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 A325192 Regular triangle read by rows where T(n,k) is the 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 k. 12
 1, 1, 0, 0, 2, 0, 0, 1, 2, 0, 1, 0, 2, 2, 0, 0, 2, 1, 2, 2, 0, 0, 3, 2, 2, 2, 2, 0, 0, 2, 4, 3, 2, 2, 2, 0, 0, 1, 7, 4, 4, 2, 2, 2, 0, 1, 0, 6, 8, 5, 4, 2, 2, 2, 0, 0, 2, 5, 11, 8, 6, 4, 2, 2, 2, 0, 0, 3, 4, 12, 12, 9, 6, 4, 2, 2, 2, 0, 0, 4, 5, 13, 17, 12, 10, 6, 4, 2, 2, 2, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 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 Wikipedia, Durfee square. EXAMPLE Triangle begins:   1   1  0   0  2  0   0  1  2  0   1  0  2  2  0   0  2  1  2  2  0   0  3  2  2  2  2  0   0  2  4  3  2  2  2  0   0  1  7  4  4  2  2  2  0   1  0  6  8  5  4  2  2  2  0   0  2  5 11  8  6  4  2  2  2  0   0  3  4 12 12  9  6  4  2  2  2  0   0  4  5 13 17 12 10  6  4  2  2  2  0   0  3  9 12 20 18 13 10  6  4  2  2  2  0   0  2 12 15 23 25 18 14 10  6  4  2  2  2  0   0  1 15 19 26 30 26 19 14 10  6  4  2  2  2  0 Row 9 counts the following partitions (empty columns not shown):    333   432    54      63       72        711       81         9          441    522     621      6111      3111111   21111111   111111111          3222   531     51111    411111          3321   5211    222111   2211111          4221   22221   321111          4311   32211                 33111                 42111 MATHEMATICA durf[ptn_]:=Length[Select[Range[Length[ptn]], ptn[[#]]>=#&]]; codurf[ptn_]:=Max[Length[ptn], Max[ptn]]; Table[Length[Select[IntegerPartitions[n], codurf[#]-durf[#]==k&]], {n, 0, 15}, {k, 0, n}] CROSSREFS Row sums are A000041. Column k = 1 is A325181. Column k = 2 is A325182. Cf. A096771, A257990, A263297, A325178, A325179, A325180, A325200. Sequence in context: A331816 A099584 A100563 * A087773 A025867 A078646 Adjacent sequences:  A325189 A325190 A325191 * A325193 A325194 A325195 KEYWORD nonn,tabl AUTHOR Gus Wiseman, Apr 08 2019 STATUS approved

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Last modified December 2 16:46 EST 2020. Contains 338877 sequences. (Running on oeis4.)