A363493 Number T(n,k) of partitions of [n] having exactly k parity changes within their blocks, n>=0, 0<=k<=max(0,n-1), read by rows.

1, 1, 1, 1, 2, 2, 1, 4, 6, 4, 1, 10, 18, 17, 6, 1, 25, 61, 68, 38, 10, 1, 75, 210, 292, 202, 83, 14, 1, 225, 778, 1252, 1116, 576, 170, 22, 1, 780, 3008, 5670, 5928, 3899, 1490, 341, 30, 1, 2704, 12219, 26114, 32382, 25320, 12655, 3856, 678, 46, 1, 10556, 52268, 126073, 177666, 163695, 98282, 39230, 9418, 1319, 62, 1

Offset

0,5

Links

Alois P. Heinz, Rows n = 0..150, flattened

Wikipedia, Partition of a set

Formula

Sum_{k=0..max(0,n-1)} k * T(n,k) = A363496(n).

Example

T(4,0) = 4: 13|24, 13|2|4, 1|24|3, 1|2|3|4.
T(4,1) = 6: 124|3, 12|3|4, 134|2, 1|23|4, 14|2|3, 1|2|34.
T(4,2) = 4: 123|4, 12|34, 14|23, 1|234.
T(4,3) = 1: 1234.
T(5,2) = 17: 1235|4, 123|4|5, 1245|3, 12|34|5, 125|3|4, 12|3|45, 1345|2, 134|25, 14|235, 14|23|5, 15|234, 1|234|5, 1|23|45, 145|2|3, 14|25|3, 1|25|34, 1|2|345.
Triangle T(n,k) begins:
     1;
     1;
     1,     1;
     2,     2,     1;
     4,     6,     4,     1;
    10,    18,    17,     6,     1;
    25,    61,    68,    38,    10,     1;
    75,   210,   292,   202,    83,    14,    1;
   225,   778,  1252,  1116,   576,   170,   22,   1;
   780,  3008,  5670,  5928,  3899,  1490,  341,  30,  1;
  2704, 12219, 26114, 32382, 25320, 12655, 3856, 678, 46, 1;
  ...

Maple

b:= proc(n, x, y) option remember; `if`(n=0, 1,
     `if`(y=0, 0, expand(b(n-1, y-1, x+1)*y*z))+
        b(n-1, y, x)*x + b(n-1, y, x+1))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(n, 0$2)):
seq(T(n), n=0..12);

Mathematica

b[n_, x_, y_] := b[n, x, y] = If[n == 0, 1,
  If[y == 0, 0, Expand[b[n - 1, y - 1, x + 1]*y*z]] +
  b[n - 1, y, x]*x + b[n - 1, y, x + 1]];
T[n_] := CoefficientList[b[n, 0, 0], z];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Sep 05 2023, after Alois P. Heinz *)

Crossrefs

Columns k=0-2 give: A124419, A363511, A363588.

Row sums give A000110.

T(n+1,n) gives A000012.

T(n+2,n) gives A027383.

T(2n+1,n) gives A363495.

Cf. A152874, A363496, A363519.

Sequence in context: A228336 A111062 A369632 * A193597 A191490 A061598

Adjacent sequences: A363490 A363491 A363492 * A363494 A363495 A363496

Keyword

nonn,tabf

Author

Alois P. Heinz, Jun 05 2023

Status

approved