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Triangle T(n,k) is the number of partitions of n with Durfee square k.
53

%I #21 Apr 13 2019 08:10:45

%S 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,

%T 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,

%U 168,110,2,0,18,204,158,5,0,19,240,221,10,0,20,285,302,20,0,21,330,407

%N Triangle T(n,k) is the number of partitions of n with Durfee square k.

%C T(n,k) is number of partitions of n-k^2 into parts of 2 kinds with at most k of each kind.

%H Alois P. Heinz, <a href="/A115720/b115720.txt">Rows n = 0..600, flattened</a>

%H Findstat, <a href="http://www.findstat.org/StatisticsDatabase/St000183/">St000183: The side length of the Durfee square of an integer partition</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DurfeeSquare.html">Durfee Square</a>

%F 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.

%e Triangle starts:

%e 1;

%e 0, 1;

%e 0, 2;

%e 0, 3;

%e 0, 4, 1;

%e 0, 5, 2;

%e 0, 6, 5;

%e 0, 7, 8;

%e 0, 8, 14;

%e 0, 9, 20, 1;

%e 0, 10, 30, 2;

%e From _Gus Wiseman_, Apr 12 2019: (Start)

%e Row n = 9 counts the following partitions:

%e (9) (54) (333)

%e (81) (63)

%e (711) (72)

%e (6111) (432)

%e (51111) (441)

%e (411111) (522)

%e (3111111) (531)

%e (21111111) (621)

%e (111111111) (3222)

%e (3321)

%e (4221)

%e (4311)

%e (5211)

%e (22221)

%e (32211)

%e (33111)

%e (42111)

%e (222111)

%e (321111)

%e (2211111)

%e (End)

%p b:= proc(n, i) option remember;

%p `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))

%p end:

%p T:= (n, k)-> add(b(m, k)*b(n-k^2-m, k), m=0..n-k^2):

%p seq(seq(T(n, k), k=0..floor(sqrt(n))), n=0..30); # _Alois P. Heinz_, Apr 09 2012

%t 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_ *)

%t durf[ptn_]:=Length[Select[Range[Length[ptn]],ptn[[#]]>=#&]];

%t Table[Length[Select[IntegerPartitions[n],durf[#]==k&]],{n,0,10},{k,0,Sqrt[n]}] (* _Gus Wiseman_, Apr 12 2019 *)

%Y 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.

%Y Cf. A008284, A115721, A115722, A257990, A325164.

%Y Related triangles are A096771, A325188, A325189, A325192, with Heinz-encoded versions A263297, A325169, A065770, A325178.

%K nonn,tabf

%O 0,5

%A _Franklin T. Adams-Watters_, Mar 11 2006