

A236543


Number of partitions of n for which (number of occurrences of the least part) = (number of occurrences of greatest part).


3



1, 2, 3, 4, 4, 8, 6, 11, 12, 18, 16, 32, 27, 44, 51, 70, 73, 114, 116, 169, 192, 250, 282, 391, 432, 559, 657, 831, 952, 1223, 1395, 1747, 2040, 2492, 2910, 3580, 4130, 5007, 5857, 7040, 8171, 9813, 11372, 13550, 15771, 18625, 21611, 25549, 29540, 34706
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OFFSET

1,2


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.


LINKS

Table of n, a(n) for n=1..50.


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

Cf. A236544, A236545, A000041.
Sequence in context: A240219 A028298 A047966 * A224038 A241037 A097093
Adjacent sequences: A236540 A236541 A236542 * A236544 A236545 A236546


KEYWORD

nonn,easy


AUTHOR

Clark Kimberling, Jan 28 2014


STATUS

approved



