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

1,3

Comments

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).

Links

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

Example

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.

Mathematica

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 *)

Crossrefs

Cf. A240674, A240675, A240450.

Sequence in context: A327188 A330270 A029341 * A079070 A354787 A096994

Adjacent sequences: A240178 A240179 A240180 * A240182 A240183 A240184

Keyword

nonn,easy,tabf

Author

Clark Kimberling, Apr 12 2014

Extensions

Name corrected by Clark Kimberling, Sep 28 2023

Status

approved