%I #10 May 19 2019 22:09:49

%S 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,

%T 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,

%U 13,20,9,5,3,2,1,1,1,0,0,1,18,25,12,8,5,2

%N 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.

%C 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.

%e The first 8 rows:

%e 1

%e 0 1

%e 0 1 1

%e 1 1 0 1

%e 1 2 1 0 1

%e 2 3 1 0 0 1

%e 3 3 2 2 0 0 1

%e 4 6 2 1 1 0 0 1

%e 5 8 4 1 2 1 0 0 1

%e For n = 5, P consists of these partitions:

%e [5], with reversal [5], thus, 1 stay number

%e [4,1], with reversal [1,4], thus 0 stay numbers

%e [3,2], with reversal [2,3], thus 0 stay numbers

%e [2,2,1], with reversal [1,2,2], thus 1 stay number

%e [2,1,1,1], with reversal [1,1,1,2], thus 2 stay numbers

%e [1,1,1,1,1], thus, 5 stay numbers.

%e As a result, row 5 of the array is 2 3 1 0 0 1

%t Map[BinCounts[#, {0, Last[#] + 1, 1}] &, Map[Map[Count[#, 0] &, # - Map[Reverse, #] &[IntegerPartitions[#]]] &, Range[0, 35]]]

%t (* _Peter J. C. Moses_, May 14 2019 *)

%Y Cf. A000041, A008483, A238479.

%K nonn,tabl,easy

%O 1,12

%A _Clark Kimberling_, May 16 2019