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%I #9 Jun 24 2015 00:44:19
%S 0,0,0,0,1,0,2,2,3,3,8,4,12,11,16,17,28,25,43,40,58,64,91,87,129,138,
%T 177,197,261,273,365,396,500,563,696,765,967,1077,1305,1472,1794,2000,
%U 2428,2725,3246,3695,4377,4920,5847,6607,7746,8788,10284,11613,13559
%N Number of partitions of n for which (number of occurrences of the least part) < (number of occurrences of greatest part).
%C The partitions of n are partitioned by the partitions counted by A236543, A236544, A236545 (see Example); consequently, A000041(n) = A236543(n) + A236544(n) + A236545(n) for n >= 1.
%e Among the 15 partitions of 7, the following 6 have #(occurrences of least part) = #(occurrences of greatest part): 7, 61, 52, 43, 421, 111111; the following 7 have " > " in place of " = ": 511, 4111, 322, 3211, 31111, 22111, 211111; and the remaining 2, have " < ": 331, 221.
%t z = 65; s = Map[Map[Length, {Select[#, First[#] == Last[#] &], Select[#, First[#] > Last[#] &], Select[#, First[#] < Last[#] &]} &[Map[{Count[#, Min[#]], Count[#, Max[#]]} &, IntegerPartitions[#]]]] &, Range[z]]; t = Flatten[s];
%t t1 = Table[t[[3 k - 2]], {k, 1, z}] (* A236543 *)
%t t2 = Table[t[[3 k - 1]], {k, 1, z}] (* A236544 *)
%t t3 = Table[t[[3 k]], {k, 1, z}] (* A236545 *)
%t (* _Peter J. C. Moses_, Jan 28 2014 *)
%Y Cf. A236543, A236545, A000041.
%K nonn,easy
%O 1,7
%A _Clark Kimberling_, Jan 28 2014
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