A363454 -id:A363454 - cp's OEIS frontend

A363451 Number of partitions of [n] such that the number of blocks containing only odd elements equals the number of blocks containing only even elements.

1, 0, 2, 2, 9, 23, 99, 353, 1778, 7927, 45273, 238797, 1526331, 9215950, 65020448, 439742641, 3388075807, 25270974635, 210763775071, 1713657668021, 15359474721088, 134902169999841, 1291589459223627, 12165062702520422, 123780591852786693, 1242763745129587332

Offset: 0

Alois P. Heinz, Jun 02 2023

nonn

			a(0) = 1: () the empty partition.
a(1) = 0.
a(2) = 2: 12, 1|2.
a(3) = 2: 123, 13|2.
a(4) = 9: 1234, 12|34, 12|3|4, 13|24, 14|23, 1|23|4, 14|2|3, 1|2|34, 1|2|3|4.
a(5) = 23: 12345, 123|45, 123|4|5, 125|34, 12|345, 125|3|4, 12|35|4, 134|25, 134|2|5, 135|24, 13|25|4, 13|2|45, 13|2|4|5, 145|23, 14|235, 15|23|4, 1|235|4, 145|2|3, 14|2|35, 15|2|34, 1|2|345, 15|2|3|4, 1|2|35|4.

Alois P. Heinz, Table of n, a(n) for n = 0..250
Wikipedia, Partition of a set

Cf. A000110, A363434, A363435, A363454.

b:= proc(n, x, y, m) option remember; `if`(n=0, `if`(x=y, 1, 0),
      `if`(x+m>0, b(n-1, y, x, m)*(x+m), 0)+b(n-1, y, x+1, m)+
      `if`(y>0, b(n-1, y-1, x, m+1)*y, 0))
    end:
a:= n-> b(n, 0$3):
seq(a(n), n=0..28);

b[n_, x_, y_, m_] := b[n, x, y, m] = If[n == 0, If[x == y, 1, 0], If[x + m > 0, b[n - 1, y, x, m]*(x + m), 0] + b[n - 1, y, x + 1, m] + If[y > 0, b[n - 1, y - 1, x, m + 1]*y, 0]];
a[n_] := b[n, 0, 0, 0];
Table[a[n], {n, 0, 28}] (* Jean-François Alcover, Oct 20 2023, after Alois P. Heinz *)

A363472 Total number of blocks in all partitions of [n] where each block has at least one odd element and at least one even element.

Original entry on oeis.org

0, 0, 1, 1, 5, 13, 55, 193, 941, 4081, 22351, 113761, 694565, 4030153, 27107095, 175738753, 1289775821, 9209233921, 73147903471, 568928274961, 4857161139365, 40796613003433, 372190216061335, 3352314486348433, 32518958606637101, 312271731474218881

Offset: 0

Alois P. Heinz, Jun 05 2023

nonn

All positive terms are odd.

			a(2) = 1: 12.
a(3) = 1: 123.
a(4) = 5 = 1 + 2 + 2: 1234, 12|34, 14|23.
a(5) = 13 = 1 + 2 + 2 + 2 + 2 + 2 + 2: 12345, 123|45, 125|34, 12|345, 134|25, 145|23, 14|235.

Alois P. Heinz, Table of n, a(n) for n = 0..580
Wikipedia, Partition of a set

Cf. A124425, A362495, A363454.

a:= n-> (h-> add(k*Stirling2(h, k)*Stirling2(n-h, k)*k!, k=0..h))(floor(n/2)):
seq(a(n), n=0..30);

a(n) = Sum_{k=0..floor(n/2)} k * Stirling2(floor(n/2),k) * Stirling2(ceiling(n/2),k) * k!.

cp's OEIS Frontend

A363451 Number of partitions of [n] such that the number of blocks containing only odd elements equals the number of blocks containing only even elements.

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