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

Author

Alois P. Heinz, Jun 07 2023

Keywords

• nonn
• tabf

Comments

The blocks are ordered with increasing least elements.

Examples

			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;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..27, flattened
• Wikipedia, Partition of a set

Crossrefs

Column k=1 gives A363550.

Row sums give A000110.

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

Cf. A363493, A363549.

Programs

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 *)

Formula

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