Identifier
Identifier
Values
[] generating graphics... => 0
[1] generating graphics... => 1
[2] generating graphics... => 1
[1,1] generating graphics... => 1
[3] generating graphics... => 1
[2,1] generating graphics... => 1
[1,1,1] generating graphics... => 1
[4] generating graphics... => 1
[3,1] generating graphics... => 1
[2,2] generating graphics... => 2
[2,1,1] generating graphics... => 1
[1,1,1,1] generating graphics... => 1
[5] generating graphics... => 1
[4,1] generating graphics... => 1
[3,2] generating graphics... => 2
[3,1,1] generating graphics... => 1
[2,2,1] generating graphics... => 2
[2,1,1,1] generating graphics... => 1
[1,1,1,1,1] generating graphics... => 1
[6] generating graphics... => 1
[5,1] generating graphics... => 1
[4,2] generating graphics... => 2
[4,1,1] generating graphics... => 1
[3,3] generating graphics... => 2
[3,2,1] generating graphics... => 2
[3,1,1,1] generating graphics... => 1
[2,2,2] generating graphics... => 2
[2,2,1,1] generating graphics... => 2
[2,1,1,1,1] generating graphics... => 1
[1,1,1,1,1,1] generating graphics... => 1
[7] generating graphics... => 1
[6,1] generating graphics... => 1
[5,2] generating graphics... => 2
[5,1,1] generating graphics... => 1
[4,3] generating graphics... => 2
[4,2,1] generating graphics... => 2
[4,1,1,1] generating graphics... => 1
[3,3,1] generating graphics... => 2
[3,2,2] generating graphics... => 2
[3,2,1,1] generating graphics... => 2
[3,1,1,1,1] generating graphics... => 1
[2,2,2,1] generating graphics... => 2
[2,2,1,1,1] generating graphics... => 2
[2,1,1,1,1,1] generating graphics... => 1
[1,1,1,1,1,1,1] generating graphics... => 1
[8] generating graphics... => 1
[7,1] generating graphics... => 1
[6,2] generating graphics... => 2
[6,1,1] generating graphics... => 1
[5,3] generating graphics... => 2
[5,2,1] generating graphics... => 2
[5,1,1,1] generating graphics... => 1
[4,4] generating graphics... => 2
[4,3,1] generating graphics... => 2
[4,2,2] generating graphics... => 2
[4,2,1,1] generating graphics... => 2
[4,1,1,1,1] generating graphics... => 1
[3,3,2] generating graphics... => 2
[3,3,1,1] generating graphics... => 2
[3,2,2,1] generating graphics... => 2
[3,2,1,1,1] generating graphics... => 2
[3,1,1,1,1,1] generating graphics... => 1
[2,2,2,2] generating graphics... => 2
[2,2,2,1,1] generating graphics... => 2
[2,2,1,1,1,1] generating graphics... => 2
[2,1,1,1,1,1,1] generating graphics... => 1
[1,1,1,1,1,1,1,1] generating graphics... => 1
[9] generating graphics... => 1
[8,1] generating graphics... => 1
[7,2] generating graphics... => 2
[7,1,1] generating graphics... => 1
[6,3] generating graphics... => 2
[6,2,1] generating graphics... => 2
[6,1,1,1] generating graphics... => 1
[5,4] generating graphics... => 2
[5,3,1] generating graphics... => 2
[5,2,2] generating graphics... => 2
[5,2,1,1] generating graphics... => 2
[5,1,1,1,1] generating graphics... => 1
[4,4,1] generating graphics... => 2
[4,3,2] generating graphics... => 2
[4,3,1,1] generating graphics... => 2
[4,2,2,1] generating graphics... => 2
[4,2,1,1,1] generating graphics... => 2
[4,1,1,1,1,1] generating graphics... => 1
