A308151 Triangular array: each row partitions the partitions of n into n parts; of which the k-th part is the number of partitions having stay number k-1; see Comments.

1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 2, 3, 1, 0, 0, 1, 3, 3, 2, 2, 0, 0, 1, 4, 6, 2, 1, 1, 0, 0, 1, 5, 8, 4, 1, 2, 1, 0, 0, 1, 8, 10, 4, 4, 1, 1, 1, 0, 0, 1, 10, 14, 8, 3, 2, 2, 1, 1, 0, 0, 1, 13, 20, 9, 5, 3, 2, 1, 1, 1, 0, 0, 1, 18, 25, 12, 8, 5, 2

Offset

1,12

Comments

The stay number of a partition P is defined as follows. Let U be the ordering of the parts of P in nonincreasing order, and let V be the reverse of U. The stay number of P is the number of numbers whose position in V is the same as in U. (1st column) = A238479. When the rows of the array are read in reverse order, it appears that the limiting sequence is A008483.

Links

Table of n, a(n) for n=1..84.

Example

The first 8 rows:
  1
  0   1
  0   1   1
  1   1   0   1
  1   2   1   0   1
  2   3   1   0   0   1
  3   3   2   2   0   0   1
  4   6   2   1   1   0   0   1
  5   8   4   1   2   1   0   0   1
For n = 5, P consists of these partitions:
[5], with reversal [5], thus, 1 stay number
[4,1], with reversal [1,4], thus 0 stay numbers
[3,2], with reversal [2,3], thus 0 stay numbers
[2,2,1], with reversal [1,2,2], thus 1 stay number
[2,1,1,1], with reversal [1,1,1,2], thus 2 stay numbers
[1,1,1,1,1], thus, 5 stay numbers.
As a result, row 5 of the array is 2 3 1 0 0 1

Mathematica

Map[BinCounts[#, {0, Last[#] + 1, 1}] &,  Map[Map[Count[#, 0] &, # - Map[Reverse, #] &[IntegerPartitions[#]]] &, Range[0, 35]]]
  (* Peter J. C. Moses, May 14 2019 *)

Crossrefs

Cf. A000041, A008483, A238479.

Sequence in context: A168318 A174203 A353642 * A114219 A119339 A037835

Adjacent sequences: A308148 A308149 A308150 * A308152 A308153 A308154

Keyword

nonn,tabl,easy

Author

Clark Kimberling, May 16 2019

Status

approved