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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 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.)