A277032 Number of permutations of [n] such that the minimal cyclic distance between elements of the same cycle equals one, a(1)=1 by convention.

1, 1, 5, 20, 109, 668, 4801, 38894, 353811, 3561512, 39374609, 474132730, 6179650125, 86676293916, 1301952953989, 20852719565694, 354771488612075, 6389625786835184, 121456993304945749, 2429966790591643402, 51042656559451380013, 1123165278137918510772

Offset: 1

Author

Alois P. Heinz, Sep 25 2016

Keywords

• nonn

Examples

			a(2) = 1: (1,2).
a(3) = 5: (1,2,3), (1,3,2), (1)(2,3), (1,2)(3), (1,3)(2).
		

Links

• Per Alexandersson et al., d-regular partitions and permutations, MathOverflow, 2014

Crossrefs

Column k=1 of A277031.

Cf. A002467, A180191, A276975.

Programs

Maple

b:= proc(n, i, l) option remember; `if`(n=0, mul(j!, j=l),
      (m-> add(`if`(i=j or n*j=1, 0, b(n-1, j, `if`(j>m,
      [l[], 0], subsop(j=l[j]+1, l)))), j=1..m+1))(nops(l)))
    end:
a:= n-> `if`(n=1, 1, n!-b(n-1, 1, [0])):
seq(a(n), n=1..15);

Mathematica

b[n_, i_, l_] := b[n, i, l] = If[n == 0, Product[j!, {j, l}], With[{m = Length[l]}, Sum[If[i == j || n*j == 1, 0, b[n-1, j, If[j>m, Append[l, 0], ReplacePart[l, j -> l[[j]]+1]]]], {j, 1, m+1}]]];
a[n_] := If[n == 1, 1, n! - b[n-1, 1, {0}]];
Array[a, 15] (* Jean-François Alcover, Mar 13 2021, after Alois P. Heinz *)