A325192 -id:A325192 - cp's OEIS frontend

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

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.

A051924 a(n) = binomial(2n,n) - binomial(2n-2,n-1); or (3n-2)*C(n-1), where C = Catalan numbers (A000108).

Original entry on oeis.org

1, 4, 14, 50, 182, 672, 2508, 9438, 35750, 136136, 520676, 1998724, 7696444, 29716000, 115000920, 445962870, 1732525830, 6741529080, 26270128500, 102501265020, 400411345620, 1565841089280, 6129331763880, 24014172955500, 94163002754652, 369507926510352

Offset: 1

Barry E. Williams, Dec 19 1999

Number of partitions with Ferrers plots that fit inside an n X n box, but not in an n-1 X n-1 box. - Wouter Meeussen, Dec 10 2001
From Benoit Cloitre, Jan 29 2002: (Start)
Let m(1,j)=j, m(i,1)=i and m(i,j) = m(i-1,j) + m(i,j-1); then a(n) = m(n,n):
  1  2  3  4 ...
  2  4  7 11 ...
  3  7 14 25 ...
  4 11 25 50 ... (End)
This sequence also gives the number of clusters and non-crossing partitions of type D_n. - F. Chapoton, Jan 31 2005
If Y is a 2-subset of a 2n-set X then a(n) is the number of (n+1)-subsets of X intersecting Y. - Milan Janjic, Nov 18 2007
Prefaced with a 1: (1, 1, 4, 14, 50, ...) and convolved with the Catalan sequence = A097613: (1, 2, 7, 25, 91, ...). - Gary W. Adamson, May 15 2009
Total number of up steps before the second return in all Dyck n-paths. - David Scambler, Aug 21 2012
Conjecture: a(n) mod n^2 = n+2 iff n is an odd prime. - Gary Detlefs, Feb 19 2013
First differences of A000984 and A030662. - J. M. Bergot, Jun 22 2013
From R. J. Mathar, Jun 30 2013: (Start)
Equivalent to the Meeussen comment and the Bergot comment: The array view of A007318 is
  1,   1,   1,   1,   1,   1,
  1,   2,   3,   4,   5,   6,
  1,   3,   6,  10,  15,  21,
  1,   4,  10,  20,  35,  56,
  1,   5,  15,  35,  70, 126,
  1,   6,  21,  56, 126, 252,
and a(n) are the hook sums Sum_{k=0..n} A(n,k) + Sum_{r=0..n-1} A(r,n). (End)
From Gus Wiseman, Apr 12 2019: (Start)
Equivalent to Wouter Meeussen's comment, a(n) is the number of integer partitions (of any positive integer) such that the maximum of the length and the largest part is n. For example, the a(1) = 1 through a(3) = 14 partitions are:
  (1)  (2)   (3)
       (11)  (31)
       (21)  (32)
       (22)  (33)
             (111)
             (211)
             (221)
             (222)
             (311)
             (321)
             (322)
             (331)
             (332)
             (333)
(End)
Coxeter-Catalan numbers for Coxeter groups of type D_n [Armstrong]. - N. J. A. Sloane, Mar 09 2022
a(n+1) is the number of ways that a best of n pairs contest with early termination can go. For example, the first stage of an association football (soccer) penalty-kick shoot out has n=5 pairs of shots and there are a(6)=672 distinct ways it can go. For n=2 pairs, writing G for goal and M for miss, and listing the up-to-four shots in chronological order with teams alternating shots, the n(3)=14 possibilities are MMMM, MMMG, MMGM, MMGG, MGM, MGGM, MGGG, GMMM, GMMG, GMG, GGMM, GGMG, GGGM, and GGGG. Not all four shots are taken in two cases because it becomes impossible for one team to overcome the lead of the other team. - Lee A. Newberg, Jul 20 2024

			Sums of {1}, {2, 1, 1}, {2, 2, 3, 3, 2, 1, 1}, {2, 2, 4, 5, 7, 6, 7, 5, 5, 3, 2, 1, 1}, ...

