A115995 -id:A115995 - cp's OEIS frontend

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))).

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

A114088 Triangle read by rows: T(n,k) is number of partitions of n whose tail below its Durfee square has k parts (n >= 1; 0 <= k <= n-1).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 3, 3, 2, 1, 1, 1, 3, 4, 3, 2, 1, 1, 1, 4, 5, 5, 3, 2, 1, 1, 1, 5, 6, 6, 5, 3, 2, 1, 1, 1, 6, 8, 8, 7, 5, 3, 2, 1, 1, 1, 7, 10, 10, 9, 7, 5, 3, 2, 1, 1, 1, 9, 13, 13, 12, 10, 7, 5, 3, 2, 1, 1, 1, 10, 16, 17, 15, 13, 10, 7, 5, 3, 2, 1, 1, 1, 12, 20, 22, 20, 17

Offset: 1

Emeric Deutsch, Feb 12 2006

From Gus Wiseman, May 21 2022: (Start)
Also the number of integer partitions of n with k parts below the diagonal. For example, the partition (3,2,2,1) has two parts (at positions 3 and 4) below the diagonal (1,2,3,4). Row n = 8 counts the following partitions:
  8    71    611    5111    41111    311111   2111111   11111111
  44   332   2222   22211   221111
  53   422   3221   32111
  62   431   3311
       521   4211
Indices of parts below the diagonal are also called strong nonexcedances.
(End)

			T(7,2)=3 because we have [5,1,1], [3,2,1,1] and [2,2,2,1] (the bottom tails are [1,1], [1,1] and [2,1], respectively).
Triangle starts:
  1;
  1, 1;
  1, 1, 1;
  2, 1, 1, 1;
  2, 2, 1, 1, 1;
  3, 3, 2, 1, 1, 1;
  3, 4, 3, 2, 1, 1, 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).

John Tyler Rascoe, Rows n = 1..140, flattened
Findstat, St000183: The side length of the Durfee square of an integer partition

Cf. A114087, A114089, A115995, A116365.
Row sums: A000041.
Column k = 0: A003114.
Weak opposite: A115994.
Permutations: A173018, weak A123125.
Ordered: A352521, rank stat A352514, weak A352522.
Opposite ordered: A352524, first col A008930, rank stat A352516.
Weak opposite ordered: A352525, first col A177510, rank stat A352517.
Weak: A353315.
Opposite: A353318.
A000700 counts self-conjugate partitions, ranked by A088902.
A115720 counts partitions by Durfee square, rank stat A257990.
A352490 gives the (strong) nonexcedance set of A122111, counted by A000701.
Cf. A002620, A006918, A219282, A238352, A238874, A325039, A330644, A352487.

g:=sum(z^(k^2)/product((1-z^j)*(1-t*z^j),j=1..k),k=1..20): gserz:=simplify(series(g,z=0,30)): for n from 1 to 14 do P[n]:=coeff(gserz,z^n) od: for n from 1 to 14 do seq(coeff(t*P[n],t^j),j=1..n) od; # yields sequence in triangular form

subdiags[y_]:=Length[Select[Range[Length[y]],#>y[[#]]&]];
Table[Length[Select[IntegerPartitions[n],subdiags[#]==k&]],{n,1,15},{k,0,n-1}] (* Gus Wiseman, May 21 2022 *)

T_qt(max_row) = {my(N=max_row+1, q='q+O('q^N), h = sum(k=1,N, q^(k^2)/prod(j=1,k, (1-q^j)*(1-t*q^j))) ); for(i=1, N-1, print(Vecrev(polcoef(h, i))))}
T_qt(10) \\ John Tyler Rascoe, Oct 24 2024

G.f. = Sum_{k>=1} q^(k^2) / Product_{j=1..k} (1 - q^j)*(1 - t*q^j).

Sum_{k=0..n-1} k*T(n,k) = A114089(n).

A209616 Sum of positive Dyson's ranks of all partitions of n.

