A263987 -id:A263987 - cp's OEIS frontend

A320843 Number of permutations sigma of {1,2,...,n} such that sigma(i) divides i or i divides sigma(i) for 1 <= i <= n.

1, 1, 2, 3, 8, 10, 36, 41, 132, 250, 700, 750, 4010, 4237, 10680, 24679, 87328, 90478, 435812, 449586, 1939684, 3853278, 8650900, 8840110, 60035322, 80605209, 177211024, 368759752, 1380348224, 1401414640, 8892787136, 9014369784, 33923638848, 59455553072, 126536289568, 207587882368

Offset: 0

Seiichi Manyama, Dec 18 2018

nonn

			In case n = 4:
permutation
------------
[1, 2, 3, 4]
[1, 4, 3, 2]
[2, 1, 3, 4]
[2, 4, 3, 1]
[3, 2, 1, 4]
[3, 4, 1, 2]
[4, 1, 3, 2]
[4, 2, 3, 1]

Carl Pomerance, Coprime permutations, arXiv:2203.03085 [math.NT], 2022.
Carl Pomerance, Permutations with arithmetic constraints, arXiv:2206.01699 [math.NT], 2022.

Cf. A005326, A263987.

a[n_] := a[n] = If[n == 0, 1,Permanent[Table[If[Divisible[i, j] || Divisible[j, i], 1, 0], {i, n}, {j, n}]]];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 22}] (* Jean-François Alcover, Jun 25 2022 *)

a(n) = matpermanent(matrix(n, n, i, j, if (!(i%j) || !(j%i), 1, 0))); \\ Michel Marcus, Dec 30 2018

a(n) = Permanent((x_{ij})) with x_{ij} = 1 if i divides j or j divides i and x_{ij} = 0 otherwise for i,j = 1,...,n. - M. Farrokhi D. G., Dec 30 2018

a(0), a(24)-a(30) from Alois P. Heinz, Dec 19 2018

a(31)-a(35) from M. Farrokhi D. G., Dec 30 2018

A333710 Number of permutations sigma of [n] such that i! divides Product_{k=1..i} sigma(k) for 1 <= i <= n.

Original entry on oeis.org

1, 1, 2, 4, 14, 28, 212, 424, 3060, 13488, 131212, 262424, 6444376, 12888752, 145241952, 2146993212, 40313750564, 80627501128, 2265599072684, 4531198145368, 173216179971224, 3202520631881824, 42018513097187068, 84037026194374136, 7051753589203676704, 50056536119264986708

Offset: 0

Seiichi Manyama, Apr 09 2020

nonn

			a(4) = 14: 1234, 1432, 2134, 2314, 2341, 2431, 3214, 3241, 3412, 3421, 4132, 4231, 4312, 4321.
a(5) = 28: 12345, 14325, 21345, 23145, 23415, 23451, 23541, 24315, 24351, 25341, 32145, 32415, 32451, 32541, 34125, 34215, 34251, 34521, 41325, 42315, 42351, 43125, 43215, 43251, 43521, 45321, 52341, 54321.

Cf. A263987, A320843, A333892.

b:= proc(s, p) option remember; (n-> `if`(n=0, 1, add(`if`(
      irem(p*n, j, 'q')=0, b(s minus {j}, q), 0), j=s)))(nops(s))
    end:
a:= n-> b({$1..n}, 1):
seq(a(n), n=0..17);  # Alois P. Heinz, Apr 09 2020

b[s_, p_] := b[s, p] = Module[{n=Length[s], q, r}, If[n==0, 1, Sum[If[{q, r} = QuotientRemainder[p n, j]; r==0, b[s~Complement~{j}, q], 0], {j, s}]]];
a[n_] := a[n] = If[n>2 && PrimeQ[n], 2 a[n-1], b[Range[n], 1]];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 25}] (* Jean-François Alcover, Nov 16 2020, after Alois P. Heinz *)

{a(n) = if(n==0, 1, my(k=0); forperm([1..n], p, if(#Set(vector(n, i, prod(j=1, i, p[j])%i!))==1, k++)); k)}

a(p) = 2*a(p-1) if p is prime.

a(14)-a(25) from Alois P. Heinz, Apr 09 2020

cp's OEIS Frontend

A320843 Number of permutations sigma of {1,2,...,n} such that sigma(i) divides i or i divides sigma(i) for 1 <= i <= n.

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