Drew Armstrong, Generalized Noncrossing Partitions and Combinatorics of Coxeter Groups, Mem. Amer. Math. Soc. 202 (2009), no. 949, x+159. MR 2561274 16; See Table 2.8.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Drew Armstrong, Generalized Noncrossing Partitions and Combinatorics of Coxeter Groups, arXiv:math/0611106 [math.CO], 2006-2007.
Jean-Luc Baril, Pamela E. Harris, Kimberly J. Harry, Matt McClinton, and José L. Ramírez, Enumerating runs, valleys, and peaks in Catalan words, arXiv:2404.05672 [math.CO], 2024. See p. 10.
F. Chapoton, Clusters.
FindStat, St000784: The maximum of the length and the largest part of the integer partition.
Sergey Fomin and Andrei Zelevinsky, Y-systems and generalized associahedra, Ann. of Math. (2) 158 (2003), no. 3, 977-1018.
Joël Gay and Vincent Pilaud, The weak order on Weyl posets, arXiv:1804.06572 [math.CO], 2018.
Milan Janjic, Two Enumerative Functions.
Joshua Marsh and Nathan Williams, Nesting Nonpartitions, J. Int. Seq., Vol. 25 (2022), Article 22.8.8.
Sanjay Moudgalya, Abhinav Prem, Rahul Nandkishore, Nicolas Regnault, and B. Andrei Bernevig, Thermalization and its absence within Krylov subspaces of a constrained Hamiltonian, arXiv:1910.14048 [cond-mat.str-el], 2019.
Hugh Thomas, Tamari Lattices and Non-Crossing Partitions in Types B and D, arXiv:math/0311334 [math.CO], 2003-2005.
Lin Yang and Shengliang Yang, Protected Branches in Ordered Trees, J. Math. Study, Vol. 56, No. 1 (2023), 1-17.

Left-central elements of the (1, 2)-Pascal triangle A029635.
Column sums of A096771.
Cf. A000108, A024482 (diagonal from 2), A076540 (diagonal from 3), A000124 (row from 2), A004006 (row from 3), A006522 (row from 4).
Cf. A128064; first differences of A000984.
Cf. A097613.
Cf. A115720, A252464, A257990, A263297, A325189, A325192, A325193.

a051924 n = a051924_list !! (n-1)
a051924_list = zipWith (-) (tail a000984_list) a000984_list
-- Reinhard Zumkeller, May 25 2013

[Binomial(2*n, n)-Binomial(2*n-2, n-1): n in [1..28]]; // Vincenzo Librandi, Dec 21 2016

C:= n-> binomial(2*n, n)/(n+1): seq((n+1)*C(n)-n*C(n-1), n=1..25); # Emeric Deutsch, Jan 08 2008
Z:=(1-z-sqrt(1-4*z))/sqrt(1-4*z): Zser:=series(Z, z=0, 32): seq(coeff(Zser, z, n), n=1..24); # Zerinvary Lajos, Jan 01 2007
a := n -> 2^(-2+2*n)*GAMMA(-1/2+n)*(3*n-2)/(sqrt(Pi)*GAMMA(1+n)):
seq(simplify(a(n)), n=1..24); # Peter Luschny, Dec 14 2015

Table[Binomial[2n,n]-Binomial[2n-2,n-1],{n,30}] (* Harvey P. Dale, Jan 15 2012 *)

a(n)=binomial(2*n,n)-binomial(2*n-2,n-1) \\ Charles R Greathouse IV, Jun 25 2013

{a(n)=polcoeff((1-x) / sqrt(1-4*x +x*O(x^n)) - 1,n)}
for(n=1,30,print1(a(n),", ")) \\ Paul D. Hanna, Nov 08 2014