Original entry on oeis.org

0, 1, 2, 4, 7, 12, 18, 29, 42, 63, 89, 128, 176, 246, 333, 453, 603, 807, 1058, 1393, 1807, 2346, 3011, 3867, 4915, 6248, 7879, 9926, 12421, 15529, 19297, 23954, 29585, 36486, 44802, 54937, 67096, 81831, 99459, 120700, 146026, 176410, 212512, 255636, 306734

Offset: 1

Omar E. Pol, Mar 10 2012

nonn

The Dyson's rank of a partition is the largest part minus the number of parts.

			For n = 5 we have:
--------------------------
Partitions        Dyson's
of 5               rank
--------------------------
5               5 - 1 =  4
4+1             4 - 2 =  2
3+2             3 - 2 =  1
3+1+1           3 - 3 =  0
2+2+1           2 - 3 = -1
2+1+1+1         2 - 4 = -2
1+1+1+1+1       1 - 5 = -4
--------------------------
The sum of positive Dyson's ranks of all partitions of 5 is 4+2+1 = 7 so a(5) = 7.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
G. E. Andrews, S. H. G. Chan, and B. Kim, The odd moments of ranks and cranks (See the function R_1), Journal of Combinatorial Theory, Series A, Volume 120, Issue 1, January 2013, Pages 77-91.
F. J. Dyson, Some guesses in the theory of partitions, Eureka (Cambridge) 8 (1944), 10-15.
Frank Garvan, Dyson's rank function and Andrews's SPT-function

Column 1 of triangle A208482.

Cf. A063995, A064174, A092269, A105805, A194547, A194549, A195822, A208478.

# 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);
R1:=add((-1)^(n+1)*q^(n*(3*n+1)/2)/(1-q^n),n=1..M);
R1:=series(R1/qinf,q,M);
seriestolist(%); # N. J. A. Sloane, Sep 04 2012

M = 101;
qinf = Product[1-q^i, {i, 1, M}];
qinf = Series[qinf, {q, 0, M}];
R1 = Sum[(-1)^(n+1) q^(n(3n+1)/2)/(1-q^n), {n, 1, M}];
R1 = Series[R1/qinf, {q, 0, M}];
CoefficientList[R1, q] // Rest (* Jean-François Alcover, Aug 18 2018, translated from Maple *)

my(N=50, x='x+O('x^N)); concat(0, Vec(1/prod(k=1, N, 1-x^k)*sum(k=1, N, (-1)^(k-1)*x^(k*(3*k+1)/2)/(1-x^k)))) \\ Seiichi Manyama, May 21 2023

a(n) = A115995(n) - A195012(n). - Omar E. Pol, Apr 06 2012
G.f.: (1/Product_{k>=1} (1-x^k)) * Sum_{k>=1} (-1)^(k-1) * x^(k*(3*k+1)/2) / (1-x^k). - Seiichi Manyama, May 21 2023
a(n) ~ log(2) * exp(Pi*sqrt(2*n/3)) / (Pi*2^(3/2)*sqrt(n)). - Vaclav Kotesovec, Jul 06 2025

More terms from Alois P. Heinz, Mar 10 2012

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.

Original entry on oeis.org

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

Gus Wiseman, Apr 08 2019

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.

			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

Richard P. Stanley, Enumerative Combinatorics, Volume 2, Cambridge University Press, 1999, p. 289.

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
Wikipedia, Durfee square.

Row sums are A000041. Column k = 1 is A325181. Column k = 2 is A325182.
Cf. A096771, A257990, A263297, A325178, A325179, A325180, A325200.
Cf. A115995, A368985.

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}]

row(n)={my(r=vector(n+1)); if(n==0, r[1]=1, forpart(p=n, my(c=1); while(c<#p && cAndrew Howroyd, Jan 12 2024

Sum_{k=1..n} k*T(n,k) = A368985(n) - A115995(n). - Andrew Howroyd, Jan 12 2024

A195012 Sum of positive cranks minus the sum of positive ranks of all partitions of n.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 5, 7, 10, 13, 17, 24, 31, 40, 53, 69, 88, 113, 144, 183, 231, 290, 362, 453, 563, 696, 859, 1058, 1296, 1587, 1935, 2354, 2856, 3458, 4175, 5033, 6051, 7259, 8692, 10390, 12391, 14756, 17537, 20808, 24648, 29151, 34417, 40581, 47773, 56158

