A374782 Number of partitions of n that do not have a fixed point that is also a fixed point of the conjugate partition.

1, 0, 2, 3, 4, 5, 8, 11, 17, 23, 33, 43, 60, 77, 104, 134, 177, 226, 295, 373, 480, 604, 766, 957, 1204, 1492, 1860, 2294, 2836, 3477, 4273, 5209, 6362, 7721, 9375, 11326, 13687, 16460, 19799, 23720, 28406, 33901, 40443, 48092, 57159, 67747, 80237, 94799

Offset: 0

Author

Alois P. Heinz, Jul 19 2024

Keywords

• nonn

Examples

			a(0) = 1: the empty partition.
a(2) = 2: 2, 11.
a(3) = 3: 3, 21, 111.
a(4) = 4: 4, 31, 211, 1111.
a(5) = 5: 5, 41, 311, 2111, 11111.
a(6) = 8: 6, 33, 51, 222, 411, 3111, 21111, 111111.
a(7) = 11: 7, 43, 61, 322, 331, 511, 2221, 4111, 31111, 211111, 1111111.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..2500
• Wikipedia, Partition (number theory)

Crossrefs

Cf. A000041, A064428, A114701, A188674, A352828.

Programs

Maple

b:= proc(n, i, p) option remember; `if`(n=0, 1,
     `if`(i<1, 0, b(n, i-1, p)+add(`if`(i=p+j, 0,
      b(n-i*j, min(n-i*j, i-1), p+j)), j=1..n/i)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..47);

Formula

a(n) = A000041(n) - A188674(n) for n > 0, a(0) = 1.