A114701 -id:A114701 - cp's OEIS frontend

A240674 Number of partitions p of n that are disjoint from their conjugate.

1, 0, 2, 2, 2, 2, 4, 4, 8, 10, 10, 14, 18, 18, 26, 30, 36, 44, 60, 64, 82, 96, 114, 130, 164, 176, 222, 248, 296, 338, 406, 450, 550, 620, 726, 816, 968, 1074, 1270, 1418, 1648, 1836, 2150, 2382, 2758, 3080, 3534, 3942, 4538, 5034, 5778, 6416, 7312, 8136, 9258

Offset: 0

Clark Kimberling, Apr 12 2014

First column of the array at A240181.

			a(6) counts these 4 partitions:  6, 33, 222, 111111, of which the respective conjugates are 111111, 222, 33, 6.

Cf. A240675, A114701, A240181.

z = 30; p[n_, k_] := p[n, k] = IntegerPartitions[n][[k]]; c[p_] := c[p] = Table[Count[#, ?(# >= i &)], {i, First[#]}] &[p]; b[n] := b[n] = Table[Intersection[p[n, k], c[p[n, k]]], {k, 1, PartitionsP[n]}]; Table[Count[Map[Length, b[n]], 0], {n, 1, z}] (* this sequence *)
Table[Count[Map[Length, b[n]], 1], {n, 1, z}]   (* A240675 *)

a(n) = 2*A114701(n), for n >= 1.

Name corrected by Clark Kimberling, Sep 28 2023

a(0)=1 prepended by Alois P. Heinz, Jul 19 2024

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

Original entry on oeis.org

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

Alois P. Heinz, Jul 19 2024

nonn

			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.

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

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

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);

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

cp's OEIS Frontend

A240674 Number of partitions p of n that are disjoint from their conjugate.

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