{a(n)=polcoeff( sum(m=1, n, x^m * sum(k=0, m, binomial(m, k)^2 * x^k) / (1-x +x*O(x^n))^(2*m)), n)}
for(n=1, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Nov 08 2014

a = lambda n: 2^(-2+2*n)*gamma(n-1/2)*(3*n-2)/(sqrt(pi)*gamma(1+n))
[a(n) for n in (1..120)] # Peter Luschny, Dec 14 2015

G.f.: (1-x) / sqrt(1-4*x) - 1. - Paul D. Hanna, Nov 08 2014
G.f.: Sum_{n>=1} x^n/(1-x)^(2*n) * Sum_{k=0..n} C(n,k)^2 * x^k. - Paul D. Hanna, Nov 08 2014
a(n+1) = binomial(2*n, n) + 2*Sum_{i=0..n-1} binomial(n+i, i) (V's in Pascal's Triangle). - Jon Perry Apr 13 2004
a(n) = n*C(n-1) - (n-1)*C(n-2), where C(n) = A000108(n) = Catalan(n). For example, a(5) = 50 = 5*C(4) - 4*C(3) - 5*14 - 3*5 = 70 - 20. Triangle A128064 as an infinite lower triangular matrix * A000108 = A051924 prefaced with a 1: (1, 1, 4, 14, 50, 182, ...). - Gary W. Adamson, May 15 2009
Sum of 3 central terms of Pascal's triangle: 2*C(2+2*n, n)+C(2+2*n, 1+n). - Zerinvary Lajos, Dec 20 2005
a(n+1) = A051597(2n,n). - Philippe Deléham, Nov 26 2006
The sequence 1,1,4,... has a(n) = C(2*n,n)-C(2*(n-1),n-1) = 0^n+Sum_{k=0..n} C(n-1,k-1)*A002426(k), and g.f. given by (1-x)/(1-2*x-2*x^2/(1-2*x-x^2/(1-2*x-x^2/(1-2*x-x^2/(1-.... (continued fraction). - Paul Barry, Oct 17 2009
a(n) = (3*n-2)*(2*n-2)!/(n*(n-1)!^2) = A001700(n) + A001791(n-1). - David Scambler, Aug 21 2012
D-finite with recurrence: a(n) = 2*(3*n-2)*(2*n-3)*a(n-1)/(n*(3*n-5)). - Alois P. Heinz, Apr 25 2014
a(n) = 2^(-2+2*n)*Gamma(-1/2+n)*(3*n-2)/(sqrt(Pi)*Gamma(1+n)). - Peter Luschny, Dec 14 2015
a(n) ~ (3/4)*4^n*(1-(7/24)/n-(7/128)/n^2-(85/3072)/n^3-(581/32768)/n^4-(2611/262144)/n^5)/sqrt(n*Pi). - Peter Luschny, Dec 16 2015
E.g.f.: ((1 - x)*BesselI(0,2*x) + x*BesselI(1,2*x))*exp(2*x) - 1. - Ilya Gutkovskiy, Dec 20 2016
a(n) = 2 * A097613(n) for n > 1. - Bruce J. Nicholson, Jan 06 2019
Sum_{n>=1} a(n)/8^n = 7/(4*sqrt(2)) - 1. - Amiram Eldar, May 06 2023

Edited by N. J. A. Sloane, May 03 2008, at the suggestion of R. J. Mathar

A096771 Triangle read by rows: T(n,m) is the number of partitions of n that (just) fit inside an m X m box, but not in an (m-1) X (m-1) box. Partitions of n with Max(max part, length) = m.