Offset: 1

Omar E. Pol, Jan 10 2012

nonn

It appears this is also the column 0 of triangle A195011 without the first one (see the Andrews-Garvan-Liang paper, page 16).
Is this also the ospt(n) function mentioned in the Andrews-Chan-Kim paper? Is A115995(n) the first crank momment? Is A209616(n) the first rank moment? - Omar E. Pol, Apr 07 2012
From Jeremy Lovejoy, Oct 14 2022: (Start)
a(n) is also the number of rank 0 strongly unimodal sequences of size n.   A strongly unimodal sequence is a sequence of positive integers which are strictly increasing up to a point (the peak) and then strictly decreasing thereafter.  The size is the sum of all of the parts and the rank is the number of parts to the left of the peak minus the number of parts to the right of the peak.
For example, there are 10 strongly unimodal sequences of size 6: (6), (1,5), (5,1), (2,4), (4,2), (1,4,1), (3,2,1), (1,2,3), (1,3,2), and (2,3,1). The sequences (6), (1,4,1), (1,3,2), and (2,3,1) have rank 0, and so a(6) = 4. (End)

			For n = 6 we have:
------------------------------------------------
Partitions
of 6                  Crank             Rank
------------------------------------------------
6                           6        6 - 1 =  5
3+3                         3        3 - 2 =  1
4+2                         4        4 - 2 =  2
2+2+2                       2        2 - 3 = -1
5+1                1 - 1 =  0        5 - 2 =  3
3+2+1              2 - 1 =  1        3 - 3 =  0
4+1+1              1 - 2 = -1        4 - 3 =  1
2+2+1+1            0 - 2 = -2        2 - 4 = -2
3+1+1+1            0 - 3 = -3        3 - 4 = -1
2+1+1+1+1          0 - 4 = -4        2 - 5 = -3
1+1+1+1+1+1        0 - 6 = -6        1 - 6 = -5
------------------------------------------------
The sum of positive cranks is 6+3+4+2+1 = 16 and the sum of positive ranks is 5+1+2+3+1 = 12 therefore a(6) = 16 - 12 = 4.

G. E. Andrews, S. H. G. Chan, and B. Kim, The odd moments of ranks and cranks (This is the function C_1 - R_1 of that paper), Journal of Combinatorial Theory, Series A, Volume 120, Issue 1, January 2013, Pages 77-91.
G. E. Andrews, F. G. Garvan, and J. Liang, Combinatorial interpretation of congruences for the spt-function, The Ramanujan Journal, December 2012, Volume 29, Issue 1-3, pp 321-338.
A. O. L. Atkin and F. G. Garvan, Relations between the ranks and cranks of partitions, arXiv:math/0208050 [math.NT], 2002.
K. Bringmann, C. Jennings-Shaffer, K. Mahlburg, and R. Rhoades, Peak positions of strongly unimodal sequences, Trans. Amer. Math. Soc. 372 (2019), 7087-7109.
Frank Garvan, Dyson's rank function and Andrews's SPT-function [Broken link?]
K. Hikami and J. Lovejoy, Torus knots and quantum modular forms, Res. Math. Sci. 2, Article 2 (2015).

Cf. A092269, A115995, A195011, A208482, A209616.

Cf. A059618.

# 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);
R1:=add((-1)^(n+1)*q^(n*(3*n+1)/2)/(1-q^n),n=1..M);
R1:=series(R1/qinf,q,M);
series(C1-R1,q,M);
seriestolist(%); # N. J. A. Sloane, Sep 04 2012

