A240180 Number of partitions of n such that (least part) = (multiplicity of greatest part).

0, 1, 0, 1, 3, 3, 5, 7, 12, 16, 24, 30, 45, 57, 81, 104, 143, 179, 243, 304, 399, 504, 650, 809, 1039, 1286, 1622, 2006, 2508, 3077, 3822, 4666, 5747, 6995, 8552, 10353, 12603, 15189, 18371, 22071, 26570, 31785, 38104, 45419, 54213, 64426, 76596, 90710

Offset

0,5

Comments

Also the number of partitions p of n such that min(p) = min(conjugate(p)). Example:a(7) counts these 7 partitions: 61, 511, 421, 4111, 3211, 31111, 211111, of which the respective conjugates are 211111, 31111, 3211, 4111, 421, 511, 61. - Clark Kimberling, Apr 11 2014

Links

Table of n, a(n) for n=0..47.

Formula

a(n) = A240179(n) - A240178(n), for n >= 0.

a(n) + 2*A240178(n) = A000041(n) for n >= 0.

Example

a(6) counts these 5 partitions:  51, 411, 321, 3111, 21111.

Mathematica

z = 60; f[n_] := f[n] = IntegerPartitions[n];
Table[Count[f[n], p_ /; Min[p] < Count[p, Max[p]]], {n, 0, z}]  (* A240178 *)
Table[Count[f[n], p_ /; Min[p] <= Count[p, Max[p]]], {n, 0, z}] (* A240179 *)
Table[Count[f[n], p_ /; Min[p] == Count[p, Max[p]]], {n, 0, z}] (* A240180 *)
Table[Count[f[n], p_ /; Min[p] > Count[p, Max[p]]], {n, 0, z}]  (* A240178, n>0 *)
Table[Count[f[n], p_ /; Min[p] >= Count[p, Max[p]]], {n, 0, z}] (* A240179, n>0 *)

Crossrefs

Cf. A240178, A240179, A000041.

Sequence in context: A001588 A107029 A352912 * A169942 A215777 A147095

Adjacent sequences: A240177 A240178 A240179 * A240181 A240182 A240183

Keyword

nonn,easy

Author

Clark Kimberling, Apr 02 2014

Status

approved