Original entry on oeis.org

1, 0, 2, 0, 1, 2, 0, 1, 2, 2, 0, 0, 3, 2, 2, 0, 0, 3, 4, 2, 2, 0, 0, 2, 5, 4, 2, 2, 0, 0, 1, 7, 6, 4, 2, 2, 0, 0, 1, 6, 9, 6, 4, 2, 2, 0, 0, 0, 7, 11, 10, 6, 4, 2, 2, 0, 0, 0, 5, 14, 13, 10, 6, 4, 2, 2, 0, 0, 0, 5, 15, 19, 14, 10, 6, 4, 2, 2, 0, 0, 0, 3, 17, 22, 21, 14, 10, 6, 4, 2, 2, 0, 0, 0, 2, 17, 29

Offset: 1

Wouter Meeussen, Aug 21 2004

Row sums are A000041. Columns are finite and sum to A051924. The final floor(n/2) terms of each row are the reverse of the initial terms of 2*A000041.

			T(5,3)=3, counting 32, 311 and 221.
From _Gus Wiseman_, Apr 12 2019: (Start)
Triangle begins:
  1
  0  2
  0  1  2
  0  1  2  2
  0  0  3  2  2
  0  0  3  4  2  2
  0  0  2  5  4  2  2
  0  0  1  7  6  4  2  2
  0  0  1  6  9  6  4  2  2
  0  0  0  7 11 10  6  4  2  2
  0  0  0  5 14 13 10  6  4  2  2
  0  0  0  5 15 19 14 10  6  4  2  2
  0  0  0  3 17 22 21 14 10  6  4  2  2
  0  0  0  2 17 29 27 22 14 10  6  4  2  2
  0  0  0  1 17 33 36 29 22 14 10  6  4  2  2
  0  0  0  1 15 39 45 41 30 22 14 10  6  4  2  2
  0  0  0  0 14 41 57 52 43 30 22 14 10  6  4  2  2
  0  0  0  0 11 47 67 69 57 44 30 22 14 10  6  4  2  2
  0  0  0  0  9 46 81 85 76 59 44 30 22 14 10  6  4  2  2
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)
Findstat, FindStat - St000784: The maximum of the length and the largest part of the integer partition

A version with reflected rows is A338621.
Cf. A051924, A096272, A096597, A115994, A252464, A263297, A325193, A368985.
Related triangles are A115720, A325188, A325189, A325192, A325200, with Heinz-encoded versions A257990, A325169, A065770, A325178, A325195.

Table[Count[Partitions[n], q_List /; Max[Length[q], Max[q]]===k], {n, 16}, {k, n}]

row(n)={my(r=vector(n)); forpart(p=n, r[max(#p,p[#p])]++); r} \\ Andrew Howroyd, Jan 12 2024

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

A325178 Difference between the length of the minimal square containing and the maximal square contained in the Young diagram of the integer partition with Heinz number n.

Original entry on oeis.org

0, 0, 1, 1, 2, 1, 3, 2, 0, 2, 4, 2, 5, 3, 1, 3, 6, 1, 7, 2, 2, 4, 8, 3, 1, 5, 1, 3, 9, 1, 10, 4, 3, 6, 2, 2, 11, 7, 4, 3, 12, 2, 13, 4, 1, 8, 14, 4, 2, 1, 5, 5, 15, 2, 3, 3, 6, 9, 16, 2, 17, 10, 2, 5, 4, 3, 18, 6, 7, 2, 19, 3, 20, 11, 1, 7, 3, 4, 21, 4, 2, 12

Offset: 1

Gus Wiseman, Apr 08 2019

nonn

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.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The partition (3,3,2,1) has Heinz number 150 and diagram
  o o o
  o o o
  o o
  o
containing maximal square
  o o
  o o
and contained in minimal square
  o o o o
  o o o o
  o o o o
  o o o o
so a(150) = 4 - 2 = 2.

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

Wikipedia, Durfee square.

Positions of zeros are A062457. Positions of 1's are A325179. Positions of 2's are A325180.

Cf. A001222, A046660, A051924, A056239, A061395, A093641, A096771, A115994, A243055, A257990, A263297, A325192, A325195.

durf[n_]:=Length[Select[Range[PrimeOmega[n]],Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]][[#]]>=#&]];
codurf[n_]:=If[n==1,0,Max[PrimeOmega[n],PrimePi[FactorInteger[n][[-1,1]]]]];
Table[codurf[n]-durf[n],{n,100}]

a(n) = A263297(n) - A257990(n).

