A240674 -id:A240674 - cp's OEIS frontend

A240181 Array: t(n,k) is the number of partitions p of n such that the number of distinct numbers in the intersection of p and its conjugate is k, for k >= 0, n >= 1.

0, 1, 2, 2, 0, 1, 2, 3, 2, 4, 1, 4, 6, 0, 1, 4, 8, 3, 8, 8, 5, 1, 10, 9, 11, 10, 22, 8, 1, 1, 14, 22, 17, 3, 18, 34, 19, 5, 1, 18, 50, 21, 12, 26, 60, 34, 13, 2, 30, 74, 52, 19, 0, 1, 36, 105, 57, 29, 4, 44, 120, 93, 34, 5, 1, 60, 144, 128, 40, 13, 64, 186

Offset: 1

Clark Kimberling, Apr 12 2014

First two columns are A240674 and A240675. Sum of numbers in row n is A000041(n), for n >= 1. Number of numbers in row n is A240450(n).

			First 15 rows:
  0 ... 1
  2
  2 ... 0 ... 1
  2 ... 3
  2 ... 4 ... 1
  4 ... 6 ... 0 ...1
  4 ... 8 ... 3
  8 ... 8 ... 5 ... 1
  10 .. 9 ... 11
  10 .. 22 .. 8 ... 1 ... 1
  14 .. 22 .. 17 .. 3
  18 .. 34 .. 19 .. 5 ... 1
  18 .. 50 .. 21 .. 12
  26 .. 60 .. 34 .. 13 .. 2
  30 .. 74 .. 52 .. 19 .. 0 .. 1
In the following table, p and c(p) denote a partition of 6 and its conjugate:
  p ........ c(p)
  6 ........ 111111
  51 ....... 21111
  42 ....... 2211
  411 ...... 3111
  33 ....... 222
  321 ...... 321
  3111 ..... 411
  222 ...... 33
  2211 ..... 42
  21111 .... 51
  111111 ... 6
Let I(p) be number of numbers in the intersection of c and c(p);  Then I(p) = 0 for 4 choices of p, I(p) = 1 for 6 choices, I(p) = 2 for 0 choices, and I(p) = 3 for 1 choice.  Thus, row 6 is 4 6 0 1.

Clark Kimberling, Table of n, a(n) for n = 1..200

Cf. A240674, A240675, A240450.

z = 30; conjugatePartition[part_] := Table[Count[#, _?(# >= i &)], {i, First[#]}] &[part]; c = Map[BinCounts[#, {0, 1 + Max[#]}] &[Map[Length, Map[Intersection[#, conjugatePartition[#]] &, IntegerPartitions[#]]]] &, Range[z]]; Flatten[c]  (* this sequence *)
Table[Length[c[[n]]], {n, 1, z}] (* A240450 *) (* Peter J. C. Moses, Apr 10 2014 *)

Name corrected by Clark Kimberling, Sep 28 2023

A240675 Number of partitions p of n such that exactly one number is in both p and its conjugate.

Original entry on oeis.org

1, 0, 0, 3, 4, 6, 8, 8, 9, 22, 22, 34, 50, 60, 74, 105, 120, 144, 186, 234, 280, 358, 440, 524, 665, 782, 954, 1150, 1354, 1630, 1944, 2258, 2666, 3170, 3728, 4365, 5128, 5976, 6978, 8144, 9488, 10952, 12700, 14716, 16932, 19558, 22434, 25764, 29505, 33782

Offset: 1

Clark Kimberling, Apr 12 2014

Second column of the array at A240181. Multiplicities greater than 1 are not counted; e.g. there is exactly one number that is in both {4,1,1} and {3,1,1,1}.

			a(6) counts these 6 partitions:  51, 42, 411, 3111, 2211, 21111, of which the respective conjugates are 21111, 2211, 3111, 411, 42, 51.

Manfred Scheucher, Table of n, a(n) for n = 1..65
Manfred Scheucher, Sage Script

Cf. A240674, 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}] (* A240674 *)
Table[Count[Map[Length, b[n]], 1], {n, 1, z}]   (* A240675 *)

conjug(v) = {my(m = matrix(#v, vecmax(v))); for (i=1, #v, for (j=1, v[i], m[i, j] = 1;);); vector(vecmax(v), i, sum(j=1, #v, m[j, i]));}
a(n) = {my(v = partitions(n)); my(nb = 0); for (k=1, #v, if (#setintersect(Set(v[k]), Set(conjug(v[k]))) == 1, nb++);); nb;} \\ Michel Marcus, Jun 02 2015

More terms from Manfred Scheucher, Jun 01 2015

cp's OEIS Frontend

A240181 Array: t(n,k) is the number of partitions p of n such that the number of distinct numbers in the intersection of p and its conjugate is k, for k >= 0, n >= 1.

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