A115722 -id:A115722 - cp's OEIS frontend

A331553 Irregular triangle T(n,k) = A115722(n,k)^2 - n.

0, 0, 1, 1, 2, 2, 2, 3, 3, 0, 3, 3, 4, 4, 1, 4, 1, 4, 4, 5, 5, 2, 5, 2, 2, 5, 2, 2, 5, 5, 6, 6, 3, 6, 3, 3, 6, 3, 3, 3, 6, 3, 3, 6, 6, 7, 7, 4, 7, 4, 4, 7, 4, 4, 4, 4, 7, 4, 4, 4, 4, 7, 4, 4, 4, 7, 7, 8, 8, 5, 8, 5, 5, 8, 5, 5, 5, 5, 8, 5, 5, 5, 5, 5, 8, 0, 5

Offset: 0

Michael De Vlieger, Jan 20 2020

Let P be an integer partition of n, and let D be the Durfee square of P with side length s, thus area s^2. We borrow the term "square excess" from A053186(n), which is simply the difference n - floor(sqrt(n)). This sequence lists the "Durfee square excess" of P = n - s^2 for all partitions P of n in reverse lexicographic order.
Zero appears in row n for n that are perfect squares. Let r = sqrt(n). For perfect square n, there exists a partition of n that consists of a run of r parts that are each r themselves; e.g., for n = 4, we have {2, 2}, for n = 9, we have {3, 3, 3}. It is clear through the Ferrers diagram of these partitions that they are equivalent to their Durfee square, thus n - s^2 = 0.
Since the partitions of any n contain Durfee squares in the range of 1 <= s <= floor(sqrt(n)) (with perfect square n also including k = 0), the distinct Durfee square excesses must be the differences n - s^2 for 1 <= s <= floor(sqrt(n)).

			Table begins:
0: 0;
1: 0;
2: 1, 1;
3: 2, 2, 2;
4: 3, 3, 0, 3, 3;
5: 4, 4, 1, 4, 1, 4, 4;
6: 5, 5, 2, 5, 2, 2, 5, 2, 2, 5, 5;
7: 6, 6, 3, 6, 3, 3, 6, 3, 3, 3, 6, 3, 3, 6, 6;
...
Table of distinct terms:
1:  0;
2:  1;
3:  2;
4:  0,  3;
5:  1,  4;
6:  2,  5;
7:  3,  6;
8:  4,  7;
9:  0,  5,  8;
...
For n = 4, the partitions are {4}, {3, 1}, {2, 2}, {2, 1, 1}, {1, 1, 1, 1}. The partition {2, 2} has Durfee square s = 2; for all partitions except {2, 2}, we have Durfee square with s = 1.

Eric Weisstein's World of Mathematics, Durfee Square.

Cf. A115722, A116365.

{0}~Join~Array[Map[Total@ # - Block[{k = Length@ #}, While[Nand[k > 0, AllTrue[Take[#, k], # >= k &]], k--]; k]^2 &, IntegerPartitions[#]] &, 12] // Flatten

T(n,k) = A115722(n,k)^2 - n.

2 * A116365(n) = sum of row n.

A115994 Triangle read by rows: T(n,k) is number of partitions of n with Durfee square of size k (n>=1; 1<=k<=floor(sqrt(n))).

Original entry on oeis.org

1, 2, 3, 4, 1, 5, 2, 6, 5, 7, 8, 8, 14, 9, 20, 1, 10, 30, 2, 11, 40, 5, 12, 55, 10, 13, 70, 18, 14, 91, 30, 15, 112, 49, 16, 140, 74, 1, 17, 168, 110, 2, 18, 204, 158, 5, 19, 240, 221, 10, 20, 285, 302, 20, 21, 330, 407, 34, 22, 385, 536, 59, 23, 440, 698, 94, 24, 506, 896, 149, 25

Offset: 1

Emeric Deutsch, Feb 11 2006

Row n has floor(sqrt(n)) terms. Row sums yield A000041. Column 2 yields A006918. sum(k*T(n,k),k=1..floor(sqrt(n)))=A115995.
T(n,k) is number of partitions of n-k^2 into parts of 2 kinds with at most k of each kind.
The limit of the diagonals is A000712 (partitions into parts of two kinds). In particular, if 0<=m<=n, T(n(n+1)/2 + m, n) = A000712(m). These partitions in this range can be viewed as an equilateral right triangle of side n, with one partition appended on the top (at the left) and another appended on the right. - Franklin T. Adams-Watters, Jan 11 2006
Successive columns approach closer and closer to A000712. - N. J. A. Sloane, Mar 10 2007

			T(5,2) = 2 because the only partitions of 5 having Durfee square of size 2 are [3,2] and [2,2,1]; the other five partitions ([5], [4,1], [3,1,1], [2,1,1,1] and [1,1,1,1,1]) have Durfee square of size 1.
Triangle starts:
  1;
  2;
  3;
  4,  1;
  5,  2;
  6,  5;
  7,  8;
  8, 14;
  9, 20,  1;
  ...