A325227 Regular triangle read by rows where T(n,k) is the number of integer partitions of n such that the lesser of the maximum part and the number of parts is k.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 0, 2, 1, 0, 0, 2, 3, 0, 0, 0, 2, 4, 1, 0, 0, 0, 2, 6, 3, 0, 0, 0, 0, 2, 6, 6, 1, 0, 0, 0, 0, 2, 8, 9, 3, 0, 0, 0, 0, 0, 2, 8, 13, 6, 1, 0, 0, 0, 0, 0, 2, 10, 16, 11, 3, 0, 0, 0, 0, 0, 0, 2, 10, 20, 17, 6, 1, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Apr 12 2019

			Triangle begins:
  1
  2  0
  2  1  0
  2  3  0  0
  2  4  1  0  0
  2  6  3  0  0  0
  2  6  6  1  0  0  0
  2  8  9  3  0  0  0  0
  2  8 13  6  1  0  0  0  0
  2 10 16 11  3  0  0  0  0  0
  2 10 20 17  6  1  0  0  0  0  0
  2 12 24 25 11  3  0  0  0  0  0  0
  2 12 28 33 19  6  1  0  0  0  0  0  0
  2 14 32 44 29 11  3  0  0  0  0  0  0  0
  2 14 38 53 43 19  6  1  0  0  0  0  0  0  0
Row n = 9 counts the following partitions:
  (9)          (54)        (333)      (4221)    (51111)
  (111111111)  (63)        (432)      (4311)
               (72)        (441)      (5211)
               (81)        (522)      (6111)
               (22221)     (531)      (42111)
               (222111)    (621)      (411111)
               (2211111)   (711)
               (21111111)  (3222)
                           (3321)
                           (32211)
                           (33111)
                           (321111)
                           (3111111)

FindStat, St000533: The maximal number of non-attacking rooks on a Ferrers shape

Column k = 2 is A265283. Column k = 3 is A325228.

Cf. A051924, A096771, A115720, A263297, A325188, A325189, A325192, A325193, A325194, A325224, A325225, A325229, A325232.

Table[Length[Select[IntegerPartitions[n],Min[Length[#],Max[#]]==k&]],{n,15},{k,n}]

A325179 Heinz numbers of integer partitions such that the difference between the length of the minimal square containing and the maximal square contained in the Young diagram is 1.

Original entry on oeis.org

3, 4, 6, 15, 18, 25, 27, 30, 45, 50, 75, 175, 245, 250, 343, 350, 375, 490, 525, 625, 686, 735, 875, 1029, 1225, 1715, 3773, 4802, 5929, 7203, 7546, 9317, 11319, 11858, 12005, 14641, 16807, 17787, 18634, 18865, 26411, 27951, 29282, 29645, 41503, 43923, 46585

Offset: 1

Gus Wiseman, Apr 08 2019

nonn

The enumeration of these partitions by sum is given by A325181.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The sequence of terms together with their prime indices begins:
    3: {2}
    4: {1,1}
    6: {1,2}
   15: {2,3}
   18: {1,2,2}
   25: {3,3}
   27: {2,2,2}
   30: {1,2,3}
   45: {2,2,3}
   50: {1,3,3}
   75: {2,3,3}
  175: {3,3,4}
  245: {3,4,4}
  250: {1,3,3,3}
  343: {4,4,4}
  350: {1,3,3,4}
  375: {2,3,3,3}
  490: {1,3,4,4}
  525: {2,3,3,4}
  625: {3,3,3,3}

Gus Wiseman, Young diagrams for the first 32 terms.

