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A236545
Number of partitions of n for which (number of occurrences of the least part) < (number of occurrences of greatest part).
3
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, 177, 197, 261, 273, 365, 396, 500, 563, 696, 765, 967, 1077, 1305, 1472, 1794, 2000, 2428, 2725, 3246, 3695, 4377, 4920, 5847, 6607, 7746, 8788, 10284, 11613, 13559
OFFSET
1,7
COMMENTS
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.
EXAMPLE
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.
MATHEMATICA
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];
t1 = Table[t[[3 k - 2]], {k, 1, z}] (* A236543 *)
t2 = Table[t[[3 k - 1]], {k, 1, z}] (* A236544 *)
t3 = Table[t[[3 k]], {k, 1, z}] (* A236545 *)
(* Peter J. C. Moses, Jan 28 2014 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jan 28 2014
STATUS
approved