A363519 Number T(n,k) of partitions of [n] having exactly k parity changes within the partition, n>=0, 0<=k<=max(0,n-1), read by rows.

1, 1, 0, 2, 0, 1, 4, 0, 3, 4, 8, 0, 2, 18, 14, 18, 0, 7, 27, 87, 42, 40, 0, 5, 102, 162, 360, 147, 101, 0, 20, 179, 866, 931, 1456, 434, 254, 0, 15, 675, 1746, 5836, 4755, 5778, 1619, 723, 0, 67, 1321, 9087, 16416, 36031, 22893, 23052, 5044, 2064, 0, 52, 5216, 19863, 93452, 117172, 206570, 115178, 94210, 20271, 6586

Offset

0,4

Comments

The blocks are ordered with increasing least elements.

Links

Alois P. Heinz, Rows n = 0..27, flattened

Wikipedia, Partition of a set

Formula

Sum_{k=0..max(0,n-1)} k * T(n,k) = A363549(n).

Example

T(4,1) = 3: 134|2, 13|24, 13|2|4.
T(4,2) = 4: 124|3, 14|23, 14|2|3, 1|24|3.
T(4,3) = 8: 1234, 123|4, 12|34, 12|3|4, 1|234, 1|23|4, 1|2|34, 1|2|3|4.
T(5,2) = 18: 1245|3, 124|35, 124|3|5, 134|25, 134|2|5, 13|245, 13|24|5, 13|2|45, 13|2|4|5, 14|235, 14|23|5, 14|25|3, 14|2|35, 14|2|3|5, 15|24|3, 1|245|3, 1|24|35, 1|24|3|5.
T(5,4) = 18: 12345, 1234|5, 123|45, 123|4|5, 12|345, 12|34|5, 12|3|45, 12|3|4|5, 145|23, 1|2345, 1|234|5, 1|23|45, 1|23|4|5, 145|2|3, 1|2|345, 1|2|34|5, 1|2|3|45, 1|2|3|4|5.
Triangle T(n,k) begins:
  1;
  1;
  0,  2;
  0,  1,    4;
  0,  3,    4,    8;
  0,  2,   18,   14,    18;
  0,  7,   27,   87,    42,    40;
  0,  5,  102,  162,   360,   147,   101;
  0, 20,  179,  866,   931,  1456,   434,   254;
  0, 15,  675, 1746,  5836,  4755,  5778,  1619,  723;
  0, 67, 1321, 9087, 16416, 36031, 22893, 23052, 5044, 2064;
  ...

Maple

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

Mathematica

b[l_, i_, t_] := b[l, i, t] = Expand[If[l == {}, 1, Sum[Function[f, b[ReplacePart[l, j -> Nothing], j, If[f, 1 - t, t]]*If[f, x, 1]][l[[j]] == t], {j, Join[{1}, Range[i, Length@l]]}]]];
T[n_] := CoefficientList[b[ Table[Mod[i, 2], {i, 2, n}], 1, 0], x];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Nov 17 2023, after Alois P. Heinz *)

Crossrefs

Column k=1 gives A363550.

Row sums give A000110.

T(n,max(0,n-1)) gives A274547.

Cf. A363493, A363549.

Sequence in context: A081265 A039991 A273821 * A108643 A133838 A182138

Adjacent sequences: A363516 A363517 A363518 * A363520 A363521 A363522

Keyword

nonn,tabf

Author

Alois P. Heinz, Jun 07 2023

Status

approved