A285853 -id:A285853 - 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 *)

A285917 Number of ordered set partitions of [n] into two blocks such that equal-sized blocks are ordered with increasing least elements.

Original entry on oeis.org

1, 6, 11, 30, 52, 126, 219, 510, 896, 2046, 3632, 8190, 14666, 32766, 59099, 131070, 237832, 524286, 956196, 2097150, 3841586, 8388606, 15425136, 33554430, 61908562, 134217726, 248377154, 536870910, 996183062, 2147483646, 3994427099, 8589934590, 16013066072

Offset: 2

Alois P. Heinz, Apr 28 2017

nonn

a(n) is odd if and only if n = 2^k with k>0.

Alois P. Heinz, Table of n, a(n) for n = 2..1000
Wikipedia, Partition of a set

Column k=2 of A285824.

Cf. A285853.

a:= n-> 2*add(binomial(n, k), k=1..n/2)-
        `if`(n::even, 3/2*binomial(n, n/2), 0):
seq(a(n), n=2..40);
# second Maple program:
a:= proc(n) option remember; `if`(n<5, [0, 1, 6, 11][n],
      (9*(n-1)*(n-4)*a(n-1)+2*(3*n^2-16*n+6)*a(n-2)
      -36*(n-2)*(n-4)*a(n-3)+8*(n-3)*(3*n-10)*a(n-4))
      /((3*n-13)*n))
    end:
seq(a(n), n=2..40);

a[n_] := 2*Sum[Binomial[n, k], {k, 1, n/2}] - If[EvenQ[n], 3/2*Binomial[n, n/2], 0];
Table[a[n], {n, 2, 40}] (* Jean-François Alcover, May 26 2018, from Maple *)

a(n) = 2*sum(k=1, n\2, binomial(n, k)) - if (!(n%2), 3*binomial(n, n/2)/2); \\ Michel Marcus, May 26 2018

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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