A236545 Number of partitions of n for which (number of occurrences of the least part) < (number of occurrences of greatest part).

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.

Links

Table of n, a(n) for n=1..55.

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. A236543, A236545, A000041.

Sequence in context: A110880 A372071 A195952 * A321327 A370898 A370904

Adjacent sequences: A236542 A236543 A236544 * A236546 A236547 A236548

Keyword

nonn,easy

Author

Clark Kimberling, Jan 28 2014

Status

approved