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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. 0
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 (list; table; graph; refs; listen; history; text; internal format)
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: A065364 A168318 A174203 * A114219 A119339 A037835

Adjacent sequences:  A308148 A308149 A308150 * A308152 A308153 A308154

KEYWORD

nonn,tabl,easy

AUTHOR

Clark Kimberling, May 16 2019

STATUS

approved

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Last modified July 31 01:15 EDT 2021. Contains 346365 sequences. (Running on oeis4.)