A363549 -id:A363549 - cp's OEIS frontend

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

Alois P. Heinz, Jun 07 2023

The blocks are ordered with increasing least elements.

			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;
  ...

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

Column k=1 gives A363550.
Row sums give A000110.
T(n,max(0,n-1)) gives A274547.
Cf. A363493, A363549.

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);

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 *)

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

A363496 Total number of parity changes within the blocks of all partitions of [n].

Original entry on oeis.org

0, 0, 1, 4, 17, 74, 356, 1808, 9923, 57442, 354407, 2296028, 15704028, 112266048, 841442105, 6564854864, 53413489773, 450789496454, 3950844987040, 35809477617544, 335901221506491, 3250110998386534, 32453151223493139, 333520967584364248, 3528754456836294712

Offset: 0

Alois P. Heinz, Jun 05 2023

nonn

			a(4) = 17 = 6*1 + 4*2 + 1*3: 124|3, 12|3|4, 134|2, 1|23|4, 14|2|3, 1|2|34, 123|4, 12|34, 14|23, 1|234, 1234.

Alois P. Heinz, Table of n, a(n) for n = 0..575
Wikipedia, Partition of a set

Cf. A000110, A077613, A363493, A363549.

b:= proc(n, x, y) option remember; `if`(n=0, [1, 0],
     `if`(y=0, 0, (p-> p+[0, p[1]])(b(n-1, y-1, x+1)*y))+
        b(n-1, y, x)*x + b(n-1, y, x+1))
    end:
a:= n-> b(n, 0$2)[2]:
seq(a(n), n=0..24);

a(n) = Sum_{k=0..max(0,n-1)} k * A363493(n,k).

cp's OEIS Frontend

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.

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