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.

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

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

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Wikipedia, Durfee square.

Formula

Sum_{k=1..n} k*T(n,k) = A368985(n) - A115995(n). - Andrew Howroyd, Jan 12 2024

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}]

Prog

(PARI) row(n)={my(r=vector(n+1)); if(n==0, r[1]=1, forpart(p=n, my(c=1); while(c<#p && c<p[#p-c], c++); r[max(#p, p[#p])-c+1]++)); r} \\ Andrew Howroyd, Jan 12 2024

Crossrefs

Row sums are A000041. Column k = 1 is A325181. Column k = 2 is A325182.

Cf. A096771, A257990, A263297, A325178, A325179, A325180, A325200.

Cf. A115995, A368985.

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