A325182 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 2.

0, 0, 0, 2, 2, 1, 2, 4, 7, 6, 5, 4, 5, 9, 12, 15, 14, 12, 10, 9, 11, 15, 21, 24, 28, 26, 24, 20, 18, 17, 19, 25, 31, 38, 42, 46, 44, 41, 36, 32, 29, 28, 31, 37, 46, 53, 62, 66, 71, 68, 65, 58, 53, 47, 44, 43, 46, 54, 63, 74, 83, 93, 98, 103, 100, 96, 88, 81

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(3) = 2 through a(14) = 12 partitions:
  3    31   311  42    43    44    432   442   533    543    544    554
  111  211       2211  421   422   441   3322  4322   4422   553    5333
                       2221  431   3222  4222  4421   5331   5332   5432
                       3211  2222  3321  4321  33311  33321  5431   5441
                             3221  4221  4411         43311  33322  5531
                             3311  4311                      33331  33332
                             4211                            43321  43322
                                                             44311  43331
                                                             53311  44321
                                                                    44411
                                                                    53321
                                                                    54311
		

References

• Richard P. Stanley, Enumerative Combinatorics, Volume 2, Cambridge University Press, 1999, p. 289.

Links

• Wikipedia, Durfee square.

Crossrefs

Cf. A006918, A096771, A257990, A263297, A325168.

Cf. A325178, A325180, A325181, A325182, A325192, A325197, A325199, A325200.

Programs

Mathematica

durf[ptn_]:=Length[Select[Range[Length[ptn]],ptn[[#]]>=#&]];
codurf[ptn_]:=Max[Length[ptn],Max[ptn]];
Table[Length[Select[IntegerPartitions[n],codurf[#]-durf[#]==2&]],{n,0,30}]