M = 101;
qinf = Product[1-q^i, {i, 1, M}];
qinf = Series[qinf, {q, 0, M}];
C1 = Sum[(-1)^(n+1) q^(n(n+1)/2)/(1-q^n), {n, 1, M}];
C1 = Series[C1/qinf, {q, 0, M}];
R1 = Sum[(-1)^(n+1) q^(n(3n+1)/2)/(1-q^n), {n, 1, M}];
R1 = Series[R1/qinf, {q, 0, M}];
CoefficientList[Series[C1-R1, {q, 0, M}], q] // Rest (* Jean-François Alcover, Aug 18 2018, translated from Maple *)

a(n) = A115995(n) - A209616(n).
From Jeremy Lovejoy, Oct 14 2022: (Start)
G.f.: (1/Product_{n>=1}(1-x^n))*Sum_{n>=1} x^(n*(n+1)/2)*(-1)^(n-1)*(1-x^(n^2))/(1-x^n).
G.f.: (1/Product_{n>=1}(1-x^n))*Sum_{n,r>=0} (-1)^(n+r)*x^(n*(3*n+5)/2+2*n*r+r*(r+3)/2). (End)
a(n) ~ exp(Pi*sqrt(2*n/3)) / (16*sqrt(3)*n). - Vaclav Kotesovec, Jul 06 2025

New name, example and more terms from Omar E. Pol, Apr 06 2012

More terms a(44)-a(50) from Alois P. Heinz, Apr 08 2012

A115722 Table of Durfee square of partitions in Mathematica 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, 1, 2, 2, 1, 2, 2, 1, 1, 1, 1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 1, 1, 1, 2, 1, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 1, 1, 1, 2, 1, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 1, 1, 1, 2, 1, 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;

Michael De Vlieger, Table of n, a(n) for n = 0..11731 (rows 0 <= n <= 26).
Eric Weisstein's World of Mathematics, Durfee Square.

Cf. A115721, A115994, A115720, A080577.

Row lengths A000041, totals A115995.

{0}~Join~Array[Map[Block[{k = Length@ #}, While[Nand[k > 0, AllTrue[Take[#, k], # >= k &]], k--]; k] &, IntegerPartitions@ #] &, 10] // Flatten (* Michael De Vlieger, Jan 17 2020 *)

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)).

A114089 Total number of parts in the tails below the Durfee squares of all partitions of n.

Original entry on oeis.org

0, 1, 3, 6, 11, 19, 31, 50, 76, 116, 169, 247, 349, 494, 682, 941, 1274, 1724, 2296, 3054, 4014, 5263, 6833, 8854, 11373, 14578, 18556, 23561, 29736, 37447, 46903, 58619, 72925, 90518, 111899, 138044, 169665, 208111, 254436, 310456, 377687, 458625

Offset: 1

Emeric Deutsch, Feb 12 2006

nonn

			a(4) = 6 because the bottom tails of the five partitions of 4, namely [4], [3,1], [2,2], [2,1,1] and [1,1,1,1], are { }, [1], { }, [1,1] and [1,1,1], respectively, having a total of 6 parts.

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 = 1..1000

Cf. A115994, A115995, A114087, A116365, A114088.

g:=sum(z^(k^2)/product((1-z^j)*(1-t*z^j),j=1..k),k=1..10): dgdt1:=simplify(subs(t=1,diff(g,t))): dgdt1ser:=series(dgdt1,z=0,55): seq(coeff(dgdt1ser,z,n),n=1..45);
# 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(j*b(n-j, j), j=1..n) -add(add(b(k, d)*b(n-d^2-k, d),
                         k=0..n-d^2)*d, d=1..floor(sqrt(n))):
seq(a(n), n=1..70);  # 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]]]]; a[n_] := Sum[j*b[n-j, j], {j, 1, n}] - Sum[Sum[b[k, d]*b[n-d^2-k, d], {k, 0, n-d^2}]*d, {d, 1, Floor[Sqrt[n]]}]; Table[a[n], {n, 1, 70}] (* Jean-François Alcover, Mar 31 2015, after Alois P. Heinz *)

a(n) = Sum_{k=0..n-1} k*A114088(n,k).
G.f.: [(d/dt){sum(q^(k^2)/product((1-q^j)(1-tq^j), j=1..k), k=1..infinity)}]_{t=1}.
a(n) = A006128(n) - A115995(n). - Vladeta Jovovic, Feb 18 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)).

A116365 Sum of the sizes of the tails below the Durfee squares of all partitions of n.

