
.1.701
.2

			Some solutions for n=4 k=4
..0..7..8.14....0..8.14..9....7..8..9.14....0..7..9.14...15..0..8.14
..1..2..4..9....1.12..7..3....0..1.12..2...15..1..3..8...11..1..2..7
.15.10.12..3...15..5..2..4...15.10..6.11....2.12.13..4....5.10.12..9
..5.11..6.13...10..6.11.13....5..3.13..4...10..5..6.11....6..3.13..4

Column 1 is A264791.

Row 1 is A264656.

A335109 Triangle read by rows: T(n,k) is the number of permutations of length n with each cycle of the permutation containing only elements that are identical (mod k), where 1 <= k <= n.

Original entry on oeis.org

1, 2, 1, 6, 2, 1, 24, 4, 2, 1, 120, 12, 4, 2, 1, 720, 36, 8, 4, 2, 1, 5040, 144, 24, 8, 4, 2, 1, 40320, 576, 72, 16, 8, 4, 2, 1, 362880, 2880, 216, 48, 16, 8, 4, 2, 1, 3628800, 14400, 864, 144, 32, 16, 8, 4, 2, 1

Offset: 1

Dennis P. Walsh, May 23 2020

Let [n] denote {1,2,...,n} and let [n](j,k) denote the subset of [n] consisting of all elements of [n] that equal j mod k. The cardinality of [n](j,k) equals ceiling(n/k) for j = 1..(n mod k) and equals floor(n/k) for j > (n mod k). Therefore, upon permuting the elements of each [n](j,k) subset, we obtain T(n,k) = (ceiling(n/k)!)^(n mod k)*(floor(n/k)!)^(k-(n mod k)).

			Triangle begins:
    1;
    2  1;
    6  2 1;
   24  4 2 1;
  120 12 4 2 1;
  ...
T(6,3) counts the 8 permutations of [6] where all cycle-mates are identical mod 3, namely, (1 4)(2 5)(3 6), (1 4)(2 5)(3)(6), (1 4)(2)(5)(3 6), (1)(4)(2 5)(3 6), (1 4)(2)(5)(3)(6), (1)(4)(2 5)(3)(6), (1)(4)(2)(5)(3 6) and (1)(2)(3)(4)(5)(6).

Per Alexandersson, Frether Getachew Kebede, Samuel Asefa Fufa, and Dun Qiu, Pattern-Avoidance and Fuss-Catalan Numbers, J. Int. Seq. (2023) Vol. 26, Art. 23.4.2.

Cf. A000142, A010551, A264557, A264635, A264656.

Cf. A275062.

seq(seq((ceil(n/k)!)^(n mod k)*(floor(n/k)!)^(k-(n mod k)), k=1..n), n=1..10);

Table[(Ceiling[n/k]!)^Mod[n, k]*(Floor[n/k]!)^(k - Mod[n, k]), {n, 10}, {k, n}] // Flatten (* Michael De Vlieger, Jun 28 2020 *)

T(n,k) = (ceiling(n/k)!)^(n mod k)*(floor(n/k)!)^(k-(n mod k)) for 1 <= k <= n.
T(n,1) = A000142(n).
T(n,2) = A010551(n) for n > 1.
T(n,3) = A264557(n) for n > 2.
T(n,4) = A264635(n) for n > 3.
T(n,5) = A264656(n) for n > 4.
T(n,k) = Product_{i=0..k-1} floor((n+i)/k)!. - Alois P. Heinz, May 23 2020

cp's OEIS Frontend

A275062 Number A(n,k) of permutations p of [n] such that p(i)-i is a multiple of k for all i in [n]; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A264692 T(n,k)=Number of nXk arrays of permutations of 0..n*k-1 with rows nondecreasing modulo 3 and columns nondecreasing modulo 5.

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A264698 T(n,k)=Number of nXk arrays of permutations of 0..n*k-1 with rows nondecreasing modulo 4 and columns nondecreasing modulo 5.

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A264714 T(n,k)=Number of nXk arrays of permutations of 0..n*k-1 with rows nondecreasing modulo 5 and columns nondecreasing modulo 6.

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A264837 T(n,k)=Number of nXk arrays of permutations of 0..n*k-1 with rows nondecreasing modulo 5 and columns nondecreasing modulo 7.

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A335109 Triangle read by rows: T(n,k) is the number of permutations of length n with each cycle of the permutation containing only elements that are identical (mod k), where 1 <= k <= n.

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Examples

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Programs

Maple

Mathematica

Formula