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
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
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Apr 08 2019
STATUS
approved