Original entry on oeis.org

0, 1, 3, 6, 11, 20, 33, 56, 86, 136, 200, 301, 429, 621, 868, 1219, 1669, 2297, 3091, 4171, 5542, 7357, 9648, 12652, 16402, 21250, 27298, 35003, 44556, 56637, 71515, 90160, 113046, 141464, 176189, 219053, 271149, 335044, 412447, 506787, 620597

Offset: 1

Emeric Deutsch, Feb 12 2006

nonn

			a(4) = 6 because the bottom tails of the five partitions of 4, namely [4], [3,1], [2,2], [2,1,1] and [1,1,1,1], are { }, [1], { }, [1,1] and [1,1,1], respectively, having total size 0+1+0+2+3=6.

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).

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)

Cf. A115994, A115995, A114087, A114088, A114089.

g:=sum(z^(k^2)/product((1-z^j)*(1-(t*z)^j),j=1..k),k=1..10): dgdt1:=simplify(subs(t=1,diff(g,t))): dgdt1ser:=series(dgdt1,z=0,55): seq(coeff(dgdt1ser,z,n),n=1..48);
# 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(k*add(b(k, d) *b(n-d^2-k, d),
            d=0..floor(sqrt(n))), k=0..n-1):
seq(a(n), n=1..40);  # Alois P. Heinz, Apr 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[k*Sum[b[k, d]*b[n-d^2-k, d], {d, 0, Floor[Sqrt[n]]}], {k, 0, n-1}]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Mar 31 2015, after Alois P. Heinz *)

a(n) = Sum_{k=0..n-1} k*A114087(n,k).
G.f.: [(d/dt){sum(q^(k^2)/product((1-q^j)(1-(tq)^j), j=1..k), k=1..oo)}]_{t=1}.
a(n) = (n*A000041(n)-A116503(n))/2. - Vladeta Jovovic, Feb 18 2006
a(n) ~ (1/(8*sqrt(3)) - sqrt(3) * (log(2))^2 / (4*Pi^2)) * exp(Pi*sqrt(2*n/3)). - Vaclav Kotesovec, Jan 03 2019

A330369 Triangle read by rows: T(n,k) (1 <= k <= n) is the total number of right angles of size k in all partitions of n.

Original entry on oeis.org

1, 0, 2, 0, 0, 3, 1, 0, 1, 4, 2, 0, 0, 2, 5, 3, 2, 0, 2, 3, 6, 4, 4, 0, 0, 4, 4, 7, 5, 6, 3, 0, 3, 6, 5, 8, 7, 8, 7, 0, 1, 6, 8, 6, 9, 9, 10, 11, 4, 0, 6, 9, 10, 7, 10, 13, 12, 15, 10, 0, 2, 11, 12, 12, 8, 11

Offset: 1

Omar E. Pol, Dec 12 2019

This triangle has the property that it contains the triangle A049597, since if we replace with zeros the positive terms before the first zero in the row n of this triangle, we get the triangle A049597.
Hence the sum of the terms after the last zero in row n equals A000041(n), the number of partitions of n (see the Example section).
Observation: at least the first 11 terms of column 1 coincide with A188674 (using the same indices).

