A277032 -id:A277032 - cp's OEIS frontend

A276975 Number of permutations of [n] such that the minimal distance between elements of the same cycle equals one, a(1)=1 by convention.

1, 1, 4, 19, 103, 651, 4702, 38413, 350559, 3539511, 39196758, 472612883, 6165080443, 86526834271, 1300282224846, 20832761552453, 354515666646827, 6386139146435035, 121406489336263622, 2429193186525638435, 51030147426536745655, 1122952442325988152627

Offset: 1

Alois P. Heinz, Sep 23 2016

nonn

			a(2) = 1: (1,2).
a(3) = 4: (1,2,3), (1,3,2), (1)(2,3), (1,2)(3).

Alois P. Heinz, Table of n, a(n) for n = 1..50
Per Alexandersson et al., d-regular partitions and permutations, MathOverflow, 2014

Column k=1 of A276974.

Cf. A002467, A180191, A277032.

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

b[n_, i_, l_] := b[n, i, l] = If[n == 0, Product[j!, {j, l}], Function[m, Sum[If[i == j, 0, b[n - 1, j, If[j > m, Append[l, 0], ReplacePart[l, j -> l[[j]] + 1]]]], {j, 1, m + 1}]][Length[l]]];
a[n_] := If[n == 1, 1, n! - b[n, 0, {}]];
Array[a, 15] (* Jean-François Alcover, Oct 28 2020, after Maple code *)

A277031 Number T(n,k) of permutations of [n] where the minimal cyclic distance between elements of the same cycle equals k (k=n for the identity permutation in S_n); triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 5, 0, 1, 0, 20, 3, 0, 1, 0, 109, 10, 0, 0, 1, 0, 668, 44, 7, 0, 0, 1, 0, 4801, 210, 28, 0, 0, 0, 1, 0, 38894, 1320, 90, 15, 0, 0, 0, 1, 0, 353811, 8439, 554, 75, 0, 0, 0, 0, 1, 0, 3561512, 63404, 3542, 310, 31, 0, 0, 0, 0, 1, 0, 39374609, 517418, 23298, 1276, 198, 0, 0, 0, 0, 0, 1

Offset: 0

Alois P. Heinz, Sep 25 2016

			T(3,1) = 5: (1,2,3), (1,3,2), (1)(2,3), (1,2)(3), (1,3)(2).
T(3,3) = 1: (1)(2)(3).
Triangle T(n,k) begins:
  1;
  0,       1;
  0,       1,     1;
  0,       5,     0,    1;
  0,      20,     3,    0,   1;
  0,     109,    10,    0,   0,  1;
  0,     668,    44,    7,   0,  0, 1;
  0,    4801,   210,   28,   0,  0, 0, 1;
  0,   38894,  1320,   90,  15,  0, 0, 0, 1;
  0,  353811,  8439,  554,  75,  0, 0, 0, 0, 1;
  0, 3561512, 63404, 3542, 310, 31, 0, 0, 0, 0, 1;
  ...

Alois P. Heinz, Rows n = 0..12, flattened
Per Alexandersson et al., d-regular partitions and permutations, MathOverflow, 2014

Columns k=0-1 give: A000007, A277032.
Row sums give A000142.
T(2n,n) = A255047(n) = A000225(n) for n>0.
Cf. A239145, A263757, A276974.

cp's OEIS Frontend

A276975 Number of permutations of [n] such that the minimal distance between elements of the same cycle equals one, a(1)=1 by convention.

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