A274581 Number T(n,k) of set partitions of [n] with alternating parity of elements and exactly k blocks; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 1, 0, 1, 5, 7, 4, 1, 0, 1, 7, 14, 12, 5, 1, 0, 1, 11, 30, 33, 19, 6, 1, 0, 1, 15, 57, 84, 62, 27, 7, 1, 0, 1, 23, 119, 222, 204, 108, 37, 8, 1, 0, 1, 31, 224, 545, 627, 409, 169, 48, 9, 1, 0, 1, 47, 460, 1425, 2006, 1558, 763, 254, 61, 10, 1

Offset

0,9

Links

Alois P. Heinz, Rows n = 0..35, flattened

Wikipedia, Partition of a set

Formula

Sum_{k=0..n} k * T(n,k) = A305823(n).

Example

T(5,1) = 1: 12345.
T(5,2) = 5: 1234|5, 123|45, 12|345, 145|23, 1|2345.
T(5,3) = 7: 123|4|5, 12|34|5, 12|3|45, 1|234|5, 145|2|3, 1|2|345, 1|23|45.
T(5,4) = 4: 12|3|4|5, 1|23|4|5, 1|2|34|5, 1|2|3|45.
T(5,5) = 1: 1|2|3|4|5.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1,  1;
  0, 1,  2,   1;
  0, 1,  3,   3,   1;
  0, 1,  5,   7,   4,   1;
  0, 1,  7,  14,  12,   5,   1;
  0, 1, 11,  30,  33,  19,   6,   1;
  0, 1, 15,  57,  84,  62,  27,   7,  1;
  0, 1, 23, 119, 222, 204, 108,  37,  8, 1;
  0, 1, 31, 224, 545, 627, 409, 169, 48, 9, 1;
  ...

Maple

b:= proc(l, i, t) option remember; `if`(l=[], x,
     `if`(l[1]=t, 0, expand(x*b(subsop(1=[][], l), 1, 1-t)
       ))+add(`if`(l[j]=t, 0, b(subsop(j=[][], l), j, 1-t)
       ), j=i..nops(l)))
    end:
T:= n-> `if`(n=0, 1, (p-> seq(coeff(p, x, j), j=0..n))(
         b([seq(irem(i, 2), i=2..n)], 1$2))):
seq(T(n), n=0..12);

Mathematica

b[l_, i_, t_] := b[l, i, t] = If[l == {}, x, If[l[[1]] == t, 0, Expand[x*b[Rest[l], 1, 1 - t]]] + Sum[If[l[[j]] == t, 0, b[Delete[l, j], j, 1 - t]], {j, i, Length[l]}]];
T[n_] := If[n==0, {1}, Function[p, Table[Coefficient[p, x, j], {j, 0, n}]][ b[Table[Mod[i, 2], {i, 2, n}], 1, 1]]];
Flatten[Table[T[n], {n, 0, 12}]] (* Jean-François Alcover, May 27 2018, from Maple *)

Crossrefs

Columns k=0-10 give: A000007, A057427, A052955(n-2) for n>1, A305777, A305778, A305779, A305780, A305781, A305782, A305783, A305784.

Diagonals include A000012, A001477, A077043.

Row sums give A274547.

T(n,ceiling(n/2)) gives A305785.

Cf. A124419, A274310 (parities alternate within blocks), A305823.

Sequence in context: A277504 A167763 A277666 * A353279 A321919 A321918

Adjacent sequences: A274578 A274579 A274580 * A274582 A274583 A274584

Keyword

nonn,tabl

Author

Alois P. Heinz, Jun 29 2016

Status

approved