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A115720 Triangle T(n,k) is the number of partitions of n with Durfee square k. 53
1, 0, 1, 0, 2, 0, 3, 0, 4, 1, 0, 5, 2, 0, 6, 5, 0, 7, 8, 0, 8, 14, 0, 9, 20, 1, 0, 10, 30, 2, 0, 11, 40, 5, 0, 12, 55, 10, 0, 13, 70, 18, 0, 14, 91, 30, 0, 15, 112, 49, 0, 16, 140, 74, 1, 0, 17, 168, 110, 2, 0, 18, 204, 158, 5, 0, 19, 240, 221, 10, 0, 20, 285, 302, 20, 0, 21, 330, 407 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
T(n,k) is number of partitions of n-k^2 into parts of 2 kinds with at most k of each kind.
LINKS
Eric Weisstein's World of Mathematics, Durfee Square
FORMULA
T(n,k) = Sum_{i=0..n-k^2} P*(i,k)*P*(n-k^2-i), where P*(n,k) = P(n+k,k) is the number of partitions of n objects into at most k parts.
EXAMPLE
Triangle starts:
1;
0, 1;
0, 2;
0, 3;
0, 4, 1;
0, 5, 2;
0, 6, 5;
0, 7, 8;
0, 8, 14;
0, 9, 20, 1;
0, 10, 30, 2;
From Gus Wiseman, Apr 12 2019: (Start)
Row n = 9 counts the following partitions:
(9) (54) (333)
(81) (63)
(711) (72)
(6111) (432)
(51111) (441)
(411111) (522)
(3111111) (531)
(21111111) (621)
(111111111) (3222)
(3321)
(4221)
(4311)
(5211)
(22221)
(32211)
(33111)
(42111)
(222111)
(321111)
(2211111)
(End)
MAPLE
b:= proc(n, i) option remember;
`if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))
end:
T:= (n, k)-> add(b(m, k)*b(n-k^2-m, k), m=0..n-k^2):
seq(seq(T(n, k), k=0..floor(sqrt(n))), n=0..30); # Alois P. Heinz, Apr 09 2012
MATHEMATICA
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, b[n-i, i]]]]; T[n_, k_] := Sum[b[m, k]*b[n-k^2-m, k], {m, 0, n-k^2}]; Table[ T[n, k], {n, 0, 30}, {k, 0, Sqrt[n]}] // Flatten (* Jean-François Alcover, Dec 03 2015, after Alois P. Heinz *)
durf[ptn_]:=Length[Select[Range[Length[ptn]], ptn[[#]]>=#&]];
Table[Length[Select[IntegerPartitions[n], durf[#]==k&]], {n, 0, 10}, {k, 0, Sqrt[n]}] (* Gus Wiseman, Apr 12 2019 *)
CROSSREFS
For a version without zeros see A115994. Row lengths are A003059. Row sums are A000041. Column k = 2 is A006918. Column k = 3 is A117485.
Related triangles are A096771, A325188, A325189, A325192, with Heinz-encoded versions A263297, A325169, A065770, A325178.
Sequence in context: A073739 A223707 A046767 * A053120 A366601 A336836
KEYWORD
nonn,tabf
AUTHOR
STATUS
approved

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Last modified February 21 03:52 EST 2024. Contains 370219 sequences. (Running on oeis4.)