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A325182
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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.
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5
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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
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OFFSET
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0,4
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COMMENTS
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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.
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REFERENCES
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Richard P. Stanley, Enumerative Combinatorics, Volume 2, Cambridge University Press, 1999, p. 289.
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LINKS
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EXAMPLE
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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
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MATHEMATICA
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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}]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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