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976 (pp. 27-28).
G. E. Andrews and K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004 (pp. 75-78).

Alois P. Heinz, Rows n = 1..1000, flattened
E. R. Canfield, From recursions to asymptotics: Durfee and dilogarithmic deductions, Advances in Applied Mathematics, Volume 34, Issue 4, May 2005, Pages 768-797
E. R. Canfield, S. Corteel, C. D. Savage, Durfee Polynomials, Electronic Journal of Combinatorics 5 (1998), #R32.
S. B. Ekhad, D. Zeilberger, A Quick Empirical Reproof of the Asymptotic Normality of the Hirsch Citation Index (First proved by Canfield, Corteel, and Savage), arXiv preprint arXiv:1411.0002, 2014.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009, page 45
Eric Weisstein's World of Mathematics, Durfee Square.

For another version see A115720. Row lengths A000196.

Cf. A115995, A115721, A115722, A008284, A006918.

g:=sum(t^k*q^(k^2)/product((1-q^j)^2,j=1..k),k=1..40): gser:=series(g,q=0,32): for n from 1 to 27 do P[n]:=coeff(gser,q^n) od: for n from 1 to 27 do seq(coeff(P[n],t^j),j=1..floor(sqrt(n))) od; # yields sequence in triangular form
# second Maple program:
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=1..floor(sqrt(n))), n=1..30); # Alois P. Heinz, Apr 09 2012

Map[Select[#,#>0&]&,Drop[Transpose[Table[CoefficientList[ Series[x^(n^2)/Product[1-x^i,{i,1,n}]^2,{x,0,nn}],x],{n,1,10}]],1]] //Grid (* Geoffrey Critzer, Sep 27 2013 *)
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, 1, 30}, {k, 1, Sqrt[n]}] // Flatten (* Jean-François Alcover, Dec 25 2015, after Alois P. Heinz *)

G.f.: sum(k>=1, t^k*q^(k^2)/product(j=1..k, (1-q^j)^2 ) ).

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.

Edited and verified by Franklin T. Adams-Watters, Mar 11 2006

A115720 Triangle T(n,k) is the number of partitions of n with Durfee square k.

Original entry on oeis.org

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

Offset: 0

Franklin T. Adams-Watters, Mar 11 2006

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

			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)

Alois P. Heinz, Rows n = 0..600, flattened
Findstat, St000183: The side length of the Durfee square of an integer partition
Eric Weisstein's World of Mathematics, Durfee Square

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.
Cf. A008284, A115721, A115722, A257990, A325164.
Related triangles are A096771, A325188, A325189, A325192, with Heinz-encoded versions A263297, A325169, A065770, A325178.

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

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 *)

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.

A115995 Sum of the sizes of the Durfee squares of all partitions of n.

Original entry on oeis.org

0, 1, 2, 3, 6, 9, 16, 23, 36, 52, 76, 106, 152, 207, 286, 386, 522, 691, 920, 1202, 1576, 2038, 2636, 3373, 4320, 5478, 6944, 8738, 10984, 13717, 17116, 21232, 26308, 32441, 39944, 48977, 59970, 73147, 89090, 108151, 131090, 158417, 191166, 230049, 276444

Offset: 0

Emeric Deutsch, Feb 11 2006

nonn

Also sum of positive cranks of all partitions of n, n>1; see A064391. - Vladeta Jovovic, Oct 20 2006

This sequence, its author and the author of the above comment were mentioned in the Andrews-Chan-Kim paper, where it is called C_1 (see the remark on page 6). - Omar E. Pol, Apr 06 2012

			a(4) = 6 because the partitions [4], [3,1], [2,2], [2,1,1] and [1,1,1,1] of 4 have Durfee squares of sizes 1,1,2,1 and 1, respectively.

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976 (pp. 27-28).
G. E. Andrews and K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004 (pp. 75-78).

Alois P. Heinz, Table of n, a(n) for n = 0..3000
George E. Andrews, Partitions and Durfee Dissection
George E. Andrews, Song Heng Chan, and Byungchan Kim, The odd moments of ranks and cranks
George E. Andrews, Frank G. Garvan, and Jie Liang, Self-conjugate vector partitions and the parity of the spt-function.
Atul Dixit, Bibekananda Maji, Partition implications of a new three parameter q-series identity, arXiv:1806.04424 [math.CO], 2018.
Eric Weisstein's World of Mathematics, Durfee Square.

