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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.

%I A325182 #6 Apr 10 2019 22:02:23
%S A325182 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,
%T A325182 18,17,19,25,31,38,42,46,44,41,36,32,29,28,31,37,46,53,62,66,71,68,65,
%U A325182 58,53,47,44,43,46,54,63,74,83,93,98,103,100,96,88,81
%N 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.
%C A325182 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.
%D A325182 Richard P. Stanley, Enumerative Combinatorics, Volume 2,  Cambridge University Press, 1999, p. 289.
%H A325182 Wikipedia, <a href="https://en.wikipedia.org/wiki/Durfee_square">Durfee square</a>.
%e A325182 The a(3) = 2 through a(14) = 12 partitions:
%e A325182   3    31   311  42    43    44    432   442   533    543    544    554
%e A325182   111  211       2211  421   422   441   3322  4322   4422   553    5333
%e A325182                        2221  431   3222  4222  4421   5331   5332   5432
%e A325182                        3211  2222  3321  4321  33311  33321  5431   5441
%e A325182                              3221  4221  4411         43311  33322  5531
%e A325182                              3311  4311                      33331  33332
%e A325182                              4211                            43321  43322
%e A325182                                                              44311  43331
%e A325182                                                              53311  44321
%e A325182                                                                     44411
%e A325182                                                                     53321
%e A325182                                                                     54311
%t A325182 durf[ptn_]:=Length[Select[Range[Length[ptn]],ptn[[#]]>=#&]];
%t A325182 codurf[ptn_]:=Max[Length[ptn],Max[ptn]];
%t A325182 Table[Length[Select[IntegerPartitions[n],codurf[#]-durf[#]==2&]],{n,0,30}]
%Y A325182 Cf. A006918, A096771, A257990, A263297, A325168.
%Y A325182 Cf. A325178, A325180, A325181, A325182, A325192, A325197, A325199, A325200.
%K A325182 nonn,look
%O A325182 0,4
%A A325182 _Gus Wiseman_, Apr 08 2019