A285855 -id:A285855 - cp's OEIS frontend

A285849 Number T(n,k) of permutations of [n] with k ordered cycles such that equal-sized cycles are ordered with increasing least elements; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 1, 1, 0, 2, 6, 1, 0, 6, 19, 18, 1, 0, 24, 100, 105, 40, 1, 0, 120, 508, 1005, 430, 75, 1, 0, 720, 3528, 6762, 6300, 1400, 126, 1, 0, 5040, 24876, 61572, 62601, 28700, 3822, 196, 1, 0, 40320, 219168, 558548, 706608, 431445, 105336, 9114, 288, 1

Offset: 0

Alois P. Heinz, Apr 27 2017

Each cycle is written with the smallest element first and equal-sized cycles are arranged in increasing order of their first elements.

			T(3,1) = 2: (123), (132).
T(3,2) = 6: (1)(23), (23)(1), (2)(13), (13)(2), (3)(12), (12)(3).
T(3,3) = 1: (1)(2)(3).
Triangle T(n,k) begins:
  1;
  0,    1;
  0,    1,     1;
  0,    2,     6,     1;
  0,    6,    19,    18,     1;
  0,   24,   100,   105,    40,     1;
  0,  120,   508,  1005,   430,    75,    1;
  0,  720,  3528,  6762,  6300,  1400,  126,   1;
  0, 5040, 24876, 61572, 62601, 28700, 3822, 196, 1;

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

Columns k=0-10 give: A000007, A104150, A285853, A285854, A285855, A285856, A285857, A285858, A285859, A285860, A285861.
Row sums give A196301.
Main diagonal and first lower diagonal give: A000012, A002411.
T(2n,n) gives A285862.
Cf. A132393, A285824.

b:= proc(n, i, p) option remember; expand(`if`(n=0 or i=1,
      (p+n)!/n!*x^n, add(b(n-i*j, i-1, p+j)*(i-1)!^j*combinat
      [multinomial](n, n-i*j, i$j)/j!^2*x^j, j=0..n/i)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2, 0)):
seq(T(n), n=0..12);

multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, p_] := b[n, i, p] = Expand[If[n == 0 || i == 1, (p + n)!/n!*x^n, Sum[b[n - i*j, i - 1, p + j]*(i - 1)!^j*multinomial[n, Join[{n - i*j}, Table[i, j]]]/j!^2*x^j, {j, 0, n/i}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, n, 0]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Apr 28 2018, after Alois P. Heinz *)

A285919 Number of ordered set partitions of [n] into four blocks such that equal-sized blocks are ordered with increasing least elements.

Original entry on oeis.org

1, 40, 350, 3080, 17129, 82488, 464650, 1901680, 8357426, 35701952, 159721016, 627687060, 2642405289, 10712590392, 45568675202, 178738923440, 736145997686, 2946913512648, 12311241803256, 48275516469180, 197284995875314, 786939537437440, 3254422571085400

Offset: 4

Alois P. Heinz, Apr 28 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 4..700
Wikipedia, Partition of a set

Column k=4 of A285824.

Cf. A285855.

b:= proc(n, i, p) option remember; series(`if`(n=0 or i=1,
      (p+n)!/n!*x^n, add(x^j*b(n-i*j, i-1, p+j)*combinat
      [multinomial](n, n-i*j, i$j)/j!^2, j=0..n/i)), x, 5)
    end:
a:= n-> coeff(b(n$2, 0), x, 4):
seq(a(n), n=4..30);

multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, p_] := b[n, i, p] = Series[If[n == 0 || i == 1, (p + n)!/n!*x^n, Sum[x^j*b[n - i*j, i - 1, p + j]*multinomial[n, Join[{n - i*j}, Table[i, j]]]/j!^2, {j, 0, n/i}]], {x, 0, 5}];
a[n_] := Coefficient[b[n, n, 0], x, 4];
Table[a[n], {n, 4, 30}] (* Jean-François Alcover, May 17 2018, translated from Maple *)

cp's OEIS Frontend

A285849 Number T(n,k) of permutations of [n] with k ordered cycles such that equal-sized cycles are ordered with increasing least elements; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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