A275062 -id:A275062 - cp's OEIS frontend

A275063 Number of permutations p of [n] such that p(i)-i is a multiple of eight for all i in [n].

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 768, 2304, 6912, 20736, 62208, 186624, 559872, 1679616, 6718464, 26873856, 107495424, 429981696, 1719926784, 6879707136, 27518828544, 110075314176, 550376570880, 2751882854400, 13759414272000

Offset: 0

Alois P. Heinz, Jul 15 2016

nonn

			a(9) = 2: 123456789, 923456781.

Alois P. Heinz, Table of n, a(n) for n = 0..665

Column k=8 of A275062.

Table[Product[Floor[(n + i)/8]!, {i, 0, 7}], {n, 0, 40}] (* Vaclav Kotesovec, Oct 02 2018 *)

a(n) = Product_{i=0..7} floor((n+i)/8)!.

a(n) ~ (2*Pi*n)^(7/2) * n! / 8^(n + 4). - Vaclav Kotesovec, Oct 02 2018

A275064 Number of permutations p of [n] such that p(i)-i is a multiple of nine for all i in [n].

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1536, 4608, 13824, 41472, 124416, 373248, 1119744, 3359232, 10077696, 40310784, 161243136, 644972544, 2579890176, 10319560704, 41278242816, 165112971264, 660451885056, 2641807540224

Offset: 0

Alois P. Heinz, Jul 15 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..683

Column k=9 of A275062.

Table[Product[Floor[(n + i)/9]!, {i, 0, 8}], {n, 0, 40}] (* Vaclav Kotesovec, Oct 02 2018 *)

a(n) = Product_{i=0..8} floor((n+i)/9)!.

a(n) ~ (2*Pi*n)^4 * n! / 9^(n + 9/2). - Vaclav Kotesovec, Oct 02 2018

A275065 Number of permutations p of [n] such that p(i)-i is a multiple of ten for all i in [n].

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 3072, 9216, 27648, 82944, 248832, 746496, 2239488, 6718464, 20155392, 60466176, 241864704, 967458816, 3869835264, 15479341056, 61917364224, 247669456896, 990677827584, 3962711310336

Offset: 0

Alois P. Heinz, Jul 15 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..699

Column k=10 of A275062.

Cf. A010879, A059995.

f:= n -> mul(floor((n+i)/10)!,i=0..9):
map(f, [$0..30]); # Robert Israel, Jul 26 2016

Table[Product[Floor[(n + i)/10]!, {i, 0, 9}], {n, 0, 40}] (* Vaclav Kotesovec, Oct 02 2018 *)

a(n) = Product_{i=0..9} floor((n+i)/10)!.
a(n) = ((m+1)!)^10/(m+1)^(10-k) where m=floor(n/10)=A059995(n) and k=n mod 10 =A010879(n). - Robert Israel, Jul 26 2016
a(n) ~ (2*Pi*n)^(9/2) * n! / 10^(n + 5). - Vaclav Kotesovec, Oct 02 2018

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

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A275063 Number of permutations p of [n] such that p(i)-i is a multiple of eight for all i in [n].

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A275064 Number of permutations p of [n] such that p(i)-i is a multiple of nine for all i in [n].

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A275065 Number of permutations p of [n] such that p(i)-i is a multiple of ten for all i in [n].

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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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