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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
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.
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
KEYWORD
nonn,tabl,easy
AUTHOR
Clark Kimberling, May 16 2019
STATUS
approved