Numbers k such that A263297(k) - A257990(k) = 1.
Positions of 1's in A325178.
Cf. A056239, A093641, A112798, A325180, A325181, A325192, A325196, A325198.

durf[n_]:=Length[Select[Range[PrimeOmega[n]],Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]][[#]]>=#&]];
codurf[n_]:=If[n==1,0,Max[PrimeOmega[n],PrimePi[FactorInteger[n][[-1,1]]]]];
Select[Range[1000],codurf[#]-durf[#]==1&]

A325180 Heinz number of integer partitions such that the difference between the length of the minimal square containing and the maximal square contained in the Young diagram is 2.

Original entry on oeis.org

5, 8, 10, 12, 20, 21, 35, 36, 42, 49, 54, 60, 63, 70, 81, 84, 90, 98, 100, 105, 126, 135, 140, 147, 150, 189, 196, 210, 225, 275, 294, 315, 385, 441, 500, 539, 550, 605, 700, 750, 770, 825, 847, 980, 1050, 1078, 1100, 1125, 1155, 1210, 1250, 1331, 1372, 1375

Offset: 1

Gus Wiseman, Apr 08 2019

nonn

The enumeration of these partitions by sum is given by A325182.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The sequence of terms together with their prime indices begins:
    5: {3}
    8: {1,1,1}
   10: {1,3}
   12: {1,1,2}
   20: {1,1,3}
   21: {2,4}
   35: {3,4}
   36: {1,1,2,2}
   42: {1,2,4}
   49: {4,4}
   54: {1,2,2,2}
   60: {1,1,2,3}
   63: {2,2,4}
   70: {1,3,4}
   81: {2,2,2,2}
   84: {1,1,2,4}
   90: {1,2,2,3}
   98: {1,4,4}
  100: {1,1,3,3}
  105: {2,3,4}

Gus Wiseman, Young diagrams corresponding to the first 96 terms.

Numbers k such that A263297(k) - A257990(k) = 2.
Positions of 2's in A325178.
Cf. A006918, A056239, A093641, A112798, A325164, A325170, A325179, A325182, A325192, A325197.

durf[n_]:=Length[Select[Range[PrimeOmega[n]],Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]][[#]]>=#&]];
codurf[n_]:=If[n==1,0,Max[PrimeOmega[n],PrimePi[FactorInteger[n][[-1,1]]]]];
Select[Range[1000],codurf[#]-durf[#]==2&]

A325193 Number of integer partitions whose sum plus co-rank is n, where co-rank is maximum of length and largest part.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 3, 2, 5, 5, 8, 8, 14, 14, 22, 24, 35, 39, 54, 62, 84, 97, 127, 148, 192, 224, 284, 334, 418, 492, 610, 716, 880, 1035, 1259, 1480, 1790, 2100, 2522, 2958, 3533, 4135, 4916, 5742, 6798, 7928, 9344, 10878, 12778, 14846, 17378, 20156, 23520

Offset: 0

Gus Wiseman, Apr 12 2019

nonn

			The a(4) = 2 through a(12) = 14 partitions:
  (2)   (21)  (3)    (31)   (4)     (33)    (5)      (43)     (6)
  (11)        (22)   (211)  (32)    (41)    (42)     (51)     (44)
              (111)         (221)   (222)   (322)    (332)    (52)
                            (311)   (321)   (331)    (421)    (333)
                            (1111)  (2111)  (411)    (2221)   (422)
                                            (2211)   (3211)   (431)
                                            (3111)   (4111)   (511)
                                            (11111)  (21111)  (2222)
                                                              (3221)
                                                              (3311)
                                                              (4211)
                                                              (22111)
                                                              (31111)
                                                              (111111)

FindStat, St000784: The maximum of the length and the largest part of the integer partition

Row sums of A325194.

Cf. A051924, A065770, A071724, A096771, A115720, A252464, A257990, A263297, A325178, A325189, A325192.

Table[Sum[Length[Select[IntegerPartitions[k],Max[Length[#],Max[#]]==n-k&]],{k,0,n}],{n,0,30}]

A325181 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 1.