Cf. A115994, A115720, A115721, A115722.

g:= add(k*z^(k^2)/mul((1-z^j)^2,j=1..k),k=1..10): gser:=series(g,z=0,56): seq(coeff(gser,z,n), n=0..52);
# second Maple program:
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:
a:= n-> add(add(b(k, d)*b(n-d^2-k, d), k=0..n-d^2)*d, d=1..isqrt(n)):
seq(a(n), n=0..70);  # Alois P. Heinz, Apr 09 2012
# Third Maple program, based on Theorem 1 of Andrews-Chan-Kim:
M:=101;
qinf:=mul(1-q^i,i=1..M);
qinf:=series(qinf,q,M);
C1:=add((-1)^(n+1)*q^(n*(n+1)/2)/(1-q^n),n=1..M);
C1:=series(C1/qinf,q,M);
seriestolist(%); # N. J. A. Sloane, Sep 04 2012

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]]]] ; a[n_] := Sum[ Sum[b[k, d]*b[n - d^2 - k, d], {k, 0, n - d^2}]*d, {d, 1, Sqrt[n]}]; Table [a[n], {n, 0, 70}] (* Jean-François Alcover, Jan 16 2015, after Alois P. Heinz *)

N=66; x='x+O('x^N); concat([0], Vec( sum(n=0,N, n*x^(n^2) / prod(k=1,n, 1-x^k)^2))) \\ Joerg Arndt, Mar 26 2014

[sum(p.frobenius_rank() for p in Partitions(n)) for n in range(45)] # Peter Luschny, Sep 15 2014

G.f.: Sum_{k>=1} (k*z^(k^2) / Product_{j=1..k} (1 - z^j)^2 ).
a(n) = Sum_{k=1..floor(sqrt(n))} k*A115994(n,k).
Convolution of A067742 and A000041. - Vladeta Jovovic, Oct 20 2006
a(n) = A195012(n) + A209616(n), n >= 1. - Omar E. Pol, Apr 06 2012
a(n) ~ log(2) * exp(Pi*sqrt(2*n/3)) / (2^(3/2)*Pi*sqrt(n)). - Vaclav Kotesovec, Jan 02 2019

Edited and verified by Franklin T. Adams-Watters, Mar 11 2006

A115721 Table of Durfee square of partitions in Abramowitz and Stegun order.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 1, 2, 2, 1, 2, 1, 1, 1, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 1, 1, 1, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 1, 2, 1, 1, 1, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 3, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 1, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 2

Offset: 0

Franklin T. Adams-Watters, Mar 11 2006

			First few rows: 0; 1,1; 1,1,1; 1,1,2,1,1; 1,1,2,1,2,1,1

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Eric Weisstein's World of Mathematics, Durfee Square.

Cf. A115722, A115994, A115720, A036036.

Row lengths A000041, totals A115995.

If partition is laid out in descending order p(1),p(2),...,p(k) without repetition factors (e.g. [3,2,2,1,1,1]), a(P) = max_k min(k,p(k)).

A330376 Irregular triangle read by rows: T(n,k) is the total number of parts in all partitions of n with Durfee square of size k (n>=1; 1<=k<=floor(sqrt(n))).

Original entry on oeis.org

1, 3, 6, 10, 2, 15, 5, 21, 14, 28, 26, 36, 50, 45, 80, 3, 55, 130, 7, 66, 190, 19, 78, 280, 41, 91, 385, 80, 105, 532, 143, 120, 700, 248, 136, 924, 399, 4, 153, 1176, 627, 9, 171, 1500, 949, 24, 190, 1860, 1397, 51, 210, 2310, 2003, 107, 231, 2805, 2823, 193

Offset: 1

Omar E. Pol, Dec 22 2019

			Triangle begins:
   1;
   3;
   6;
  10,  2;
  15,  5;
  21, 14;
  28, 26;
  36, 50;
  45, 80, 3;

Andrew Howroyd, Table of n, a(n) for n = 1..1799 (rows 1..200)
Eric Weisstein's World of Mathematics, Durfee Square

Row sums give A006128, n >= 1.
Column 1 gives A000217, n >= 1.
Cf. A114089, A115721, A115722, A115994, A116858, A118198, A208474, A330369, A330640, A330641, A330642, A330643.
Cf. A330369.

\\ by enumeration over partitions.
ds(p)={for(i=2, #p, if(p[#p+1-i]Andrew Howroyd, Feb 02 2022

\\ by generating function.
P(n,k,y)={1/prod(j=1, k, 1 - y*x^j + O(x*x^n))}
T(n,k)={my(r=n-k^2); if(r<0, 0, subst(deriv(polcoef(y^k*P(r,k,1)*P(r,k,y), r)), y, 1))}
{ for(n=1, 10, print(vector(sqrtint(n), k, T(n,k)))) } \\ Andrew Howroyd, Feb 02 2022

Terms a(10) and beyond from Andrew Howroyd, Feb 02 2022

cp's OEIS Frontend

A331553 Irregular triangle T(n,k) = A115722(n,k)^2 - n.

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A115994 Triangle read by rows: T(n,k) is number of partitions of n with Durfee square of size k (n>=1; 1<=k<=floor(sqrt(n))).

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A115720 Triangle T(n,k) is the number of partitions of n with Durfee square k.

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A115995 Sum of the sizes of the Durfee squares of all partitions of n.

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A115721 Table of Durfee square of partitions in Abramowitz and Stegun order.

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A330376 Irregular triangle read by rows: T(n,k) is the total number of parts in all partitions of n with Durfee square of size k (n>=1; 1<=k<=floor(sqrt(n))).

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