A346086 -id:A346086 - cp's OEIS frontend

A346085 Number T(n,k) of permutations of [n] such that k is the GCD of the cycle lengths; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 1, 1, 0, 4, 0, 2, 0, 15, 3, 0, 6, 0, 96, 0, 0, 0, 24, 0, 455, 105, 40, 0, 0, 120, 0, 4320, 0, 0, 0, 0, 0, 720, 0, 29295, 4725, 0, 1260, 0, 0, 0, 5040, 0, 300160, 0, 22400, 0, 0, 0, 0, 0, 40320, 0, 2663199, 530145, 0, 0, 72576, 0, 0, 0, 0, 362880

Offset: 0

Alois P. Heinz, Jul 04 2021

			T(3,1) = 4: (1)(23), (13)(2), (12)(3), (1)(2)(3).
T(4,4) = 6: (1234), (1243), (1324), (1342), (1423), (1432).
Triangle T(n,k) begins:
  1;
  0,       1;
  0,       1,      1;
  0,       4,      0,     2;
  0,      15,      3,     0,    6;
  0,      96,      0,     0,    0,    24;
  0,     455,    105,    40,    0,     0, 120;
  0,    4320,      0,     0,    0,     0,   0, 720;
  0,   29295,   4725,     0, 1260,     0,   0,   0, 5040;
  0,  300160,      0, 22400,    0,     0,   0,   0,    0, 40320;
  0, 2663199, 530145,     0,    0, 72576,   0,   0,    0,     0, 362880;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
Wikipedia, Permutation

Columns k=0-1 give: A000007, A079128.
Even bisection of column k=2 gives A346086.
Row sums give A000142.
T(2n,n) gives A110468(n-1) for n >= 1.
Cf. A057731, A145877, A346066, A359951.

b:= proc(n, g) option remember; `if`(n=0, x^g, add((j-1)!
      *b(n-j, igcd(g, j))*binomial(n-1, j-1), j=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 0)):
seq(T(n), n=0..12);

b[n_, g_] := b[n, g] = If[n == 0, x^g, Sum[(j - 1)!*
     b[n - j, GCD[g, j]] Binomial[n - 1, j - 1], {j, n}]];
T[n_] := CoefficientList[b[n, 0], x];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Aug 30 2021, after Alois P. Heinz *)

Sum_{k=1..n} k * T(n,k) = A346066(n).

Sum_{prime p <= n} T(n,p) = A359951(n). - Alois P. Heinz, Jan 20 2023

A359951 Number of permutations of [n] such that the GCD of the cycle lengths is a prime.

Original entry on oeis.org

0, 0, 1, 2, 3, 24, 145, 720, 4725, 22400, 602721, 3628800, 67692625, 479001600, 12924021825, 103953833984, 2116670180625, 20922789888000, 959231402754625, 6402373705728000, 257071215652932681, 3242340687872000000, 142597230222616430625, 1124000727777607680000

Offset: 0

Alois P. Heinz, Jan 19 2023

nonn

			a(2) = 1: (12).
a(3) = 2: (123), (132).
a(4) = 3: (12)(34), (13)(24), (14)(23).
a(5) = 24: (12345), (12354), (12435), (12453), (12534), (12543), (13245), (13254), (13425), (13452), (13524), (13542), (14235), (14253), (14325), (14352), (14523), (14532), (15234), (15243), (15324), (15342), (15423), (15432).

Alois P. Heinz, Table of n, a(n) for n = 0..451
Wikipedia, Permutation

Cf. A000040, A000142, A005225, A079128, A214003, A346085, A346086.

b:= proc(n, g) option remember; `if`(n=0, `if`(isprime(g), 1, 0),
      add(b(n-j, igcd(j, g))*(n-1)!/(n-j)!, j=1..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..23);

b[n_, g_] := b[n, g] = If[n == 0, If[PrimeQ[g], 1, 0], Sum[b[n - j, GCD[j, g]]*(n - 1)!/(n - j)!, {j, 1, n}]];
a[n_] := b[n, 0];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Dec 13 2023, after Alois P. Heinz *)

a(n) = Sum_{prime p <= n} A346085(n,p).

a(p) = (p-1)! for prime p.

cp's OEIS Frontend

A346085 Number T(n,k) of permutations of [n] such that k is the GCD of the cycle lengths; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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