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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.
13
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.
Sequence in context: A099584 A375159 A100563 * A087773 A025867 A078646
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Apr 08 2019
STATUS
approved