Original entry on oeis.org

0, 0, 2, 1, 0, 2, 3, 2, 1, 0, 2, 3, 4, 3, 2, 1, 0, 2, 3, 4, 5, 4, 3, 2, 1, 0, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, 0, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 2, 1, 0, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 3, 2, 1, 0, 2, 3, 4, 5, 6, 7, 8, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 2, 3, 4, 5, 6

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.

			The a(2) = 2 through a(15) = 1 partitions:
(2)  (21) (32)  (33)  (322) (332) (433)  (443)  (444)  (4333) (4433) (4443)
(11)      (221) (222) (331)       (3331) (3332) (3333) (4432) (4442)
                (321)                    (4331) (4332) (4441)
                                                (4431)

Giovanni Resta, Table of n, a(n) for n = 0..150

Cf. A006918, A084835, A096771, A257990, A263297, A325178, A325179, A325182, A325191, A325192, A325198.

durf[ptn_]:=Length[Select[Range[Length[ptn]],ptn[[#]]>=#&]];
codurf[ptn_]:=Max[Length[ptn],Max[ptn]];
Table[Length[Select[IntegerPartitions[n],codurf[#]-durf[#]==1&]],{n,0,30}]

More terms from Giovanni Resta, Apr 15 2019

A325182 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 2.

Original entry on oeis.org

0, 0, 0, 2, 2, 1, 2, 4, 7, 6, 5, 4, 5, 9, 12, 15, 14, 12, 10, 9, 11, 15, 21, 24, 28, 26, 24, 20, 18, 17, 19, 25, 31, 38, 42, 46, 44, 41, 36, 32, 29, 28, 31, 37, 46, 53, 62, 66, 71, 68, 65, 58, 53, 47, 44, 43, 46, 54, 63, 74, 83, 93, 98, 103, 100, 96, 88, 81

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.

			The a(3) = 2 through a(14) = 12 partitions:
  3    31   311  42    43    44    432   442   533    543    544    554
  111  211       2211  421   422   441   3322  4322   4422   553    5333
                       2221  431   3222  4222  4421   5331   5332   5432
                       3211  2222  3321  4321  33311  33321  5431   5441
                             3221  4221  4411         43311  33322  5531
                             3311  4311                      33331  33332
                             4211                            43321  43322
                                                             44311  43331
                                                             53311  44321
                                                                    44411
                                                                    53321
                                                                    54311

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

Wikipedia, Durfee square.

Cf. A006918, A096771, A257990, A263297, A325168.

Cf. A325178, A325180, A325181, A325182, A325192, A325197, A325199, A325200.

durf[ptn_]:=Length[Select[Range[Length[ptn]],ptn[[#]]>=#&]];
codurf[ptn_]:=Max[Length[ptn],Max[ptn]];
Table[Length[Select[IntegerPartitions[n],codurf[#]-durf[#]==2&]],{n,0,30}]

cp's OEIS Frontend

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

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A051924 a(n) = binomial(2*n,n) - binomial(2*n-2,n-1); or (3n-2)*C(n-1), where C = Catalan numbers (A000108).

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A096771 Triangle read by rows: T(n,m) is the number of partitions of n that (just) fit inside an m X m box, but not in an (m-1) X (m-1) box. Partitions of n with Max(max part, length) = m.

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A325178 Difference between the length of the minimal square containing and the maximal square contained in the Young diagram of the integer partition with Heinz number n.

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A325227 Regular triangle read by rows where T(n,k) is the number of integer partitions of n such that the lesser of the maximum part and the number of parts is k.

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A325179 Heinz numbers of integer partitions such that the difference between the length of the minimal square containing and the maximal square contained in the Young diagram is 1.

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A325180 Heinz number of integer partitions such that the difference between the length of the minimal square containing and the maximal square contained in the Young diagram is 2.

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A051924 a(n) = binomial(2n,n) - binomial(2n-2,n-1); or (3n-2)*C(n-1), where C = Catalan numbers (A000108).