			Triangle begins:
   1;
   0,  2;
   0,  0,  3;
   1,  0,  1,  4;
   2,  0,  0,  2,  5;
   3,  2,  0,  2,  3,  6;
   4,  4,  0,  0,  4,  4,  7;
   5,  6,  3,  0,  3,  6,  5,  8;
   7,  8,  7,  0,  1,  6,  8,  6,  9;
   9, 10, 11,  4,  0,  6,  9, 10,  7, 10;
  13, 12, 15, 10,  0,  2, 11, 12, 12,  8, 11;
Figure 1 below shows the Ferrers diagram of the partition of 24: [7, 6, 3, 3, 2, 1, 1, 1]. Figure 2 shows the right-angles diagram of the same partition. Note that in this last diagram we can see the size of the three right angles as follows: the first right angle has size 14 because it contains 14 square cells, the second right angle has size 8 and the third right angle has size 2.
.
.                                     Right-angles   Right
Part   Ferrers diagram         Part   diagram        angle
                                      _ _ _ _ _ _ _
  7    * * * * * * *             7   |  _ _ _ _ _ _|  14
  6    * * * * * *               6   | |  _ _ _ _|     8
  3    * * *                     3   | | | |           2
  3    * * *                     3   | | |_|
  2    * *                       2   | |_|
  1    *                         1   | |
  1    *                         1   | |
  1    *                         1   |_|
.
       Figure 1.                      Figure 2.
.
For n = 8 the partitions of 8 and their respective right-angles diagrams are as follows:
.
    _       _ _       _ _ _       _ _ _ _       _ _ _ _ _
  1| |8   2|  _|8   3|  _ _|8   4|  _ _ _|8   5|  _ _ _ _|8
  1| |    1| |      1| |        1| |          1| |
  1| |    1| |      1| |        1| |          1| |
  1| |    1| |      1| |        1| |          1|_|
  1| |    1| |      1| |        1|_|
  1| |    1| |      1|_|
  1| |    1|_|
  1|_|
    _ _ _ _ _ _       _ _ _ _ _ _ _       _ _ _ _ _ _ _ _
  6|  _ _ _ _ _|8   7|  _ _ _ _ _ _|8   8|_ _ _ _ _ _ _ _|8
  1| |              1|_|
  1|_|
.
    _ _       _ _ _       _ _ _ _       _ _ _ _ _       _ _ _ _ _ _
  2|  _|7   3|  _ _|7   4|  _ _ _|7   5|  _ _ _ _|7   6|  _ _ _ _ _|7
  2| |_|1   2| |_|  1   2| |_|    1   2| |_|      1   2|_|_|        1
  1| |      1| |        1| |          1|_|
  1| |      1| |        1|_|
  1| |      1|_|
  1|_|
.
    _ _       _ _ _       _ _ _       _ _ _ _       _ _ _ _       _ _ _ _ _
  2|  _|6   3|  _ _|6   3|  _ _|6   4|  _ _ _|6   4|  _ _ _|6   5|  _ _ _ _|6
  2| | |2   2| | |  2   3| |_ _|2   2| | |    2   3| |_ _|  2   3|_|_ _|    2
  2| |_|    2| |_|      1| |        2|_|_|        1|_|
  1| |      1|_|        1|_|
  1|_|
.
    _ _       _ _ _        _ _ _ _
  2|  _|5   3|  _ _|5    4|  _ _ _|5
  2| | |3   3| |  _|3    4|_|_ _ _|3
  2| | |    2|_|_|
  2|_|_|
.
There are  5 right angles of size 1, so T(8,1) = 5.
There are  6 right angles of size 2, so T(8,2) = 6.
There are  3 right angles of size 3, so T(8,3) = 3.
There are no right angle  of size 4, so T(8,4) = 0.
There are  3 right angles of size 5, so T(8,5) = 3.
There are  6 right angles of size 6, so T(8,6) = 6.
There are  5 right angles of size 7, so T(8,7) = 5.
There are  8 right angles of size 8, so T(8,8) = 8.
Hence the 8th row of triangle is [5, 6, 3, 0, 3, 6, 5, 8].
Note that the sum of the terms after the last zero is 3 + 6 + 5 + 8 = 22, equaling A000041(8) = 22, the number of partitions of 8.

G. E. Andrews, Theory of Partitions, Cambridge University Press, 1984, page 143 [Defines the right angles in the Ferrers graph of a partition. - N. J. A. Sloane, Nov 20 2020]

Row sums give A115995.
Right border gives A000027.
Cf. A000041, A049597, A083480, A083906, A188674, A259481, A325458, A330375, A330378, A330379.

cp's OEIS Frontend

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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A114088 Triangle read by rows: T(n,k) is number of partitions of n whose tail below its Durfee square has k parts (n >= 1; 0 <= k <= n-1).

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A209616 Sum of positive Dyson's ranks of all partitions of n.

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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.

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A195012 Sum of positive cranks minus the sum of positive ranks of all partitions of n.

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A115722 Table of Durfee square of partitions in Mathematica order.

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A114089 Total number of parts in the tails below the Durfee squares of all partitions of n.