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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. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

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.

REFERENCES

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

LINKS

Table of n, a(n) for n=0..67.

Wikipedia, Durfee square.

EXAMPLE

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

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}]

CROSSREFS

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

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

Sequence in context: A230290 A294783 A172021 * A215959 A209555 A059594

Adjacent sequences:  A325179 A325180 A325181 * A325183 A325184 A325185

KEYWORD

nonn,look

AUTHOR

Gus Wiseman, Apr 08 2019

STATUS

approved

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Last modified November 27 14:10 EST 2020. Contains 338683 sequences. (Running on oeis4.)