[3,3,3] generating graphics... => 3
[3,3,2,1] generating graphics... => 2
[3,3,1,1,1] generating graphics... => 2
[3,2,2,2] generating graphics... => 2
[3,2,2,1,1] generating graphics... => 2
[3,2,1,1,1,1] generating graphics... => 2
[3,1,1,1,1,1,1] generating graphics... => 1
[2,2,2,2,1] generating graphics... => 2
[2,2,2,1,1,1] generating graphics... => 2
[2,2,1,1,1,1,1] generating graphics... => 2
[2,1,1,1,1,1,1,1] generating graphics... => 1
[1,1,1,1,1,1,1,1,1] generating graphics... => 1
[10] generating graphics... => 1
[9,1] generating graphics... => 1
[8,2] generating graphics... => 2
[8,1,1] generating graphics... => 1
[7,3] generating graphics... => 2
[7,2,1] generating graphics... => 2
[7,1,1,1] generating graphics... => 1
[6,4] generating graphics... => 2
[6,3,1] generating graphics... => 2
[6,2,2] generating graphics... => 2
[6,2,1,1] generating graphics... => 2
[6,1,1,1,1] generating graphics... => 1
[5,5] generating graphics... => 2
[5,4,1] generating graphics... => 2
[5,3,2] generating graphics... => 2
[5,3,1,1] generating graphics... => 2
[5,2,2,1] generating graphics... => 2
[5,2,1,1,1] generating graphics... => 2
[5,1,1,1,1,1] generating graphics... => 1
[4,4,2] generating graphics... => 2
[4,4,1,1] generating graphics... => 2
[4,3,3] generating graphics... => 3
[4,3,2,1] generating graphics... => 2
[4,3,1,1,1] generating graphics... => 2
[4,2,2,2] generating graphics... => 2
[4,2,2,1,1] generating graphics... => 2
[4,2,1,1,1,1] generating graphics... => 2
[4,1,1,1,1,1,1] generating graphics... => 1
[3,3,3,1] generating graphics... => 3
[3,3,2,2] generating graphics... => 2
[3,3,2,1,1] generating graphics... => 2
[3,3,1,1,1,1] generating graphics... => 2
[3,2,2,2,1] generating graphics... => 2
[3,2,2,1,1,1] generating graphics... => 2
[3,2,1,1,1,1,1] generating graphics... => 2
[3,1,1,1,1,1,1,1] generating graphics... => 1
[2,2,2,2,2] generating graphics... => 2
[2,2,2,2,1,1] generating graphics... => 2
[2,2,2,1,1,1,1] generating graphics... => 2
[2,2,1,1,1,1,1,1] generating graphics... => 2
[2,1,1,1,1,1,1,1,1] generating graphics... => 1
[1,1,1,1,1,1,1,1,1,1] generating graphics... => 1
click to show generating function       
Description
The side length of the Durfee square of an integer partition.
Given a partition $\lambda = (\lambda_1,\ldots,\lambda_n)$, the Durfee square is the largest partition $(s^s)$ whose diagram fits inside the diagram of $\lambda$. In symbols, $s = \max\{ i \mid \lambda_i \geq i \}$.
This is also known as the Frobenius rank.
References
[1] wikipedia:Durfee_square
[2] Andrews, G. E., Eriksson, K. Integer partitions MathSciNet:2122332
Code
def statistic(p):
    s = 0
    while len(p) > s and p[s] >= s+1:
        s += 1
    return s

#CodeLanguage: Mathematica
SageForm[part_] := StringJoin["[", Riffle[ToString /@ part, ","], "]"]
ips = Join @@ (IntegerPartitions /@ Range[10]);
stat[p_] := Combinatorica`DurfeeSquare[p];
Print[StringJoin @@ 
   Table[StringJoin[SageForm[p], " => ", ToString[stat[p]], "\n"], {p,
      ips[[1 ;; 138]]}]];
Created
May 05, 2014 at 06:22 by Lahiru Kariyawasam
Updated
Oct 29, 2017 at 16:33 by Martin Rubey