A363496 -id:A363496 - cp's OEIS frontend

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

Alois P. Heinz, Jun 05 2023

			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;
  ...

Alois P. Heinz, Rows n = 0..150, flattened
Wikipedia, Partition of a set

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.

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);

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 *)

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

A363549 Total number of parity changes within all partitions of [n].

Original entry on oeis.org

0, 0, 2, 9, 35, 152, 690, 3476, 18362, 105507, 633439, 4077386, 27317322, 194202164, 1431942154, 11117639073, 89255122863, 750051307896, 6500488706254, 58693723144324, 545325034579386, 5258317906940219, 52072378163037347, 533479904210984930, 5603461526474415150

Offset: 0

Alois P. Heinz, Jun 09 2023

nonn

The blocks are ordered with increasing least elements.

			a(3) = 9 = 1*1 + 4*2: 13|2, 123, 12|3, 1|23, 1|2|3.
a(4) = 35 = 3*1 + 4*2 + 8*3: 134|2, 13|24, 13|2|4, 124|3, 14|23, 14|2|3, 1|24|3, 1234, 123|4, 12|34, 12|3|4, 1|234, 1|23|4, 1|2|34, 1|2|3|4.

Alois P. Heinz, Table of n, a(n) for n = 0..28
Wikipedia, Partition of a set

Cf. A000110, A363496, A363519.

a(n) = Sum_{k=0..max(0,n-1)} k * A363519(n,k).

cp's OEIS Frontend

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.

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