A325181 Number of integer partitions of n such that the difference between the length of the minimal square containing and the maximal square contained in the Young diagram is 1.

0, 0, 2, 1, 0, 2, 3, 2, 1, 0, 2, 3, 4, 3, 2, 1, 0, 2, 3, 4, 5, 4, 3, 2, 1, 0, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, 0, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 2, 1, 0, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 3, 2, 1, 0, 2, 3, 4, 5, 6, 7, 8, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 2, 3, 4, 5, 6

Offset: 0

Author

Gus Wiseman, Apr 08 2019

Keywords

• nonn
• look

Comments

The maximal square contained in the Young diagram of an integer partition is called its Durfee square, and its length is the rank of the partition.

Examples

			The a(2) = 2 through a(15) = 1 partitions:
(2)  (21) (32)  (33)  (322) (332) (433)  (443)  (444)  (4333) (4433) (4443)
(11)      (221) (222) (331)       (3331) (3332) (3333) (4432) (4442)
                (321)                    (4331) (4332) (4441)
                                                (4431)
		

Links

• Giovanni Resta, Table of n, a(n) for n = 0..150

Crossrefs

Cf. A006918, A084835, A096771, A257990, A263297, A325178, A325179, A325182, A325191, A325192, A325198.

Programs

Mathematica

durf[ptn_]:=Length[Select[Range[Length[ptn]],ptn[[#]]>=#&]];
codurf[ptn_]:=Max[Length[ptn],Max[ptn]];
Table[Length[Select[IntegerPartitions[n],codurf[#]-durf[#]==1&]],{n,0,30}]

Extensions

More terms from Giovanni Resta, Apr 15 2019